Jérome, Etienne, Kristelle, Aymen, Bilel, Gautier Rafa Muriel

Je remercie chaleureusement mon directeur de thèse Christian Klein pour le sujet qu'il m'a proposé, ainsi que pour son soutien, son écoute et sa gentillesse. Je ne remercierai jamais assez Dmitry Korotkin et Vasilisa Shramchenko pour toutes les discussions mathématiques que nous avons pu avoir, mais aussi pour tous ces instants chocolat partagés... Merci à Vladimir Matveev pour son attention et tout ce savoir qu'il partage et transmet. Je remercie B. Dubrovin et

Résumé

Un outil puissant dans le cadre des solutions algébro-géométriques des équations intégrables est l'identité de The modern theory of integrable systems has considerably enlarged the class of exactly solvable equations. It has also led to the discovery and understanding of new physical phenomena as solitons and breathers. Among the most powerful solution generating methods are techniques from algebraic geometry which lead to solutions in terms of multi-dimensional theta functions on certain Riemann surfaces. In the seventies, almost periodic solutions to the Korteweg-de Vries (KdV), the Kadomtsev-Petviashvili (KP), the sine-Gordon equations and others were constructed in terms of multi-dimensional theta functions. These are entire transcendental functions which are in general dened as Fourier series with certain periodicity properties.

What does integrability mean?

The notion of integrable system arose in the framework of mechanical systems for which the equations of motion are solvable by quadratures, i.e., only algebraic operations, integrations as well as applications of the inverse function theorem are needed. A reference of particular value is the book [START_REF] Arnold | Mathematical methods of classical mechanics[END_REF] on Hamiltonian systems. In classical Hamiltonian mechanics, a mechanical system with n degrees of freedom is described by a phase space parametrized by 2n coordinates q i , p i , i = 1, . . . , n, and the evolution of these coordinates is given by the following equations of motion:

dq i dt = ∂H ∂p i , dp i dt = - ∂H ∂q i , i = 1, . . . , n,
where H = H(q i , p i ) is called the Hamilton function of the system. The rst main result in this context is due to Liouville (who essentially applied Hamilton's result) and is known as Liouville's theorem. The latter states that if a mechanical system of the above form has n independent functions in involution (independent in the sense that their gradients are linearly independent), one of which being H, then it can be solved by quadratures. Recall that two functions f and g are said to be in Moreover, f is called a rst integral of the system if f and H are in involution.

independently developed a Lax formalism of the Toda lattice. Generalization of the inverse scattering method to systems with two spatial dimensions was made by Zakharov and Shabat [START_REF] Zakharov | A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, I[END_REF]. A typical example is the Kadomtsev-Petviashvili (KP) equation, a 2 + 1 dimensional version of the KdV equation.

Several new techniques were invented in the seventies, in particular, the direct method (bilinearization) of R. Hirota, the algebro-geometric method of B.A. Dubrovin, A.R. Its, I.M. Krichever, V.B. Matveev, P. Van Moerbeke, S.P. Novikov, as well as the group-theoretical (or Lie-algebraic) method due to M. Adler, B. Kostant, W.W. Symes, A.G. Reyman and M.A.

Semenov-Tian-Shansky which emerged in the second half of the seventies.

The discovery of solitary waves

The remarkable phenomenon of solitary wave was rst observed by J. Scott Russel [START_REF] Russell | Report on waves. Proc. Roy. Soc. Edinburgh[END_REF] in 1834:

I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with greater velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-dened heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original gure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my rst chance interview with that singular and beautiful phenomenon...

In 1895, Korteweg and de Vries [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[END_REF] derived the equation

4 u t = 6 uu x + u xxx , (1.1.1)
where u = u(x, t) is a function of two variables, to model the wave propagation of water in a shallow channel. They obtained solitary wave solutions (see e.g. [START_REF] Newell | Solitons in Mathematics and Physics[END_REF]), namely, solutions for which the shape is maintained when travelling. Such a solution is of the form u(x, t) = 2 c 2 sech 2 c (x + c 2 t) + d , for some c, d ∈ R. Viewing t as a time parameter, these solutions can be described as having a single localized hump of height 2 c 2 travelling to the left at speed c 2 , with the position at time t = 0 being determined by the value of d. Notice that waves with smaller amplitudes travel more slowly than waves with larger amplitudes.

The modern theory of soliton equations started with the famous numerical computation of the interaction of solitary waves of the KdV equation by Zabusky and Kruskal [START_REF] Zabusky | Interaction of solitons in a collisionless plasma and the recurrence of initial states[END_REF] in 1965.

The authors coined the word soliton to describe these solitary waves which behave like particles and may collide without changing shape. These observations stimulated theoretical researches and soon led in 1967 to the discovery by Gardner, Greene, Kruskal and Miura [START_REF] Gardner | Method for Solving the Korteweg-de Vries Equation[END_REF] of exact multi-soliton solutions and the Inverse Scattering Transform method (IST) that produces those solutions which have no analogue for linear partial dierential equations. The most important physical property of solitons is that they are localized wave packets which survive collisions with other solitons without change of shape. For a guide to the vast literature on solitons, see, for instance, [START_REF] Novikov | Theory of Solitons -The Inverse Scattering Method[END_REF][START_REF] Degasperis | Solitons[END_REF].

Existence of solitonic solutions to the nonlinear Schrödinger equation (NLS)

i ∂ψ ∂t + ∂ 2 ψ ∂x 2 + 2ρ |ψ| 2 ψ = 0, (1.1.2)
where ρ = ±1, was proved by Zakharov and Shabat [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media[END_REF] using a modication of the IST. The N -soliton solutions to both the (self-)focusing NLS equation (ρ = 1), as well as the defocusing NLS equation (ρ = -1), can also be computed by Darboux transformations [START_REF] Matveev | Darboux Transformations and Solitons[END_REF], Hirota's bilinear method (see, e.g., [START_REF] Hirota | Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of solitons[END_REF][START_REF] Polyanin | Handbook of Nonlinear Partial Dierential Equations[END_REF][START_REF] Chen | Introduction to Solitons[END_REF]) or Wronskian techniques (see [START_REF] Freeman | A method of obtaining the N-soliton solution of the Boussinesq equation in terms of a wronskian[END_REF][START_REF] Nimmo | The use of Bäcklund transformations in obtaining N-soliton solutions in Wronskian form[END_REF][START_REF] Freeman | Soliton Solutions of Non-linear Evolution Equations[END_REF]). Hirota's method is based on a transformation of the underlying equation to a bilinear equation. The resulting multi-soliton solutions are expressed in the form of polynomials in exponential functions.

Wronskian techniques formulate the N -soliton solutions in terms of the Wronskian determinant of N functions. This method allows a straightforward direct check that the obtained functions satisfy the equation since dierentiation of a Wronskian is simple. On the other hand, multisoliton solutions of (1.1.2) can be directly derived from algebro-geometric solutions when the associated hyperelliptic Riemann surface degenerates into a Riemann surface of genus zero, see, for instance, [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF].

An algebro-geometric approach to integrable systems

Algebraic geometry appears in the theory of integrable systems in the inverse problem for the KdV equation with periodic initial data, for which the inverse scattering transform method failed.

The inverse spectral problem for linear operators with periodic coecients (potentials) remained unsolved for a long time, and was to some extent solved by the consideration of so-called nitegap potentials. These are potentials for which the number of intervals forming the spectrum, and thus the number of gaps separating these intervals, is nite.

The history of periodic spectral theory starts with the investigations of Sturm and Liouville on the eigenvalues of certain dierential operators of second order with given boundary conditions, now referred to as Sturm-Liouville theory. Contrary to the case were the potential is rapidly decaying, in the periodic problem there is no simple asymptotic behavior, resulting in a theory which is more technical and less explicit, investigated by Dubrovin, Matveev, Novikov, Van Moerbeke, Its and others (see, for instance, [START_REF] Dubrovin | Nonlinear equations of Korteweg-de Vries type, nite-zone linear operators, and Abelian varieties[END_REF][START_REF] Novikov | Theory of Solitons -The Inverse Scattering Method[END_REF][START_REF] Mckean | The spectrum of Hill's equation[END_REF]).

Development of the nite-gap potential theory started with the works of [START_REF] Novikov | The periodic problem for the Korteweg-de Vries equation[END_REF] and [START_REF] Lax | Periodic solutions of the KdV equation[END_REF] (see [START_REF] Novikov | The periodic problem for the Korteweg-de Vries equation[END_REF] and [START_REF] Lax | Periodic solutions of the KdV equation[END_REF]). Novikov proposed a solution of the inverse periodic spectral problem for nite-gap potentials which is not eective in the sense that he described all nitegap potentials as solutions of the stationary higher KdV equations (Novikov's equation). An other approach (see [START_REF] Dubrovin | Theta functions and non-linear equations[END_REF][START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]) which is more eective consists in introducing Abelian functions, a generalization of elliptic functions to more than one variable, which are expressed as ratios of homogeneous polynomials of Riemann's theta function. It turns out that all nite-gap potentials are Abelian functions. The starting point of this approach is that the description of nite-gap potentials is reduced to a Jacobi inversion problem on a Riemann surface.

Fundamental algebraic-geometric tools in the theory of nite-gap linear operators and in the algebraic-geometric version of the inverse scattering method are the so-called Baker-Akhiezer functions which solve the auxiliary linear partial dierential equations with nite-gap coecients. Clebsh and Gordan rst considered a generalization of the exponential function on the Riemann sphere to Riemann surfaces of higher genus. Baker-Akhiezer functions are special functions with essential singularities on a Riemann surface and can be expressed in terms of the so-called theta function, which, in the present context, yields almost periodic solutions of the underlying equation. These functions were introduced in [START_REF] Krichever | Integration of nonlinear equations by the methods of algebraic geometry[END_REF] as a generalization of the analytic properties of Bloch eigenfunctions of operators with periodic and almost periodic coecients.

Let us illustrate this theory in the case of the Kadomtsev-Petviashvili (KP) equation. Two versions of this equation exist:

1. the stable version (also called the KP2 equation)

3 4 u y = w x , w y = u t - 1 4 (6 uu x + u xxx ), (1.1.3)
2. the unstable version (the KP1 equation)

3 4 u y = w x , w y = u t - 1 4 (6 uu x -u xxx ), (1.1.4)
where u, w are functions of the variables x, y, t. Notice that the change of variables (x, y, t) → (ix, iy, it) transforms the KP2 equation into the KP1. Equation (1.1.5)

The Baker-Akhiezer functions in this case are dened for each Riemann surface R g of nite genus g with a xed point p 0 on it, and a local parameter k -1 in a neighbourhood of this point, k -1 (p 0 ) = 0. For any set of points {p j } g j=1 there exists a unique function Ψ(x, y, t, p), with p ∈ R g , such that:

1. Ψ is meromorphic on R g \ {p 0 } and has no more than simple poles at the points p j (if they are distinct), 2. at the point p 0 it has an essential singularity of the form: Ψ(x, y, t, p) = 1 + ∞ n=1 ξ n (x, y, t) k -n exp kx + k 2 y + k 3 t , (1.1.6) where k = k(p) for any p lying in a neighbourhood of p 0 .

For any formal series (1.1.6) there exist unique operators L and A of the form (1.1.5) such that the following relations hold near point p 0 (∂ y -L)Ψ = O(k -1 ) exp kx + k 2 y + k 3 t , (∂ t -A)Ψ = O(k -1 ) exp kx + k 2 y + k 3 t .

(1.1.7)

From (1.1.7) it can be seen that the coecient u(x, y, t) of these operators reads u(x, y, t) = -2 ξ 1,x (x, y, t).

(1. 1.8) Now notice that the left hand sides of (1.1.7) dene functions having the same analytical properties except at p 0 as Ψ, and behave as (1.1.7) near this point. The uniqueness of the Baker-Akhiezer function Ψ implies that they are equal to zero, namely, (∂ y -L)Ψ = 0, (∂ t -A)Ψ = 0.

It follows that function u (1.1.8) is a solution of the KP equation. The explicit construction of Baker-Akhiezer functions in terms of Riemann theta functions and Abelian dierentials allows to write the solutions (1.1.8) of the KP equation in terms of theta functions:

u(x, y, t) = 2 ∂ 2 x ln Θ(Z(x, y, t) -d) + γ, (1.1.9)
for some d ∈ C g , where the vector Z ∈ C g is a linear function of the variables x, y, t whose vector coecients depend on the associated Riemann surface and the marked point p 0 , as well as the constant γ ∈ C. For any z ∈ C g , Θ(z) denotes the multi-dimensional theta function (2.2.1) with zero characteristic.

The construction was proposed in [START_REF] Krichever | Integration of nonlinear equations by the methods of algebraic geometry[END_REF][START_REF] Krichever | Algebro-geometric construction of the Zakharov-Shabat equations and their periodic solutions[END_REF] and was developed in dierent ways for various types of integrable equations in remarkable works by Novikov, Dubrovin, Matveev, Its, Van Moerbeke, who constructed algebraic geometrical solutions of the KdV equation, sine-Gordon equation and some other Lax-type equations (see, for instance, [START_REF] Krichever | Methods of algebraic geometry in the theory of non-linear equations[END_REF][START_REF] Dubrovin | Theta functions and non-linear equations[END_REF][START_REF] Krichever | Holomorphic bundles over Riemann surfaces and the Kadomtsev-Petviashvili (KP) equation. I[END_REF][START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]).

Parallely, in 1973, Fay [START_REF] Fay | Theta functions on Riemann surfaces[END_REF] discovered the so-called trisecant identity which is an important and special identity satised by theta functions. It is a far-reaching generalization of the addition theorem for elliptic theta functions. This identity states that, for any points a, b, c, d on a compact Riemann surface of genus g > 0, and for any z ∈ C g , there exist constants γ 1 , γ 2 and γ 3 such that

γ 1 Θ z + a c Θ z + d b + γ 2 Θ z + a b Θ z + d c = γ 3 Θ(z) Θ z + a+d c+b ; (1.1.10)
here and below we use the notation b a for the Abel map (2.1.13) between a and b. This identity plays an important role in various domains of mathematics, as for example in the theory of Jacobian varieties [START_REF] Arbarello | Fay's trisecant formula and a characterization of Jacobian Varieties[END_REF], in conformal eld theory [START_REF] Raina | Fay's Trisecant Identity and Conformal Field Theory[END_REF], and in operator theory [START_REF] Mccullough | The trisecant identity and operator theory[END_REF].

A few years later, Mumford [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF] realized that theta-functional solutions of certain integrable equations such as KdV, KP, and Sine-Gordon, may be derived from Fay's trisecant identity and its degenerations. It is a remarkable fact that algebro-geometric solutions of these integrable equations, previously obtained by the use of Baker-Akhiezer functions, arise as a degeneration of a purely algebro-geometric identity. The aim of this thesis is to extend this approach to other integrable equations such as the Camassa-Holm and Dym type equations, as well as generalizations of the NLS equation (1.1.2) namely, the multi-component NLS equation and the Davey-Stewartson equations.

Reality problems in soliton theory

Solutions of integrable equations (or soliton equations) are physically meaningful if they can be applied to describe actual physical phenomena. This is not the case in general without restrictions on the associated data. Therefore, it is often necessary to select physically or geometrically relevant classes of solutions corresponding to the studied problem: for instance solutions that satisfy certain reality conditions, smooth solutions, or bounded solutions. Let us mention a few famous examples in the algebro-geometric framework.

The KP1 (1.1.4) and KP2 (1.1.3) equations may be applied to the description of various interesting phenomena in plasma physics and hydrodynamics. In both cases only real nonsingular solutions are physically relevant. A solution is called non-singular if it is non-singular on the whole real Abel torus. This problem was solved by Dubrovin and Natanzon in [START_REF] Dubrovin | Real theta function solutions of the Kadomtsev-Petviashvili equation[END_REF]. It turns out that the Riemann surface R g of genus g associated to algebro-geometric solutions is real, namely there exists an anti-holomorphic involution τ : R g → R g , τ 2 = id on it. Moreover, Dubrovin and Natanzon proved that solutions of KP1 and KP2 are non-singular for a given topological type of the associated real Riemann surface.

The nonlinear Schrödinger equation (1.1.2) can be solved by introducing its complexied

version i ∂ψ ∂t + ∂ 2 ψ ∂x 2 + 2 ψ 2 ψ * = 0, -i ∂ψ * ∂t + ∂ 2 ψ * ∂x 2 + 2 ψ * 2 ψ = 0, (1.1.11)
where ψ and ψ * are complex valued functions of the real variables x and t. This system has two natural real reductions: the defocusing NLS equation, obtained by imposing the reality condition ψ * =ψ, and the focusing NLS equation under the condition ψ * = ψ. These reality conditions constrain the associated (hyperelliptic) spectral curve to be real. More precisely, it was proved by Its [START_REF] Its | Inversion of hyperelliptic integrals and integration of nonlinear dierential equations[END_REF] (see also [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]) that the spectral curve associated to solutions of the defocusing NLS equation is a hyperelliptic curve with real branch points only, whereas the hyperelliptic curve for the focusing case has pairwise conjugate branch points being all non-real. Physical applications such as light propagation in ber optics, where the sign + oris determined by the dispersion relation, require non-singular solutions.

The sine-Gordon equation was derived in the end of the nineteenth century. It describes for instance immersions of negative curvature surfaces into R 3 . In the light-cone variables (ξ, η), the sine-Gordon equation reads u ξη = sin(u).

(1.1.12)

Assume that an asymptotic coordinate system is chosen (a coordinate system such that coordinate lines have zero normal curvature). Thus the angle between the coordinate lines satises (1.1.12). This means that only real non-singular solutions such that u(ξ, η) = 0 (mod π) are relevant. According to [START_REF] Kozel | Almost periodic solutions of the equation u ttu xx + sin(u) = 0[END_REF] and [START_REF] Its | Explicit formulas for solutions of the Nonlinear Schrödinger equation[END_REF], the spectral curve associated to real-valued solutions consists of a hyperelliptic curve having either only nonpositive real branch points or pairwise conjugate non-real branch points. However, no ideas were proposed where the poles are located on the Riemann surface. For this reason, the theory of periodic nite-gap solutions of the sine-Gordon equation lacked applications for a long time. The problem was solved by Cherednik [START_REF] Cherednik | Reality conditions in nite-zone integration[END_REF] in 1980;

he proved that any real solution is in fact non-singular.

Outline of the thesis

Chapter 2, 3. Chapter 2 contains various algebraic tools related to the theory of Riemann surfaces, theta functions and Abelian dierentials. For further details, the reader is referred, for instance, to [START_REF]Computational Approach to Riemann Surfaces[END_REF][START_REF] Farkas | Riemann surfaces[END_REF][START_REF] Dubrovin | Theta functions and non-linear equations[END_REF][START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]]. Fay's identity obtained in [START_REF] Fay | Theta functions on Riemann surfaces[END_REF] is presented as a generalization to Riemann surfaces of the well-known identity for the cross-ratio function in the complex plane. Following Mumford's approach [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF], we explain how algebro-geometric solutions of the KdV, KP, sine-Gordon and the Toda lattice equations arise from Fay's identity.

As mentioned in Section 1.1.4, various tools dealing with the theory of real Riemann surfaces are necessary to construct physically meaningful algebro-geometric solutions of integrable systems. In Chapter 3, we recall basic facts concerning the topological type of a real Riemann surface. We introduce a canonical homology basis (used in [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF]) adapted to the anti-holomorphic involution τ . This canonical homology basis has particular properties with respect to τ and will be used in the whole thesis. An explicit description of the real and imaginary parts of the Jacobian is given according to the works of Vinnikov [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF] and Dubrovin-Natanzon [START_REF] Dubrovin | Real theta function solutions of the Kadomtsev-Petviashvili equation[END_REF]. Moreover, we study both actions of the anti-holomorphic involution τ on the homology group H 1 (R g \ {a, b}) of the punctured Riemann surface R g \ {a, b}, and on its dual, the relative homology group H 1 (R g , {a, b}), where a, b are two distinct points lying on R g . These technical results allow to construct in Chapter 4 real-valued and smooth algebro-geometric solutions of the Camassa-Holm and Dym type equations, as well as smooth algebro-geometric solutions of the multi-component nonlinear Schrödinger equation and the Davey-Stewartson equations in Chapter 6.

Chapter 4, 5. Algebro-geometric solutions of the Camassa-Holm and Dym type equations are obtained from theta-functional identities. Real-valuedness and smoothness of the solutions are studied from a topological point of view. In Chapter 5 these solutions are considered on surfaces degenerated to genus 0 to get special solutions such as solitons, cuspons as well as peakons.

At the end of the twentieth century, special attention was given to a shallow water equation (rst discovered by means of geometric considerations by Fokas and Fuchssteiner [START_REF] Fokas | Symplectic structures, their Bäcklund transformations and hereditary symmetries[END_REF]) investigated by Camassa and Holm in [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF] (see also [START_REF] Camassa | A new integrable shallow water equation[END_REF]):

u t + 3 uu x = u xxt + 2 u x u xx + uu xxx -2k u x , (1.2.1)
where the function u of the real variables x and t represents the uid velocity in x-direction measured at time t by an observer moving at speed k. In this context, only real-valued solutions are physically meaningful. They showed that for all k, the Camassa-Holm equation (CH) (1.2.1) is integrable, and for k = 0, it has travelling solutions of the form u(x, t) = c exp{-|x -vt|} (v is the speed), which are called peakons because they have a discontinuous rst derivative at the wave peak. In particular, they described the dynamics of the peakons in terms of a nite-dimensional completely integrable Hamiltonian system, namely, each peakon solution is associated to a mechanical system of moving particles. The class of mechanical systems of this type was further extended by Calogero and Françoise in [START_REF] Calogero | An integrable Hamiltonian system[END_REF][START_REF] Calogero | Solvable quantum version of an integrable Hamiltonian system[END_REF]. Multi-peakon solutions were studied using dierent approaches in a series of papers [START_REF] Beals | Acoustic scattering and the extended Korteweg de Vries hierarchy[END_REF][START_REF] Beals | Multi-peakons and a theorem of Stietjes[END_REF][START_REF] Beals | Multipeakons and the classical moment[END_REF][START_REF] Camassa | Characteristic variables for a completely integrable shallow water equation[END_REF]. Periodic solutions of the shallow water equation were discussed in [START_REF] Mckean | A shallow water equation on the circle[END_REF].

In Chapter 4, we also study the Dym type equation (DH)

u xxt + 2 u x u xx + uu xxx -2k u x = 0, (1.2.2)
which is a member of the Dym hierarchy (see [START_REF] Hunter | On a completely integrable nonlinear hyperbolic variational equation[END_REF] and [START_REF] Alber | The geometry of peaked solitons and billiard solutions of a class of integrable PDE's[END_REF][START_REF] Alber | On the link between umbilic geodesics and soliton solutions of nonlinear PDE's[END_REF]). An important feature of equations (1.2.1) and (1.2.2) is that the corresponding AbelJacobi mapping is not standard. Namely, the holomorphic dierentials that are involved do not form a basis on the hyperelliptic spectral curve, moreover, it involves a meromorphic dierential. In contrast to the well known cases of KdV, NLS and sine-Gordon equations, complex solutions of the CH and Dym equations are not meromorphic functions of (x, t) but have several branches. This is due to the presence of an implicit function y(x, t) of the variables x and t in the argument of the theta function appearing in the solutions. This monodromy eect is present in the prole of real-valued solutions such as cusps and peakons solutions.

Algebro-geometric solutions of the Camassa-Holm equation and their properties are studied in [START_REF] Alber | N-component integrable systems and geometric asymptotics[END_REF][START_REF] Alber | On billiard solutions of nonlinear PDE's[END_REF][START_REF] Alber | The geometry of peaked solitons and billiard solutions of a class of integrable PDE's[END_REF][START_REF] Alber | On billiard solutions of nonlinear PDE's[END_REF][START_REF] Alber | The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type[END_REF][START_REF] Alber | Wave solutions of evolution equations and Hamiltonian ows on nonlinear subvarieties of generalized Jacobians[END_REF][START_REF] Alber | Algebraic geometrical solutions for certain evolution equations and Hamiltonian ows on nonlinear subvarieties of generalized Jacobians[END_REF]. Finite-gap solutions of the Dym equation were studied by Dmitrieva [START_REF] Dmitrieva | Finite-gap solutions of the Harry Dym equation[END_REF] and Novikov [START_REF] Novikov | Algebraic-geometrical solutions of the HarryDym equations[END_REF] in relation to the KdV equation, by introducing an additional phase function. Soliton solutions of Dym type equations were studied by Dmitrieva [START_REF] Dmitrieva | The higher-times approach to multisoliton solutions of the Harry Dym equation[END_REF]. Notice that the solitary waves are orbitally stable, i.e., their shape is stable under small perturbations, both for the smooth solitons and for the peakons.

Our own approach to the construction of algebro-geometric solutions of the CH equation (resp.

Dym equation) diers from the ones studied in [START_REF] Alber | The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type[END_REF][START_REF] Alber | Wave solutions of evolution equations and Hamiltonian ows on nonlinear subvarieties of generalized Jacobians[END_REF][START_REF] Alber | Algebraic geometrical solutions for certain evolution equations and Hamiltonian ows on nonlinear subvarieties of generalized Jacobians[END_REF]. While the latter references employ the trace formula for the function u in terms of (projections of ) auxiliary divisors, and used generalized theta functions and generalized Jacobians (going back to investigations of Clebsch and Gordan [START_REF] Clebsch | Theorie der Abelschen Funktionen[END_REF]), we derive our solutions from a purely algebraic identity satised by the theta function, namely, Fay's trisecant identity. The spectral data consist of a hyperelliptic curve of the form µ 2 = 2g+2 j=1 (λλ j ), and a set of marked points. For the CH equation, there are three marked points: two of them are interchanged under the involution σ(λ, µ) = (λ, -µ), and the third is a ramication point (λ j 0 , 0); whereas for the Dym equation, these marked points consist of two ramication points of the underlying hyperelliptic curve. We start by introducing new identities satised by the theta function associated to these spectral data. These identities, corollaries of Fay's famous trisecant identity (1.1.10), allow us to derive and describe complex nite-gap solutions of equations (1.2.1) and (1.2.2).

Our construction of real-valued solutions is based on the description of the real and imaginary part of the Jacobian associated to a real hyperelliptic curve (i.e., the branch points λ j are real or pairwise conjugate non-real). It turns out that real-valued solutions are either smooth or admit an innite number of singularities which correspond to cusps.

In Chapter 5 we consider dierent degenerations of algebro-geometric solutions of the CH and Dym equations when the spectral curve becomes singular and its genus drops to zero. The solutions are then expressed in terms of purely elementary functions which by an appropriate choice of parameters describe interactions between solitons and cuspons. This representation also yields the existence of peakons in a special limiting case. Chapter 6. We prove a new degenerated version of Fay's trisecant identity (1.1.10). Following Mumford's approach, the new identity is applied to construct new algebro-geometric solutions of the multi-component nonlinear Schrödinger equation. This approach also provides an independent derivation of known algebro-geometric solutions to the Davey-Stewartson equations.

The rst main result of this chapter is a new degeneration of Fay's identity (1.1.10). This new identity holds for two distinct points a, b on a compact Riemann surface of genus g > 0:

D a ln Θ(z + b a ) Θ(z) + D 2 a ln Θ(z + b a ) Θ(z) + D a ln Θ(z + b a ) Θ(z) -K 1 2 + 2 D 2 a ln Θ(z) + K 2 = 0, (1.2.3)
for any z ∈ C g , where K 1 and K 2 are scalars independent of z but dependent on the points a and b; here D a and D a denote operators of directional derivatives along the vectors V a and W a (2.3.3). In particular, this identity implies that the following function of the variables x and t

ψ(x, t) = A Θ(Z -d + b a ) Θ(Z -d) exp {i (-K 1 x + K 2 t)} , (1.2.4)
where Z = i V a x + i W a t, and where A ∈ C, d ∈ C g are constant, is a solution of the linear Schrödinger equation Shabat in [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media[END_REF]. Algebro-geometric solutions of (1.1.2) were found by Its in [START_REF] Its | Inversion of hyperelliptic integrals and integration of nonlinear dierential equations[END_REF]; the geometric theory of these solutions was developed by Previato [START_REF] Previato | Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation[END_REF].

i ∂ψ ∂t + ∂ 2 ψ ∂x 2 + 2 u ψ = 0, (1.2
There exist various ways to generalize the NLS equation. The rst is to increase the number of dependent variables in (1.1.2). This leads to the multi-component nonlinear Schrödinger

equation i ∂ψ j ∂t + ∂ 2 ψ j ∂x 2 + 2 n k=1 s k |ψ k | 2 ψ j = 0, j = 1, . . . , n, (1.2.6)
denoted by n-NLS s , where s = (s 1 , . . . , s n ), s k = ±1. Here ψ j (x, t) are complex valued functions of the real variables x and t. The case n = 1 corresponds to the NLS equation. The two-component NLS equation (n = 2) is relevant in the study of electromagnetic waves in optical media in which the electric eld has two nontrivial components. Integrability of the two-component NLS equation in the case s = (1, 1) was rst established by Manakov [START_REF] Manakov | On the theory of two-dimensional stationary self-focusing of electromagnetic waves[END_REF]. In optical bers, for arbitrary n ≥ 2, the components ψ j in (1.2.6) correspond to components of the electric eld transverse to the direction of wave propagation. These components of the transverse eld form a basis of the polarization states. Integrability for the multi-component case with any n ≥ 2 and s k = ±1 was established in [START_REF] Radhakrishnan | Integrability and singularity structure of coupled nonlinear Schrödinger equations[END_REF]. Algebro-geometric solutions of the two-component NLS equation with signature (1, 1) were investigated in [START_REF] Elgin | Eective integration of the nonlinear vector Schrödinger equation[END_REF] using the Lax formalism and Baker-Akhiezer functions; these solutions are expressed in terms of theta functions of special trigonal spectral curves.

The second main result of this chapter is the construction of smooth algebro-geometric solutions of the multi-component nonlinear Schrödinger equation (1.2.6) for arbitrary n ≥ 2, obtained by using (1.2.3). We rst nd solutions to the complexied system

i ∂ψ j ∂t + ∂ 2 ψ j ∂x 2 + 2 n k=1 ψ k ψ * k ψ j = 0, -i ∂ψ * j ∂t + ∂ 2 ψ * j ∂x 2 + 2 n k=1 ψ k ψ * k ψ * j = 0, j = 1, . . . , n, (1.2.7)
where ψ j (x, t) and ψ * j (x, t) are complex valued functions of the real variables x and t. This system reduces to the n-NLS s equation (1.2.6) under the reality conditions ψ * j = s j ψ j , j = 1, . . . , n.

(1.2.8)

Algebro-geometric data associated to the solutions of (1.2.7) are given by {R g , f, z a }, where R g is a compact Riemann surface of genus g > 0, f is a meromorphic function of degree n + 1 on R g and z a ∈ CP 1 is a non critical value of the meromorphic function f such that f -1 (z a ) = {a 1 , . . . , a n+1 }. Then the solutions ψ j , ψ * j , j = 1, . . . , n, of system (1.2.7) read

ψ j (x, t) = A j Θ(Z -d + a j a n+1 ) Θ(Z -d) exp {i (-E j x + F j t)} , ψ * j (x, t) = q 2 (a n+1 , a j ) A j Θ(Z -d - a j a n+1 ) Θ(Z -d) exp {i (E j x -F j t)} ,
where the scalars E j , F j , q 2 (a n+1 , a j ) depend on the points a n+1 , a j ∈ R g , and where A j ∈ C, d ∈ C g are constant; here the g-dimensional vector Z is a linear function of the variables x and t. Imposing the reality conditions (1.2.8), we describe explicitly solutions for the focusing case s = (1, . . . , 1) and the defocusing case s = (-1, . . . , -1) associated to a real branched covering of the Riemann sphere. In particular, our solutions of the focusing case are associated to a covering without real branch point. Our general construction, being applied to the two-component case, gives solutions with more independent parameters than in [START_REF] Elgin | Eective integration of the nonlinear vector Schrödinger equation[END_REF] for xed genus of the spectral curve. Moreover, we provide smoothness conditions for our solutions.

Another way to generalize the NLS equation is to increase the number of spatial dimensions to two. This leads to the Davey-Stewartson equations (DS),

i ψ t + ψ xx -α 2 ψ yy + 2 (Φ + ρ |ψ| 2 ) ψ = 0, Φ xx + α 2 Φ yy + 2ρ |ψ| 2 xx = 0, (1.2.9)
where α = i or α = 1, and ρ = ±1; ψ(x, y, t) and Φ(x, y, t) are functions of the real variables x, y and t, the latter being real-valued and the former being complex valued. In what follows, DS1 ρ denotes the Davey-Stewartson equation when α = i, and DS2 ρ the Davey-Stewartson equation when α = 1. The Davey-Stewartson equation (1.2.9) was introduced in [START_REF] Davey | On three-dimensional packets of surface waves[END_REF] to describe the evolution of a three-dimensional wave package on water of nite depth. Complete integrability of the equation was shown in [START_REF] Anker | On the Soliton Solutions of the Davey-Stewartson Equation for Long Waves[END_REF]. If solutions ψ and Φ of (1.2.9) do not depend on the variable y the rst equation in (1.2.9) reduces to the NLS equation (1.1.2) under appropriate boundary conditions for the function Φ + ρ |ψ| 2 when x tends to innity.

Algebro-geometric solutions of the Davey-Stewartson equations (1.2.9) were previously obtained in [START_REF] Malanyuk | Finite-gap solutions of the Davey-Stewartson equations[END_REF] using the formalism of Baker-Akhiezer functions. In both [START_REF] Malanyuk | Finite-gap solutions of the Davey-Stewartson equations[END_REF] and the present thesis, solutions of (1.2.9) are constructed from solutions of the complexied system which, after the change of coordinates ξ = 1 2 (xiα y) and η = 1 2 (x + iα y), with α = i or 1, reads

i ψ t + 1 2 (ψ ξξ + ψ ηη ) + 2 ϕ ψ = 0, -i ψ * t + 1 2 (ψ * ξξ + ψ * ηη ) + 2 ϕ ψ * = 0, (1.2.10) ϕ ξη + 1 2 ((ψψ * ) ξξ + (ψψ * ) ηη ) = 0,
where ϕ := Φ + ψψ * . This system reduces to (1.2.9) under the reality condition:

ψ * = ρ ψ. (1.2.11)
The third main result of this chapter is an independent derivation of the solutions [START_REF] Malanyuk | Finite-gap solutions of the Davey-Stewartson equations[END_REF] using the degenerated Fay identity (1.2.3). Algebro-geometric data associated to these solutions are {R g , a, b, k a , k b }, where R g is a compact Riemann surface of genus g > 0, a and b are two distinct points on R g , and k a , k b are arbitrary local parameters near a and b. These solutions read

ψ(ξ, η, t) = A Θ(Z -d + b a ) Θ(Z -d) exp i -G 1 ξ -G 2 η + G 3 t 2 , ψ * (ξ, η, t) = - κ 1 κ 2 q 2 (a, b) A Θ(Z -d - b a ) Θ(Z -d) exp i G 1 ξ + G 2 η -G 3 t 2 , ϕ(ξ, η, t) = 1 2 (ln Θ(Z -d)) ξξ + 1 2 (ln Θ(Z -d)) ηη + h 4 ,
where the scalars G i , q 2 (a, b) depend on the points a, b ∈ R g , and κ 1 , κ 2 , A, h ∈ C, d ∈ C g are arbitrary constants; the g-dimensional vector Z is a linear function of the variables ξ, η and t. The reality condition (1.2.11) imposes constraints on the associated algebro-geometric data. In particular, the Riemann surface R g has to be real. The approach used in [START_REF] Malanyuk | Finite-gap solutions of the Davey-Stewartson equations[END_REF] to study reality conditions (1.2.11) is based on properties of Baker-Akhiezer functions. Our present approach based on identity (1.2.3) allows to construct solutions of DS1 ρ and DS2 ρ corresponding to Riemann surfaces of more general topological type than in [START_REF] Malanyuk | Finite-gap solutions of the Davey-Stewartson equations[END_REF].

Chapter 7. In this chapter, we construct solutions in terms of elementary functions of the multi-component NLS equation (1.2.6), and the Davey-Stewartson equation (1.2.9). The solutions of n-NLS and DS presented here are obtained by degenerating algebro-geometric solutions constructed in Chapter 6. This method for nding solutions in terms of elementary functions has not been applied to n-NLS and DS so far. It provides a unied approach to various solutions of n-NLS and DS expressed in terms of a simple determinantal form, and allows to present new solutions to the multi-component NLS equation in terms of elementary functions.

Multi-soliton solutions of (1.2.6) were considered in a series of papers, see for instance [START_REF] Manakov | On the theory of two-dimensional stationary self-focusing of electromagnetic waves[END_REF][START_REF] Radhakrishnan | Bright and dark soliton solutions to coupled nonlinear Schrödinger equations[END_REF][START_REF] Radhakrishnan | Inelastic Collision and Switching of Coupled Bright Solitons in Optical Fibers[END_REF][START_REF] Kanna | Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations[END_REF][START_REF] Ablowitz | Integrable Nonlinear Schrödinger Systems and their Soliton Dynamics[END_REF]. Here we present a family of dark and bright multi-solitons, breather and rational breather solutions to the multi-component NLS equation. It appears to be the rst time that breathers and rational breathers are given for the multi-component case. The notion of a dark soliton refers to the fact that the solution tends asymptotically to a non-zero constant, i.e., it describes a darkening on a bright background, whereas the bright soliton is a localized bright spot being described by a solution that tends asymptotically to zero. The name 'breather' reects the behavior of the prole which is periodic in time or space and localized in space or time. It is remarkable that degenerations of algebro-geometric solutions to the multi-component NLS equation lead to the breather solutions, well known in the context of the one-component case as the soliton on a nite background [START_REF] Akhmediev | First-order exact solutions of the nonlinear Schrödinger equation[END_REF] (breather periodic in space), the Ma breather [START_REF] Ma | The perturbed plane-wave solutions of the cubic Schrödinger equation[END_REF] (breather periodic in time) and the rational breather [START_REF] Peregrine | Water waves, nonlinear Schrödinger equations and their solutions[END_REF]. In the NLS framework, these solutions have been suggested as models for a class of extreme, freak or rogue wave events (see e.g. [START_REF] Henderson | Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation[END_REF][START_REF] Osborne | The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains[END_REF][START_REF] Andonowati | Extreme wave phenomena in down-stream running modulated waves[END_REF]). A family of rational solutions to the focusing NLS equation was constructed in [START_REF] Eleonskii | Rational multisoliton solutions to the NLS equation, Soviet Doklady 1986 sect[END_REF] and was rediscovered recently in [START_REF] Dubard | On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation[END_REF] via Wronskian techniques. Here we give for the rst time a family of breather and rational breather solutions of the multi-component NLS equation.

For the one component case, our solutions consist of the well known breather and Peregrine breather of the focusing NLS equation. For the multi-component case, we nd new proles of breathers and rational breathers which do not exist in the scalar case.

The second main result of this chapter is the construction of particular solutions of the DS equations such as solitons, breathers, dromion and lump, obtained by degenerating the algebrogeometric solutions given in Chapter 6. A main feature of equations in 1 + 1 dimensions is the existence of soliton solutions which are localized in one dimension. Solutions of the 2 + 1

dimensional integrable equations which are localized only in one dimension (plane solitons) were constructed in [START_REF] Ablowitz | Solitons and the Inverse Scattering Transform[END_REF][START_REF] Arkadiev | Inverse scattering transform and soliton solutions for Davey-Stewartson II equation[END_REF]. Moreover, various recurrent solutions (the growing-and-decaying mode, breather and rational growing-and-decaying mode solutions) were investigated in [START_REF] Tajiri | Periodic soliton solutions to the Davey-Stewartson equation[END_REF]. The spectral theory of soliton type solutions to the DS1 equation (called dromions) with exponential fall o in all directions on the plane, and their connection with the initial-boundary value problem, have been studied by dierent methods in a series of papers [START_REF] Boiti | Scattering of localized solitons in the plane[END_REF][START_REF] Fokas | Dromions and a boundary value problem for the Davey-Stewartson 1 equation[END_REF][START_REF] Santini | Energy exchange of interacting coherent structures in multidimensions[END_REF][START_REF] Radha | Localized Coherent Structures and Integrability in a Generalized (2 + 1)-Dimensional Nonlinear Schrödinger Equation[END_REF]. The lump solution (a rational non-singular solution) to the DS2

equation was discovered in [START_REF] Arkadiev | Inverse scattering transform and soliton solutions for Davey-Stewartson II equation[END_REF].

Here we present a family of dark multi-soliton solutions of the DS1 and DS2 + equations, as well as a family of bright multi-solitons for the DS1 and DS2

equations, obtained by degenerat- ing algebro-geometric solutions. Moreover, a class of breather and rational breather solutions of the DS1 equation is given. These solutions have a very similar appearance to those in 1 + 1 dimensions. Moreover, it is shown how the simplest solutions, the dromion and the lump solutions can be derived from algebro-geometric solutions.

Chapter 8. It is well known that physically meaningful almost periodic solutions to certain integrable partial dierential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. In this chapter we study typical analytical problems in the numerical evaluation of these solutions. To illustrate that, we discuss solutions to the multicomponent nonlinear Schrödinger equation as well as solutions to the Davey-Stewartson equations, investigated in Chapter 6.

Novikov criticized the practical relevance of theta functions since no numerical algorithms existed at the time to actually compute the found solutions. He suggested an eective treatment of theta functions (see, for instance, [START_REF] Dubrovin | Theta functions and non-linear equations[END_REF]) by a suitable parametrization of the characteristic quantities of a Riemann surface, i.e., the periods of holomorphic and certain meromorphic differentials on the given surface. This program is limited to genera smaller than 4 since so-called Schottky relations exist for higher genus between the components of the period matrix of a Riemann surface. The task to nd such relations is known as the Schottky problem. This led to the famous Novikov conjecture for the Schottky problem that a Riemann matrix (a symmetric matrix with negative denite real part) is the matrix of B-periods of the normalized holomorphic dierentials of a Riemann surface if and only if Krichever's formula with this matrix yields a solution to the KP equation. The conjecture was nally proven by Shiota [START_REF] Shiota | Characterization of Jacobian varieties in terms of soliton equations[END_REF].

First plots of KP solutions appeared in [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF] and via Schottky uniformizations in [START_REF] Bordag | Periodic Multiphase Solutions of the Kadomtsev-Petviashviliequation[END_REF]. Since all compact Riemann surfaces can be dened via non-singular plane algebraic curves of the form

F (x, y) := N n=1 M m=1 a mn x m y n = 0, x, y ∈ C, (1.2.12)
with constant complex coecients a nm , Deconinck and van Hoeij developed an approach to the symbolic-numerical treatment of algebraic curves. This approach is distributed as the algcurves package with Maple, see [START_REF] Deconinck | Computing Riemann matrices of algebraic curves[END_REF][START_REF] Deconinck | Computing Riemann Theta Functions[END_REF][START_REF] Deconinck | Computing with plane algebraic curves and Riemann surfaces: the algorithms of the Maple package algcurves[END_REF]. A purely numerical approach to real hyperelliptic Riemann surfaces was given in [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods[END_REF][START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods: KdV and KP solutions[END_REF], and for general Riemann surfaces in [START_REF] Frauendiener | Algebraic curves and Riemann surfaces in Matlab[END_REF]. For a review of computational approaches to Riemann surfaces the reader is referred to [START_REF]Computational Approach to Riemann Surfaces[END_REF].

In this chapter we want to address typical analytical problems appearing in the numerical study of theta-functional solutions to integrable PDEs, and we present the state of the art of the eld by considering concrete examples. The case of hyperelliptic Riemann surfaces (N = 2 in (1.2.12)) is the most accessible, since equation (1.2.12) can be solved explicitly for y, and since a basis for dierentials and homology can be given a priori. Families of hyperelliptic curves can be conveniently parametrized by their branch points. The codes [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods[END_REF][START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods: KdV and KP solutions[END_REF] are in principle able to treat numerically collisions of branch points, a limit in which certain periods of the corresponding hyperelliptic surface diverge. If the limiting Riemann surface has genus 0, the theta series breaks down to a nite sum which, for an appropriate choice of the characteristic, gives well known solitonic solutions to the studied equation.

For solutions dened on general real algebraic curves, i.e., curves (1.2.12) with all a nm real, an important point in applications are reality and smoothness conditions. These are conveniently formulated for a homology basis for which the A-cycles are invariant under the action of the anti-holomorphic involution. However, the existing algorithms for the computational treatment of algebraic curves produce a basis of the homology that is in general not related to possible automorphisms of the curve. To implement the reality and smoothness requirements, a transformation to the basis for which the conditions are formulated has to be constructed. We study the necessary symplectic transformations and give explicit relations for so-called M-curves, curves with the maximum number of real ovals.

To illustrate these concepts, we have studied for the rst time numerically theta-functional solutions to integrable equations from the family of NLS equations, namely, the multi-component nonlinear Schrödinger equation (1.2.6) and the Davey-Stewartson equations (1.2.9). To ensure the correct numerical implementation of the formulae obtained in Chapter 6, we check for each point in the spacetime whether certain identities for theta functions are satised. Since these identities are not used in the code, they provide a strong test for the computed quantities.

Numerically the identities are never exactly satised, but to high precision. The code reports a warning if the residual of the test relations is larger than 10 -6 which is well below plotting accuracy. Typically the conditions are satised to machine precision 1 . In addition we compute the solutions on a numerical grid and numerically dierentiate them. We check in this way for low genus that the solutions to n-NLS s and DS in terms of multi-dimensional theta functions satisfy the respective equations to better than 10 -6 . These two completely independent tests ensure that the presented plots are showing the correct solutions to better than plotting accuracy. Schrödinger and sine-Gordon equations. For the history of the topic the reader may refer, for instance, to the reviews [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF] and [START_REF] Dubrovin | Theta functions and non-linear equations[END_REF]. It turns out that most theta-functional solutions of integrable equations can be directly derived from Fay's identity, an identity satised by the multi-dimensional theta function.

Riemann surfaces

A Riemann surface is a connected one-complex-dimensional analytic manifold. The most important examples of Riemann surfaces consist of non-singular algebraic curves, since one can introduce a natural complex structure on it. The theory of compact Riemann surfaces and the theory of algebraic curves over C are very well developed and exceedingly rich. In this part we follow the line of the Chapter 1 in [START_REF]Computational Approach to Riemann Surfaces[END_REF].

Riemann surfaces and non-singular algebraic curves

An algebraic curve C is a subset in C 2 dened by

C = {(λ, µ) ∈ C 2 | P (λ, µ) = 0},
where P is an irreducible polynomial in λ and µ of the form

P (λ, µ) = N i=0 M j=0 p ij µ i λ j , with coecients p ij ∈ C. The curve C is called non-singular if ∇P (λ, µ) C = 0, where ∇P = ∂P ∂λ , ∂P ∂µ .
Chapter 2. Theta functions and Fay's identity On a non-singular algebraic curve, one can introduce a complex structure as follows: in a neighbourhood of the points where ∂P/∂µ = 0, one chooses the local parameter to be the projection on the λ-plane, and near the points where ∂P/∂λ = 0, one chooses the local parameter to be the projection on the µ-plane. Holomorphic compatibility of the underlying local parameters can be checked using a complex version of the implicit function theorem.

Example 2.1.1. Let N ∈ N such that N ≥ 3. The polynomial equation

µ 2 = N j=1 (λ -λ j ), (2.1.1)
where λ j ∈ C, describes a non-singular algebraic curve C if λ i = λ k for i = k; such a curve is called hyperelliptic. The local parameter in a neighbourhood of a point (λ 0 , µ 0 ) with λ 0 = λ j is given by the homeomorphism (λ, µ) → λ, and the local parameter in a neighbourhood of each point (λ j , 0) is dened by the homeomorphism (λ, µ) → (λλ j )

N = 2g + 1 or N = 2g + 2 equals g.
It is a remarkable but nontrivial result that any compact Riemann surface can be realized via a compactied algebraic curve. In the whole thesis we deal with compact Riemann surfaces of genus g that we denote by R g . If not stated otherwise, we let R g denote both the non-singular compactied algebraic curve and its compact Riemann surface.

Holomorphic mappings and coverings

A mapping f : M → N between Riemann surfaces is called holomorphic if, for every local parameter (U, z) on M and every local parameter (V, w) on N , with (λ j , 0), j = 1, . . . , N and ∞ for N = 2g + 1, (λ j , 0), j = 1, . . . , N for N = 2g + 2,

U ∩ f -1 (V ) = ∅, the mapping w • f • z -1 : z(U ∩ f -1 (V )) -→ w(V ) is holomorphic. Moreover, locally the mapping F = w • f • z -1 behaves as F (z) = z n ,
with the branch number b f = 1 at these points.

Homology group and intersection index

In this work we use the notion of an intersection index on a compact Riemann surface. Intersection indices are well dened by introducing equivalence relations related to homology groups.

First homology group. Here we consider an oriented triangulation of a compact Riemann surface R. Formal sums of points p i , n i p i , of oriented edges γ i , γ = n i γ i , and of oriented triangles D i , D = n i D i , with n i ∈ Z, are called 0-chains, 1-chains and 2-chains respectively. Let us denote these sets by C 0 , C 1 and C 2 . A 1-chain γ with δ γ = 0 is called a cycle, and a 1-chain γ of the form γ = δD is called a boundary; here δ denotes the boundary operator. Denote by Z the subgroup of cycles and by B the subgroup of boundaries:

Z = {γ ∈ C 1 | δ γ = 0}, B = δC 2 .
Due to δ 2 = 0, every boundary is a cycle and we have B ⊂ Z ⊂ C 1 . One can introduce an equivalence relation between elements of C 1 : two 1-chains are called homologous if their dierence is a boundary, namely,

γ 1 ∼ γ 2 ⇐⇒ (γ 1 -γ 2 ∈ B, i.e. ∃ D ∈ C 2 : δD = γ 1 -γ 2 ) , for any γ 1 , γ 2 ∈ C 1 . The factor group H 1 (R) = H 1 (R, Z) = Z/B
is called the rst homology group of R. For a compact Riemann surface R g of genus g, this is a free abelian group of rank 2g. A standard set of generators for this group can be obtained using a certain representation of R g as a polygon with 4g sides, appropriately identied in pairs. We denote these generators by A 1 , B 1 , . . . , A g , B g .

Intersection index. To introduce intersection numbers of elements of the rst homology group, it is convenient to represent them by smooth cycles. Moreover, given two elements of H 1 (R g ), one can represent them by smooth cycles intersecting transversally in a nite number of points. Let γ 1 and γ 2 be two smooth cycles intersecting transversally at the point p. One associates to this point a number (γ 1 •γ 2 ) p = ±1, where the sign is determined by the orientation of the basis (γ 1 (p), γ 2 (p)); here γ i (p) denotes the tangent vector of γ i at p. More generally, let γ 1 , γ 2 be two smooth cycles intersecting transversally at the nite set of their intersection points.

Then the intersection index of γ 1 and γ 2 is dened by

γ 1 • γ 2 = p∈Intersection set (γ 1 • γ 2 ) p .
The intersection number is a bilinear skew symmetric map:

• : H 1 (R g ) × H 1 (R g ) -→ Z.
Canonical homology basis. A homology basis (A, B) := (A 1 , . . . , A g , B 1 , . . . , B g ) of a compact Riemann surface R g of genus g with the following intersection indices

A i • B j = δ i, j A i • A j = B i • B j = 0,
is called canonical basis of cycles. If the Riemann surface R g is cut along this basis, one obtains a 4g-gon where each cycle corresponds to a pair of the sides A i , A -1 i , B i , B -1 i which are identied on R g . In the whole thesis, the notation A (resp. B) will also be used to denote the vector (A 1 , . . . , A g ) t (resp. (B 1 , . . . , B g ) t ).

Change of canonical homology basis. Canonical homology basis are related via a symplectic transformation. Let (A, B) and ( Ã, B) be arbitrary canonical homology basis dened on R g , represented here by 2g-dimensional vectors. Then there exists a symplectic matrix

M = A B C D ∈ Sp(2g, Z) such that A B C D Ã B = A B . (2.1.2)
Recall that a symplectic matrix M ∈ Sp(2g, Z) satises M t J g M = J g , with the matrix J g given by J g = 0 I g -I g 0 , where I g denotes the g × g unit matrix. Symplectic matrices

M = A B C D ∈ Sp(2g, Z
) are characterized by the following system of matrix equations:

A t D -C t B = I g , (2.1.3) A t C = C t A, (2.1.4) D t B = B t D.
(2.1.5)

Moreover, the inverse matrix M -1 is given by M

-1 = D t -B t -C t A t .

Abelian dierentials and their periods

An Abelian dierential on a compact Riemann surface R g is a meromorphic 1-form ω given on R g . In other words, locally ω reads f (z) dz, where f is a meromorphic function of z in its domain.

Zeros and poles of Abelian dierentials are well dened, as well as the notions of multiplicity and residue.

First, second and third kind dierentials. One distinguishes three kinds of Abelian differentials: holomorphic dierentials (rst kind), meromorphic dierentials with residues equal to zero at all singular points (second kind), and meromorphic dierentials of general form (third kind). Note that for dierentials of the third kind, the sum of residues equals zero. Let a ∈ R g , and N ∈ N with N > 1. We denote by Ω (N ) a the dierential of the second kind having only one singularity at point a of the form

Ω (N ) a (p) = 1 k a (p) N + O(1) dk a (p), p ∈ R g , (2.1.6)
where k a is a local parameter in a neighbourhood of a. Note that the dierential Ω (N ) a depends on the choice of the local parameter k a near the point a. Now assume that a, b ∈ R g are connected by a contour which does not intersect basic cycles. Hence we can dene the meromorphic dierential of the third kind Ω b-a which has residue 1 at b and residue -1 at a. Theorem 2.1.3. Let R g be a compact Riemann surface of genus g > 0. The dimension of the complex vector space of holomorphic dierentials given on R g equals g.

Example 2.1.4. For a hyperelliptic curve of the form (2.1.1), a basis of holomorphic dierentials is given by

ω j = λ j-1 µ dλ, j = 1, . . . , g, (2.1.7)
where g, the genus of the surface, is equal to N/2 -1 for even N , and to (N -1)/2 for odd N .

Periods of Abelian dierentials. Let (A, B) be a canonical homology basis on R g . Periods of an Abelian dierential ω of the rst or second kind are dened by

A k = A k ω, B k = B k ω, k = 1, . . . , g.
For an Abelian dierential Ω of the third kind having simple poles at p j ∈ R g , j = 1, . . . , n, there exist additional periods called polar periods, dened by

C j = γ j Ω, j = 1, . . . , n,
where γ j is a small cycle homologous to zero in H 1 (R g ) encircling the pole p j only. For any two Abelian dierentials ω and ω , the following Riemann's bilinear identity holds: and Ω b-a are related to normalized holomorphic dierentials via the Riemann's bilinear identity. Namely, the choice ω = ω j and ω = Ω

Rg ω ∧ ω = g k=1 A k B k -A k B k , ( 2
(N ) a with N > 1 in (2.1.8) leads to B j Ω (N ) a = 1 N -1 α j,N -2 , j = 1, . . . , g, (2.1.10)
where the scalar α j,N -2 is dened by the following expansion of the normalized holomorphic dierential ω j near the point a:

ω j = ∞ i=0 α j,i k i a dk a ,
where k a denotes a local parameter in a neighbourhood of a. For the choice ω = ω j and ω = Ω b-a one gets

B j Ω b-a = b a ω j , j = 1, . . . , g, (2.1.11)
where the integration path connecting a and b does not cross the A and B-cycles.

Riemann matrix. The g ×g matrix of B-periods B of the normalized holomorphic dierentials ω 1 , . . . , ω g , with entries given by It follows that the Abel map denes a map on linear equivalence classes of divisors.

B kj = B k ω j , j, k = 1, . . . ,

Multi-dimensional theta functions

This part is devoted to the theory of theta functions associated to Riemann surfaces, and explores some of its numerous applications. Following the line of the Chapter 2 in [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF], we recall how meromorphic functions and normalized Abelian dierentials on a compact Riemann surface of arbitrary genus are constructed explicitly in terms of the multi-dimensional theta functions. We discuss also the important role of the zeros of the theta function in the Jacobi inversion problem.

The question is whether we can nd a divisor, i.e., the preimage under the Abel map, for an arbitrary point in the Jacobian. Theta functions are of great importance in mathematical physics due to their role in the inverse problem for periodic and quasi-periodic ows.

Denition and properties

The theta function with (half integer

) characteristic δ = [δ 1 , δ 2 ] is dened by Θ B [δ](z) = m∈Z g exp 1 2 B(m + δ 1 ), m + δ 1 + m + δ 1 , z + 2iπδ 2 (2.2.1)
for any z ∈ C g ; here B is the Riemann matrix with entries (2.1.12), and δ 1 , δ 2 ∈ 0,

Θ[δ](z) = Θ(z + 2iπδ 2 + Bδ 1 ) exp 1 2 Bδ 1 , δ 1 + z + 2iπδ 2 , δ 1 . (2.2.2)
The theta function Θ[δ](z) is even if the characteristic δ is even, i.e, 4 δ 1 , δ 2 is even, and odd if the characteristic δ is odd, i.e., 4 δ 1 , δ 2 is odd. An even characteristic is called non-singular if Θ[δ](0) = 0, and an odd characteristic is called non-singular if the gradient ∇Θ[δ](0) is non-zero.

Algebro-geometric solutions of integrable systems are expressed in terms of Abelian functions (i.e., meromorphic functions in g complex variables having 2g independent periods) on the Jacobian J(R g ) with period lattice Λ. The theta function with characteristic (2.2.1) is not an Abelian function on J(R g ) since it has the following quasi-periodicity property with respect to the lattice Λ:

Θ[δ](z + 2iπN + BM) = Θ[δ](z) exp -1 2 BM, M -z, M + 2iπ( δ 1 , N -δ 2 , M ) , (2.2.3)
for any N, M ∈ Z g . It turns out that from (2.2.3), we construct Abelian functions on J(R g ) as a quotient of theta functions, namely, the function

f (z) = n i=1 Θ(z + a i ) Θ(z + b i ) , (2.2.4)
where the vectors a i , b i ∈ C g are such that n i=1 a i ≡ n i=1 b i (mod 2iπZ g ), is meromorphic on the torus J(R g ) = C g /Λ.

For the formulation of solutions to physically relevant integrable equations in terms of multidimensional theta functions, there is typically a preferred homology basis in which the solution takes a simple form. Let (A, B) and ( Ã, B) be arbitrary canonical homology basis dened on R g .

Under the change of homology basis (2.1.2), the theta function transforms as:

Θ B [δ](z) = κ det K exp 1 2 zt ( Kt ) -1 B z Θ B[ δ](z),
B = 2iπ (2iπ C + D B) K-1 , (2.2.6) z = (2iπ) -1 Kt z, (2.2.7) δ 1 δ 2 = A -B -C D δ1 δ2 + 1 2 Diag B A t D C t , (2.2.8)
for any z ∈ C g , where Diag(.) denotes the column vector of the diagonal entries of the matrix.

Here κ is a constant independent of z and B (the exact value of κ is not needed for our purposes).

Theta divisor and Jacobi inversion problem

The general Jacobi inversion problem consists, for a given point ζ = (ζ 1 , . . . , ζ g ) t ∈ J(R g ), in nding g points p 1 , . . . , p g on R g such that the following identity holds in the Jacobian:

g k=1 p k p 0 ω ≡ ζ ;
(2.2.9)

here ω = (ω 1 , . . . , ω g ) t denotes the vector of normalized holomorphic dierentials and p 0 ∈ R g is an arbitrary point.

In the case g = 1, this problem reduces to the inversion of an elliptic integral, which leads to elliptic functions, i.e., doubly periodic meromorphic functions having poles at a lattice of points in the complex plane. The theory of elliptic functions and their relation to the elliptic theta function was developed by Jacobi in his great treatise Fundamenta nova theoriae functionum ellipticarum (1829), and later papers. In his work, Jacobi discovered that elliptic functions can be expressed as quotients of theta functions.

In 1832, Jacobi formulated the inversion problem for hyperelliptic integrals [83], i.e., integrals over a rational function R(x, W (x)), where W (x) is the square root of a polynomial of degree 2g+1 or 2g+2 with g ≥ 2.

In the case g = 2, this problem was solved by Göpel [72] and Rosenhain [138], generalizing Jacobi's theta functions to theta functions depending on two variables.

The basic problem for a more complicated algebraic integral was to nd an expression for the inverse of the transformation dened by a complete set of integrals of the rst kind. This problem is known as the Jacobi inversion problem, and its solution, involving theta functions of several variables, was found by Riemann and Weierstrass (see, for instance, [START_REF] Stahl | Theorie der Abel'schen Functionen[END_REF][START_REF] Krazer | Lehrbuch der Thetafunktionen[END_REF][START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF]). The following theorem by Riemann gives a solution to the Jacobi inversion problem (2.2.9):

Theorem 2.2.1. Let (A, B) be a canonical homology basis and choose p 0 ∈ R g . Denote by K = (K 1 , . . . , K g ) t the vector of Riemann constants dened by

K j = 2iπ + B jj 2 - 1 2i π k =j A k ω k (p) p p 0 ω j , j = 1, . . . , g. (2.2.10)
Let ζ ∈ J(R g ) such that the Riemann theta function, dened by F (p) = Θ(Π(p)ζ -K) for any p ∈ R g , does not vanish identically on R g . Then the function F (p) has on R g exactly g zeros p 1 , . . . , p g that give a solution to the Jacobi inversion problem (2.2.9). Moreover, the points p 1 , . . . , p g are dened by (2.2.9) uniquely up to a permutation.

This theorem provides also information on the theta divisor, the set of zeros of the theta function:

Corollary 2.2.1. For a non-special divisor D = p 1 + . . . + p g (i.e., there exists a non-constant function f with (f ) ≥ -D), the function F dened by

F (p) = Θ (Π(p) -Π(D) -K)
for any p ∈ R g , has exactly g zeros p 1 , . . . , p g .

Construction of functions and dierentials on a Riemann surface

In this part we construct meromorphic functions and normalized Abelian dierentials of the second and third kinds in terms of the multi-dimensional theta function (2.2.1). The results presented here are based on Riemann's fundamental theorem (Theorem 2.2.1). This representation of Abelian dierentials in terms of theta functions is crucial for a numerical study of the solutions of the studied integrable equations.

Let us rst introduce the prime form which is a generalization of the function xy on C 2 to a Riemann surface of genus g > 0. Let a, b ∈ R g be distinct points. Then the prime form is dened by

E(a, b) = Θ[δ]( a b ) h δ (a)h δ (b) ; (2.2.11)
here h δ (a) is a spinor given by h 2 δ (a) =

g j=1 ∂Θ[δ]
∂z j (0) ω j (a), where δ is a non-singular odd characteristic. Note that the prime form is independent of the choice of the characteristic δ.

From the basic properties of the prime form (see [START_REF] Fay | Theta functions on Riemann surfaces[END_REF] for more details), it can be checked that: i) if the divisor D = n i=1 a ib i is principal, then the map f : R g → CP 1 dened by

f (p) = n i=1 E(p, a i ) E(p, b i ) (2.2.12)
is a single-valued meromorphic function on R g having zeros at a i and poles at b i ; ii) for two distinct points a, b ∈ R g , the normalized third kind dierential Ω b-a reads

Ω b-a (p) = d p ln E(p, b) E(p, a) , p ∈ R g ; (2.2.13)
iii) the bidierential Ω(p, q) which has zero A-periods and is decomposed according to

Ω(p, q) = 1 (k(p) -k(q)) 2 + O(1) dk(p) dk(q),
for any p, q in the same neighbourhood, k being a local parameter dened on this neighbourhood, admits the representation: Ω(p, q) = d p d q ln E(p, q);

(2.2.14) iv) the normalized second kind dierential Ω (N ) a is thus given by

Ω (N ) a (p) = ∂ kx Ω(p, x) x=a (2.2.15)
for any p ∈ R g , where k x denotes a local parameter in a neighbourhood of x ∈ R g .

Fay's identity

The well known trisecant identity discovered by Fay is a far-reaching generalization of the addition theorem for elliptic theta functions to multi-dimensional theta functions (see [START_REF] Fay | Theta functions on Riemann surfaces[END_REF]). It can also be seen as a generalization of the identity satised by the cross-ratio to Riemann surfaces.

Fay's identity plays an important role in various domains of mathematics, for instance, in the theory of Jacobian varieties [START_REF] Arbarello | Fay's trisecant formula and a characterization of Jacobian Varieties[END_REF], in conformal eld theory [START_REF] Raina | Fay's Trisecant Identity and Conformal Field Theory[END_REF], and in operator theory [START_REF] Mccullough | The trisecant identity and operator theory[END_REF].

Moreover, as discovered by Mumford [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF], theta-functional solutions of certain integrable equations such as Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP), or Sine-Gordon equations may be derived from Fay's trisecant identity and its degenerations.

Fay's identity and generalization of the cross-ratio to Riemann surfaces

Let a, b, c, d ∈ R g be arbitrary points on a compact Riemann surface R g of genus g > 0. Then for any z ∈ C g , Fay's trisecant identity has the form

E(a, b)E(c, d) Θ z + a c Θ z + d b + E(a, c)E(d, b) Θ z + a b Θ z + d c = E(a, d)E(c, b) Θ(z) Θ z + a c + d b , (2.3.1)
where all integration contours do not intersect cycles of the canonical homology basis (recall that b a denotes the Abel map (2.1.13) between a and b); here Θ denotes the theta function (2.2.1) with zero characteristic, and E(a, b) the prime form introduced in (2.2.11). Proofs of this formula may be found in Fay [START_REF] Fay | Theta functions on Riemann surfaces[END_REF], Mumford [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF], Farkas [START_REF] Farkas | On Fay's trisecant formula[END_REF], or Gunning [START_REF] Gunning | Some identities for Abelian integrals[END_REF].

More recently [START_REF] Poor | Fay's trisecant formula and cross-ratios[END_REF], identity (2.3.1) was proved using cross-ratio functions on Riemann surfaces. This point of view allows to consider Fay's identity as a generalization of the identity satised by the cross-ratio on the sphere to Riemann surfaces. Namely, let us denote by

λ 0 (a, b, c, d) = (a -b) (c -d) (a -d) (c -b)
the cross-ratio of four points a, b, c, d on the Riemann sphere. Then it is well known that λ 0 satises the following identity:

λ 0 (a, b, c, d) + λ 0 (a, c, b, d) = 1. (2.3.2)
The cross-ratio function on an arbitrary compact Riemann surface R g of genus g > 0 is in fact characterized by its divisor, its symmetries, and its transformation laws (see [START_REF] Poor | Fay's trisecant formula and cross-ratios[END_REF] for more details). We denote the cross-ratio function on R g by λ g (a, b, c, d), for a, b, c, d ∈ R g . Therefore, according to the properties of the prime form and characterizations of the cross-ratio function, the function λ g (a, b, c, d) can be written with Riemann's theorem as

λ g (a, b, c, d) = E(a, b) E(c, d) E(a, d) E(c,

b) .

The cross-ratio functions λ g (a, b, c, d) and λ g (a, c, b, d) are then related by identity (2.3.1) which thus generalizes identity (2.3.2) to Riemann surfaces.

Previously known degenerated versions of Fay's identity

Let us now discuss degenerations of identity (2.3.1). Let k a (p) denote a local parameter near a ∈ R g , where p lies in a neighbourhood of a. For j = 1, . . . , g, consider the following expansion of the normalized holomorphic dierentials ω j near a:

ω j (p) = V a, j + W a, j k a (p) + U a, j k a (p) 2 2! + o k a (p) 2 dk a (p), (2.3.3)
where V a, j , W a, j , U a, j ∈ C. Let us denote by D a the operator of directional derivative along the vector V a = (V a,1 , . . . , V a,g ) t :

D a F (z) = g j=1 ∂ z j F (z) V a,j = ∇F (z), V a , (2.3.4)
where F : C g -→ C is an arbitrary function; denote by D a the operator of directional derivative along the vector W a = (W a,1 , . . . , W a,g ) t , and by D a the directional derivative along the vector

U a = (U a,1 , . . . , U a,g ) t .
Dierentiating identity (2.3.1) with respect to the local parameter k d near point d, and considering the limit when d tends to b, one gets the rst degenerated version of Fay's identity derived in [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF]:

D b ln Θ(z + a c ) Θ(z) = p 1 (a, b, c) + p 2 (a, b, c) Θ(z + a b ) Θ(z + b c ) Θ(z + a c ) Θ(z) , (2.3.5)
which holds for any distinct points a, b, c ∈ R g and any z ∈ C g ; functions p 1 and p 2 of the variables a, b, c are given by:

p 1 (a, b, c) = D b ln Θ[δ]( b c ) Θ[δ]( b a ) , (2.3.6) p 2 (a, b, c) = Θ[δ]( a c ) Θ[δ]( a b ) Θ[δ]( c b ) D b Θ[δ](0), (2.3.7)
where δ is a non-singular odd characteristic. Note that the p i for i = 1, 2, depend on the choice of local parameters near the points a, b, c. From representations (2.2.13) and (2.2.11) in terms of multi-dimensional theta functions of the dierential Ω c-a and the prime form, one gets:

Ω c-a (b) = p 1 (a, b, c) dk b , (2.3.8) E(c, a) E(b, a) E(c, b) = p 2 (a, b, c) dk b , (2.3.9)
where k b denotes a local parameter near point b.

Analogously, dierentiating (2.3.5) with respect to the local parameter k c near point c, and considering the limit when c tends to a, one gets

D a D b ln Θ(z) = q 1 (a, b) + q 2 (a, b) Θ(z + b a ) Θ(z - b a ) Θ(z) 2 , (2.3.10)
which holds for any distinct points a, b ∈ R g and any z ∈ C g ; functions q 1 and q 2 of the variables a and b depend on the choice of local parameters near these points and are given by:

q 1 (a, b) = D a D b ln Θ[δ]( b a ) , (2.3.11) q 2 (a, b) = D a Θ[δ](0) D b Θ[δ](0) Θ[δ]( b a ) 2 , (2.3.12)
where δ is a non-singular odd characteristic. From (2.2.14) and (2.2.11) one easily gets:

Ω(a, b) = -q 1 (a, b) dk a dk b , (2.3.13) E(a, b) -2 = q 2 (a, b) dk a dk b , (2.3.14)
where k c denotes a local parameter near point c ∈ {a, b}.

The third degenerated version of Fay's identity derived in [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF] is obtained from (2.3.10) in the limit when b tends to a. Namely, let b be in a neighbourhood of a, put k a (b) = t, and expand identity (2.3.10) in terms of the variable t. Then it can be checked that the rst nontrivial term, which is the t 4 term, reads

D 4 a ln Θ(z) + 6 D 2 a ln Θ(z) 2 -2D a D a ln Θ(z) + 3 D a 2 ln Θ(z) + c 1 D 2 a ln Θ(z) + c 2 = 0, (2.3.15)
where the scalars c 1 and c 2 depend on the coecients appearing in the expansion with respect to the variable t of q 1 (a, b) and q 2 (a, b) (knowledge of the exact value of c 1 and c 2 is not needed for our purpose).

In Section 6.1, we shall present a new degenerated version of Fay's identity obtained by dierentiating identity (2.3.1) twice (instead of once) with respect to the local parameter k d near point d, and by considering the limit when d tends to b.

Applications to integrable systems -Mumford's approach

In the early 80's, Mumford observed [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF] that integrable equations like KdV, KP or sine-Gordon, are `hidden' in Fay's trisecant formula (2.3.1). Mumford's approach, based on degenerated versions of Fay's identity, shows how algebro-geometric solutions of integrable equations arise from a degeneration of a purely algebro-geometric identity.

KP and KdV equations. Historically, the Korteweg-de Vries equation (KdV)

4 u t = 6 uu x + u xxx , (2.3.16)
and its generalization to two spatial variables, the Kadomtsev-Petviashvili equation (KP2),

3 u yy = (4 u t -(6 uu x + u xxx )) x , (2.3.17)
were the most important examples of applications of algebraic-geometric methods in the 1970's, see for instance [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]. Algebro-geometric solutions of the KP equation were rst constructed by Krichever [START_REF] Krichever | Algebro-geometric construction of the Zakharov-Shabat equations and their periodic solutions[END_REF]. Here we show how these solutions can be derived from Fay's identity following Mumford's approach.

Let R g be a compact Riemann surface of genus g > 0 and take a ∈ R g . Choose d ∈ C g and dene the function

u(x, y, t) = 2 D 2 a ln Θ(V a x + W a y + U a t/2 -d) + c 1 /6, (2.3.18)
where the scalar c 1 was introduced in (2.3.15), and where the vectors V a , W a , U a are dened in (2.3.3). Then the function u satises the KP equation (2.3.17). Indeed, this can be checked directly by applying the second directional derivative D 2 a to identity (2.3.15). Finally, solutions (2.3.18) of the KP equation obtained via Mumford's approach coincide with the ones (1.1.9)

constructed by using Baker-Akhiezer functions.

Chapter 2. Theta functions and Fay's identity

Algebro-geometric solutions of the KdV equation are deduced from (2.3.18) when the Riemann surface is hyperelliptic. Namely, assume that R g is a hyperelliptic curve of the form (2.1.1), and denote by σ the hyperelliptic involution dened by σ(λ, µ) = (λ, -µ). Let a be one of the ramication points of the curve and choose a local parameter k a near a satisfying k a (σ(p)) =k a (p) for any p lying in a neighbourhood of a. Let us check that the vector W a vanishes. Denote by σ * the action of σ lifted to the space of holomorphic dierentials, namely, σ * ω(p) = ω(σp) for any p ∈ R g . Then, using the following action of σ on the normalized holomorphic dierentials ω j , Sine-Gordon equation. Analogously, let us show how algebro-geometric solutions of the sine-Gordon equation arise from Fay's identity. In light-cone coordinates (x, t) = (ξ + η, ξη), the sine-Gordon equation

σ * ω j = -ω j , j = 1, . . . , g, (2.3
u tt -u xx = sin(u), (2.3.20)
has the form:

u ξη = sin(u). (2.3.21)
Theta-functional solutions of the sine-Gordon equation were found in [START_REF] Kozel | Almost periodic solutions of the equation u ttu xx + sin(u) = 0[END_REF] and [START_REF] Its | Explicit formulas for solutions of the Nonlinear Schrödinger equation[END_REF] following the line of the spectral approach applied to the KdV equation.

Let R g be a hyperelliptic curve of the form (2.1.1) and let a, b ∈ R g be two distinct ramication points. First let us prove that the vector r = (2.3.23) Since the contour + ˜ is closed, it consists of a linear combination of A and B-cycles. In other words, there exist n, m ∈ Z g such that

+ ˜ ω = 2iπn + Bm,
which by (2.3.23) leads to (2.3.22). Therefore, from identity (2.3.10) and the quasi-periodicity (2.2.3) of the theta function, we deduce

D a D b ln Θ(z + r) Θ(z) = q 2 (a, b) e -1 2 Bm,m Θ(z) 2 Θ(z + r) 2 e -φ(z) - Θ(z + r) 2 Θ(z) 2 e φ(z) , (2.3.24)
for any z ∈ C g , where φ(z) = z + r, m . Let us dene the function

u(z) = 2i ln Θ(z + r) Θ(z) + iφ(z). (2.3.25) Thus (2.3.24) becomes D a D b u(z) = c 2i e iu(z) -e -iu(z) = c sin(u(z)), (2.3.26
)

where c = q 2 (a, b) exp{-1 2 Bm, m }. Hence, putting z = c -1/2 (V a ξ + V b η) -d in (2.3.25)
, where vectors V a , V b are dened in (2.3.3), and d ∈ C g is arbitrary, one gets that function u(ξ, η) is a solution of the sine-Gordon equation (2.3.21).

Toda lattice equations. The Toda lattice equation (1D Toda)

∂ 2 x u n = e u n-1 -un -e un-u n+1 , (2.3.27)
is one of the most famous integrable nonlinear lattice equations, see [START_REF] Flaschka | The Toda lattice[END_REF] and [START_REF] Toda | Theory of Nonlinear Lattices[END_REF]; here u n (x), for x ∈ R and n ∈ Z, is a dynamical function on a one-dimensional lattice. The Toda lattice equation describes the dynamics of a one-dimensional physical lattice, the masses of which are subject to an interaction potential of exponential type. This equation admits numerous extensions having applications in various other physical and mathematical contexts, see, e.g., [START_REF] Toda | Theory of Nonlinear Waves and Solitons[END_REF][START_REF] Eilenberger | An Introduction to Dierence Equations[END_REF][START_REF] Faddeev | Hamiltonian Methods in the Theory of Solitons[END_REF][START_REF] Teschl | Jacobi Operators and Completely Integrable Nonlinear Lattices[END_REF]. It should also be mentioned that the Toda equation can be viewed as a discrete version of the KdV equation (see [START_REF] Toda | Theory of Nonlinear Lattices[END_REF] or [START_REF] Palais | The symmetries of solitons[END_REF]). Periodic problems for the Toda chain and a discrete version of the KdV equation were studied by Kac and Van Moerbeke in a series of papers (see, for instance, [START_REF] Kac | A complete solution of the periodic Toda problem[END_REF]).

Equation (2.3.27) admits a two-dimensional integrable version, the 2D Toda lattice equation:

∂ x ∂ y u n = e u n-1 -un -e un-u n+1 , (2.3.28)
where the function u n (x, y) depends on two dynamical variables x, y and a discrete variable n ∈ Z. Note that 2-periodic solutions of (2.3.28), namely, solutions which satisfy u n+2 = u n , are solutions of the sine-Gordon equation (2.3.21) after a suitable change of variables.

The rst results on algebro-geometric solutions of the Toda equation were given by Date and Tanaka [START_REF] Date | Analogue of inverse scattering theory for the discrete Hill's equation and exact solutions for the periodic Toda lattice[END_REF]. Further important contributions were made by Krichever [START_REF] Krichever | Algebraic curves and nonlinear dierence equations[END_REF]- [START_REF] Krichever | Algebraic-geometrical methods in the theory of integrable equations and their perturbations[END_REF], Van Moerbeke and Mumford [START_REF] Van Moerbeke | The spectrum of dierence operators and algebraic curves[END_REF][START_REF] Mumford | An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related non-linear equations[END_REF]. Algebro-geometric solutions of the 2D Toda lattice are constructed from the algebro-geometric data {R g , a, b}, where R g is a compact Riemann surface of genus g and a, b ∈ R g are two distinct points. Analogously to the KP and KdV equations, if we restrict these algebro-geometric data to {R g , σ, a, b} where R g is a hyperelliptic curve with σ the hyperelliptic involution, and a, b ∈ R g satisfy σ(a) = b, the 2D Toda lattice reduces to the 1D Toda lattice. Now let us construct algebro-geometric solutions of (2.3.28) using Mumford's approach. From (2.3.10) one has:

D a D b ln Θ(z + (n -1) r) = q 1 + q 2 Θ(z + n r) Θ(z + (n -2) r) Θ(z + (n -1) r) 2 , (2.3.29) D a D b ln Θ(z + n r) = q 1 + q 2 Θ(z + (n + 1) r) Θ(z + (n -1) r) Θ(z + n r) 2 , (2.3.30)
for any z ∈ C g , n ∈ Z; here q i for i = 1, 2 denotes the scalar q i (a, b) and r =

D a D b ln Θ(z + (n -1) r) Θ(z + n r) = (2.3.31) q 2 Θ(z + n r) Θ(z + (n -1) r) Θ(z + (n -2) r) Θ(z + (n -1) r) -q 2 Θ(z + (n + 1) r) Θ(z + n r) Θ(z + (n -1) r) Θ(z + n r) ,
which can be rewritten as

D a D b ln Θ(z + (n -1) r) Θ(z + n r) = (2.3.32) exp ln q 2 + ln Θ(z + (n -2) r) Θ(z + (n -1) r) -ln Θ(z + (n -1) r) Θ(z + n r) -exp ln q 2 + ln Θ(z + (n -1) r) Θ(z + n r) -ln Θ(z + n r) Θ(z + (n + 1) r)
.

Now let us dene the function Chapter 3

u n (z) = α -n ln q 2 + ln Θ(z + (n -1) r) Θ(z + n r) (2.3.33) for some z ∈ C g , n ∈ Z, and α ∈ C. Thus (2.3.32) reads D a D b u n = e u n-1 -un -e un-u n+1 . Putting z = V a x + V b y -d in (2.

Real Riemann surfaces

In this chapter we recall some facts from the theory of real compact Riemann surfaces R g .

Following the line of [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF], we introduce a canonical homology basis adapted to the real structure of R g . For this choice of basic cycles, we study reality properties of various objects dened on the real Riemann surface R g .

Topological type of a real Riemann surface

A Riemann surface R g is called real if it admits an anti-holomorphic involution τ : R g → R g , τ 2 = id. For this real structure, we denote by R g (R) the set of xed points. The connected components of the set of xed points of the anti-involution τ are called real ovals of τ . According to Harnack's inequality [START_REF] Harnack | Ueber die Vieltheiligkeit der ebenen algebraischen Curven[END_REF], the number k of real ovals of a real Riemann surface of genus g cannot exceed g + 1: 0 ≤ k ≤ g + 1. Curves with the maximal number k = g + 1 of real ovals are called M-curves.

The complement R g \ R g (R) has either one or two connected components. The curve R g is called a dividing curve if R g \ R g (R) has two components, and R g is called non-dividing if R g \ R g (R) is connected (notice that an M-curve is always a dividing curve).

Example 3.1.1. Consider the hyperelliptic curve of genus g dened by the equation

µ 2 = 2g+2 i=1 (λ -λ i ), (3.1.1)
where the branch points λ i ∈ R are ordered such that λ 1 < . . . < λ 2g+2 . On such a curve, we can dene two anti-holomorphic involutions τ 1 and τ 2 , given respectively by τ 1 (λ, µ) = (λ, µ) and τ 2 (λ, µ) = (λ, -µ). Projections of real ovals of τ 1 on the λ-plane coincide with the segments [λ 2g+2 , λ 1 ], . . . , [λ 2g , λ 2g+1 ], whereas projections of real ovals of τ 2 on the λ-plane coincide with the segments [λ 1 , λ 2 ], . . . , [λ 2g+1 , λ 2g+2 ]. Hence the curve (3.1.1) is an M-curve with respect to both anti-involutions τ 1 and τ 2 .

Example 3.1.2. The hyperelliptic curve of genus g dened by the equation

µ 2 = - g+1 i=1 (λ -λ i )(λ -λ i ), λ i ∈ C \ R, (3.1.2)
does not have real ovals with respect to the anti-holomorphic involution τ (λ, µ) = (λ, µ).

3.2

Action of τ on the homology group H 1 (R g ) Denote by (A, B) the set of generators of the homology group H 1 (R g ). According to Proposition 2.2 in [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF] (see also [START_REF] Gross | Real algebraic curves[END_REF]), there exists a canonical homology basis (that we call for simplicity 'Vinnikov basis' in the following) such that

τ A τ B = I g 0 H -I g A B , (3.2.1)
where I g is the g × g unit matrix, and H is a block diagonal g × g matrix dened as follows:

1) if R g (R) = ∅, H =               0 1 1 0 . . . 0 1 1 0 0 . . . 0               if R g is dividing, H =           1 . . . 1 0 . . . 0           if R g is non-dividing; rank(H) = g + 1 -k in both cases;
2) if R g (R) = ∅, (i.e., the curve does not have real oval), then

H =        0 1 1 0 . . . 0 1 1 0        or H =          0 1 1 0 . . . 0 1 1 0 0          ; rank(H) = g if g is even, rank(H) = g -1 if g is odd.
Let us choose the homology basis satisfying (3.2.1), and let us study the action of τ on the normalized holomorphic dierentials, as well as the action of the complex conjugation on the theta function with zero characteristics.

Denote by τ * the action of τ lifted to the space of holomorphic dierentials: τ * ω(p) = ω(τ p) for any p ∈ R g . By (3.2.1) the A-cycles of the homology basis are invariant under τ . Due to the normalization conditions (2.1.9), this leads to the following action of τ on the normalized holomorphic dierentials:

τ * ω j = -ω j , (3.2.2)
Using (3.2.1) and (3.2.2) we get the following reality property for the Riemann matrix B:

B = B -2iπ H. (3.2.3) Moreover, according to Proposition 2.3 in [150], for any z ∈ C g , relation (3.2.3) implies Θ(z) = κ Θ(z -iπ diag(H)), (3.2.4)
where κ is a complex scalar of modulus one which depends on the matrix H (the exact value of κ is not needed for our purpose); here diag(H) denotes the vector of diagonal elements of the matrix H.

3.3

Action of τ on H 1 (R g \ {a, b}) and H 1 (R g , {a, b})

Let a, b ∈ R g be two distinct points. Here we study the action of τ on the homology group H 1 (R g \ {a, b}) of the punctured Riemann surface R g \ {a, b}, and the action of τ on its dual, the relative homology group H 1 (R g , {a, b}).

Denote by (A, B, ) the standard generators of the relative homology group H 1 (R g , {a, b}),

where is an oriented contour between a and b which does not intersect the canonical homology basis (A, B), and denote by (A, B, S b ) the generators of the homology group H 1 (R g \ {a, b}), where S b is a positively oriented small contour around b such that S b • = 1.

In what follows, we assume that the basis of cycles (A, B) of the rst homology group satises (3.2.1). We consider the case where τ a = b, and the case where τ a = a, τ b = b.

Case τ a = b

In this part we assume that τ a = b. Proposition 3.3.1. The action of the anti-holomorphic involution τ on:

1. the generators (A, B, ) of the relative homology group H 1 (R g , {a, b}) is given by

  τ A τ B τ   =   I g 0 0 H -I g 0 N t 0 -1     A B   , (3.3.1)
for some vector N ∈ Z g ;

2. the generators (A, B, S b ) of the homology group H 1 (R g \ {a, b}) is given by   τ A τ B τ S b   =   I g 0 0 H -I g N 0 0 1     A B S b   , (3.3.2)
where the vector N ∈ Z g is the same as in (3.3.1).

Proof. The action of τ on the A and B-cycles in (3.3.1) coincides with the one (3.2.1) in H 1 (R g ). From (3.2.1), one sees that any contour in H 1 (R g ) which is invariant under τ is a combination of A-cycles only. In particular, the τ -invariant closed contour τ + ∈ H 1 (R g ) can be written as

τ + = N t A, (3.3.3)
for some N ∈ Z g . This proves (3.3.1). Now let us prove (3.3.2). By (3.2.1), the cycles τ A admit the following decomposition in

H 1 (R g \ {a, b}): τ A = A + n S b , (3.3.4)
for some n ∈ Z g . Since τ changes the orientation of R g , all intersection indices change their sign under the action of τ . We get from (3.3.4)

0 = A • = -τ A • τ = -(A + n S b ) • τ = -(A + n S b ) • (-+ N t A), (3.3.5)
where N ∈ Z g is dened by (3.3.1). The last intersection index in (3.3.5) equals n, which implies τ A = A. According to (3.2.1), the action of τ on the B-cycles in H 1 (R g \ {a, b}) is given by

τ B = -B + HA + m S b , (3.3.6)
for some m ∈ Z g . Then

0 = B • = -τ B • τ = -(-B + HA + m S b ) • τ = -(-B + HA + m S b ) • (-+ N t A), (3.3.7)
where N is dened by (3. 1. the generators (A, B, ) of the relative homology group H 1 (R g , {a, b}) is given by

  τ A τ B τ   =   I g 0 0 H -I g 0 N t M t 1     A B   , (3.3.9)
where N, M ∈ Z g are related by 2 N + HM = 0;

(3.3.10) 2. the generators (A, B, S b ) of the homology group H 1 (R g \ {a, b}) is given by   τ A τ B τ S b   =   I g 0 -M H -I g N 0 0 -1     A B S b   , (3.3.11)
where the vectors N, M ∈ Z g are the same as in (3.3.9).

Proof. The action of τ on the A and B-cycles in (3.3.9) coincides with the one (3.2.1) in H 1 (R g ).

From (3.2.1), one sees that each contour C ∈ H 1 (R g ) which satises τ C = -C, can be represented by C = Ñt A + Mt B, (3.3.12)
where Ñ, M ∈ Z g are related by 2 Ñ + H M = 0. In particular, the closed contour τ -∈ H 1 (R g , {a, b}) can be written as

τ -= N t A + M t B, (3.3.13)
where N, M ∈ Z g are related by 2 N + HM = 0. This proves (3.3.9). Now let us prove (3.3.11). By (3.2.1), the cycles τ A admit the following decomposition in

H 1 (R g \ {a, b}) τ A = A + n S b , (3.3.14)
for some n ∈ Z g . Therefore, we get from (3.3.14)

0 = A • = -τ A • τ = -(A + n S b ) • τ = -(A + n S b ) • ( + N t A + M t B), (3.3.15)
where the vectors N, M ∈ Z g are dened by (3.3.9). The last intersection index in (3.3.15) equals

-(n + M), which gives τ A = A -M S b . According to (3.2.1), the action of τ on the B-cycles in H 1 (R g \ {a, b}) is given by τ B = -B + HA + m S b , (3.3.16)
for some m ∈ Z g . Then

0 = B • = -τ B • τ = -(-B + HA + m S b ) • τ = -(-B + HA + m S b ) • ( + N t A + M t B), (3.3.17)
where N, M ∈ Z g are dened by (3.3.9). The last intersection index in (3.3.17 Analogously to the case where τ a = b, considering the action (3.3.11) of τ on the A-cycles, we obtain the following action of τ on the normalized dierential of the third kind Ω b-a :

τ * Ω b-a = Ω b-a + g k=1 M k ω k , (3.3.18)
where ω k , k = 1, . . . , g, are the normalized holomorphic dierentials.

Action of τ on the Jacobian and theta divisor

In this part, we review known results [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF], [START_REF] Dubrovin | Real theta function solutions of the Kadomtsev-Petviashvili equation[END_REF] about the theta divisor of real Riemann surfaces.

For a given real Riemann surface R g , let us choose the canonical homology basis satisfying (3.2.1), and consider the Jacobian J = J(R g ).

The anti-holomorphic involution τ dened on R g gives rise to an anti-holomorphic involution on the Jacobian: if D is a positive divisor of degree n on R g , then τ D is the class of the point τ D n τ p 0 ω = D n p 0 τ * ω in the Jacobian. Therefore, by (3.2.2) τ lifts to the anti-holomorphic involution on J, denoted also by τ , given by

τ ζ = -ζ + n ζ Π(τ p 0 ),
S 1 = {ζ ∈ J; ζ + τ ζ = iπ diag(H)}, (3.4.2) S 2 = {ζ ∈ J; ζ -τ ζ = iπ diag(H)}, (3.4.3)
where the matrix H was introduced in Section 3.2. Below we study their intersections S 1 ∩ (Θ) and S 2 ∩ (Θ) with the theta divisor (Θ), the set of zeros of the theta function.

Let us introduce the following notation: (e i ) k = δ ik , B i = B e i . The following proposition was proved in [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF].

Proposition 3.4.1. The set S 1 is the disjoint union of the tori T v dened by

T v = {ζ ∈ J; ζ = iπ (diag(H)/2 + v 1 e r+1 + . . . + v g-r e g ) + β 1 Re(B 1 ) + . . . + β g Re(B g ) , β 1 , . . . , β r ∈ R/2Z , β r+1 , . . . , β g ∈ R/Z}, (3.4.4) where v = (v 1 , . . . , v g-r ) ∈ (Z/2Z) g-r and r is the rank of the matrix H. Moreover, if R g (R) = ∅, then T v ∩ (Θ) = ∅ if and only if the curve is dividing and v = 0.
The last statement means that among all curves which admit real ovals, the only torus T v which does not intersect the theta divisor is the torus T 0 corresponding to dividing curves. This torus is given by

T 0 = {ζ ∈ J; ζ = β 1 Re(B 1 ) + . . . + β g Re(B g ), β 1 , . . . , β r ∈ R/2Z , β r+1 , . . . , β g ∈ R/Z}. (3.4.5)
It is straightforward to prove that the set S 1 is the disjoint union of the tori (3.4.4). Moreover, in the case where R g (R) = ∅ and R g is non-dividing, for all v the torus T v contains an odd half-period which yields T v ∩ (Θ) = ∅. The same holds for all v = 0 in the case where the curve is dividing or does not have real ovals. Corollary 4.3 in [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF] proves that if R g is dividing then T 0 ∩ (Θ) = ∅.

The following proposition was proved in [START_REF] Dubrovin | Real theta function solutions of the Kadomtsev-Petviashvili equation[END_REF]. Proposition 3.4.2. The set S 2 is the disjoint union of the tori Tv dened by

Tv = {ζ ∈ J ; ζ = 2iπ (α 1 e 1 + . . . + α g e g ) + (v 1 /2) B r+1 + . . . + (v g-r /2) B g , α 1 , . . . , α g ∈ R/Z}, (3.4.6) where v = (v 1 , . . . , v g-r ) ∈ (Z/2Z) g-r and r is the rank of the matrix H. Moreover, if R g (R) = ∅, then Tv ∩ (Θ) = ∅ if and only if the curve is an M-curve and v = 0.
The rst step of the proof consists in showing that, for ζ ∈ S 2 , in the case where the number k of real ovals satises 1 ≤ k ≤ g, the divisor D of degree g dened by ζ = Π(D) + K can be deformed in such a way that it contains the point p 0 , the base point of the Abel map with τ p 0 = p 0 (here K denotes the vector of Riemann constants). In this case one gets Θ(ζ) = 0. It follows that the intersection set between S 2 and (Θ) can be empty only for curves with g + 1 real ovals. The proof is completed by noticing that in the case k = g + 1, there exists only one connected component of the Jacobian such that Θ(ζ) does not vanish; it consists of divisors of degree g having one point on each real oval which does not contain p 0 . The image Π(D) + K of such divisors in the Jacobian consists of vectors in T0 . We give here an independent derivation of algebro-geometric solutions to the Camassa-Holm and Dym-type equations based on identities between multi-dimensional theta functions, which naturally arise from Fay's identity (2.3.1). We also give conditions for the solutions to be real and smooth. In what follows R g denotes a hyperelliptic curve of genus g > 0, written as

µ 2 = 2g+2 i=1 (λ -λ i ), (4.0.1)
where the branch points λ i ∈ C satisfy the relations λ i = λ j for i = j. We denote by σ the hyperelliptic involution dened by σ(λ, µ) = (λ, -µ).

Solutions of the Camassa-Holm equation

The Camassa-Holm equation (CH),

u t + 3 uu x = u xxt + 2 u x u xx + uu xxx -2k u x , (4.1.1)
where k ∈ C, was rst obtained by Fuchssteiner and Fokas [START_REF] Fokas | Symplectic structures, their Bäcklund transformations and hereditary symmetries[END_REF] using the method of recursion operators. Camassa and Holm [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF] derived this equation by means of physical principles, showing that it models the propagation of unidirectional waves on shallow water, and found a Lax pair of the system. Note that the CH equation can be expressed in the following simple form,

m t + u m x + 2 m u x = 0, (4.1.2)
where we put m := uu xx + k.

Theta-functional solutions of the CH equation (4.1.2) were rst given in [START_REF] Alber | Algebraic geometrical solutions for certain evolution equations and Hamiltonian ows on nonlinear subvarieties of generalized Jacobians[END_REF] by solving a generalized Jacobi inversion problem. Our goal here is to give an independent derivation of such solutions via Fay's trisecant identity (2.3.1). In this way, one gets purely transcendental conditions on the parameters (i.e., without reference to a divisor dened by the solution of a Jacobi inversion problem), so that solutions are real-valued and smooth.

Identities between theta functions

From corollaries (2.3.5) and (2.3.10) of Fay's identity, we derive the following identities, necessary in our approach to construct algebro-geometric solutions of the Camassa-Holm equation. namely, e = (λ j , 0) for some j ∈ {1, . . . , 2g + 2}. Denote by g 1 and g 2 the following functions of the variable z ∈ C g :

g 1 (z) = Θ z + r 2 Θ(z) , g 2 (z) = Θ z -r 2 Θ(z) , (4.1.3)
where r = b a ω and ω is the vector of normalized holomorphic dierentials. Then the two following identities hold:

D b D e ln g 1 g 2 = - p 2 g 1 g 2 D b ln g 1 g 2 , (4.1.4) D b D e ln(g 1 g 2 ) = q2 p2 1 g 1 g 2 D b ln g 1 g 2 -2 p1 -2 q2 g 1 g 2 . (4.1.5)
Here we used the following notation:

p 2 = p 2 (b, e, a), pi = p i (e, b, a), q2 = q 2 (b, e), (4.1.6)
where the scalars q 2 (a, b) and p i (a, b, c), i = 1, 2, are dened in (2.3.12) and (2.

D b D e ln g 1 g 2 = -p 2 D b (g 1 g 2 ) (g 1 g 2 ) 2 = - p 2 g 1 g 2 D b ln g 1 g 2 ,
D b ln g 1 = p1 + p2 g 2 Θ(z + r) Θ z + r 2 .
Changing z to -z in the last equality, one gets

D b ln g 2 = -p1 -p2 g 1 Θ(z -r) Θ z -r 2 .
From both previous identities, it can be deduced that

Θ(z + r) Θ z + r 2 = (p 2 g 2 ) -1 (D b ln g 1 -p1 ), (4.1.8) Θ(z -r) Θ z -r 2 = -(p 2 g 1 ) -1 (D b ln g 2 + p1
).

D b D e ln(g 1 g 2 ) = q2 g 1 Θ(z + r) Θ z + r 2 + q2 g 2 Θ(z -r) Θ z -r 2 -2 q2 g 1 g 2 , ( 4
x + α 1 y + α 2 t + ζ = ln Θ Z -d + r 2 Θ Z -d -r 2 , (4.1.11)
where r = ω. Here the vector Z is given by Z(x, t) = V e y(x, t) + V b t, where the function p 1 is dened in (2.3.6). Then the following function of the variables x and t is solution of the CH equation:

u(x, t) = D b ln Θ Z -d + r 2 Θ Z -d -r 2 -α 2 .
u(x, t) = β D b ln Θ Z -d + r 2 Θ Z -d -r 2 + δ = β D b ln g 1 g 2 + δ, (4.1.15)
where Z(x, t) is dened in (4.1.12), and the functions g 1 , g

y x = D e ln g 1 g 2 -α 1 -1
.

Assuming α 1 = p 1 (b, e, a), by (4.1.7) the function y x becomes

y x = g 1 g 2 p 2 . (4.1.16)
Analogously it can be checked that

y t = -y x u β - δ β -α 2 . ( 4
u x = -β D b ln(g 1 g 2 ).
u xx = β D b ln g 1 g 2 -2 p1 -2β p2 (g 1 g 2 ) 2 ; (4.1.19)
here we used the identity q2 = -p2 p

m(x, t) = δ + k + 2β p1 + 2β p2 (g 1 g 2 ) 2 . (4.1.20)
Taking the derivative of m with respect to x, and the derivative of m with respect to t, one gets respectively:

m x (x, t) = 4β p2 (g 1 g 2 ) 2 y
x D e ln(g 1 g 2 ),

2 p2 D e ln(g 1 g 2 ) y x (g 1 g 2 ) 2 u 1 - 1 β + δ β + α 2 -D b ln(g 1 g 2 ) δ + k + 2β p1 + 2 p2 (g 1 g 2 ) 2 (β -1) = 0.
In particular, the previous equality holds for β = 1, δ = -2 p1 -k and α 2 =δ, which completes the proof.

Real-valued solutions and smoothness conditions

In this part, we identify real-valued and smooth solutions of the CH equation. Assume that R g is a real hyperelliptic curve which admits real ovals with respect to an anti-holomorphic involution τ (see Chapter 3). Let us choose the homology basis satisfying (3.2.1). Recall that R g (R) denotes the set of xed points of the anti-holomorphic involution τ .

The following propositions provide reality and smoothness conditions on solutions u(x, t) (4.1.14) in the case where points a and b are stable under τ , and in the case where τ a = b. In particular, it is proved that, for xed t 0 ∈ R, the function u(x, t 0 ) is smooth with respect to the real variable x when a and b are stable under τ . In the case where τ a = b, the function u(x, t 0 ) is smooth, otherwise it has cusp-like singularities. where the vector Z is dened in (4.1.12). In fact, using the expansion (2.

α 1 = α 1 , α 2 = α 2 . ( 4
h(y, t) = Θ Z + d + r 2 + iπ diag(H) Θ Z + d -r 2 + iπ diag(H) . ( 4
h(y, t) = Θ Z -d + r 2 Θ Z -d -r 2 exp{-r, L },
where we used the quasi-periodicity property (2.2.3) of the theta function. Therefore, the function Since the theta function is entire, singularities of the solution u are situated at the zeros of its denominator. By the same argument used previously to prove the real-valuedness of u, if the curve is an M-curve and d ∈ iR g , the functions Θ(Zd ± r

h is real if L = 0, that is d ∈ iR g .
2 ) and Θ(Zd) do not vanish. In this case, the function u is smooth with respect to the real variable y. Now let us prove that u is smooth with respect to the real variable x. By (4.1.11), the function x(y) is smooth with respect to the variable y. Moreover, it can be seen from (4.1.11) and (4.1.7) that

x y (y) = p 2 g 1 g 2 = p 2 Θ(Z -d) 2 Θ Z -d + r 2 Θ Z -d -r 2 . (4.1.30)
Since the functions Θ(Zd ± r 2 ) and Θ(Zd) do not vanish, we deduce that x(y) is a strictly monotonic real function, and thus the inverse function y(x) has the same property. Therefore, the function u(x, t 0 ) = u(y(x)) is a smooth real-valued function with respect to the real variable x. for some n ∈ N \ {0} and x 0 ∈ R, which correspond to cusps.

Proof. Analogously to the case where a and b are stable under τ , let us prove that solutions u (4.1.14) are real-valued. First let us check that vector Z (4.1.12) satisfy:

Z = Z.
α 1 = α 1 , α 2 = α 2 . ( 4
h(y, t) = Θ Z -d + r 2 + p Θ Z -d -r 2 + p , (4.1.34)
where p = -iπNiπ diag(H). Let us choose the vector d ∈ C g such that

d ≡ d + p (mod 2iπZ g + BZ g ),
which is, since dd and p are purely imaginary, equivalent to d = d + p + 2iπT, for some T ∈ Z g . Here we used the action (3.2.3) of the complex conjugation on the Riemann matrix B, and the fact that B has a negative denite real part. Hence, the vector d can be written as 1.35). With this choice of vector T, the imaginary part of the vector Zd -r 2 equals -iπN. Therefore, the vector Zd -r 2 is real modulo 2iπZ g if all components of the vector N are even. To summarize, the function h(y, t) dened in (4.1.27) is a real-valued function with constant sign if the hyperelliptic curve is dividing (i.e., all ramication points are stable under τ , since the ramication point e is stable under τ ), if T = 0 and N = 2L for some L ∈ Z g , where vector N ∈ Z g is dened in (3.3.1). Analogously to the proof of Proposition 4.1.3, we conclude that x is a real-valued continuous function of the real variables y and t, and thus solutions u(x, t) (4.1.14) are real-valued functions of the real variables x and t. Now let us study smoothness conditions for xed t 0 ∈ R. Notice that function u(y) (4.1.14) is a smooth function of the real variable y since the denominator does not vanish, as we have seen before. Put z = Zd. Let us consider the function x y (y) given in (4.1.30) in both cases: N = 0 and N = 0.

d = d R + iπ 2 (N + diag(H) -2 T), (4.1
Z -d + r 2 is real if T = 0 in (4.
-If N = 0, the function x y (y) does not vanish, since in this case z ∈ R g which implies that the function Θ(z) does not vanish. Hence, analogously to the case where a and b are stable under τ , for xed t 0 ∈ R, the function u(x, t 0 ) is smooth with respect to the real variable x.

-If N = 0, the function Θ(z) vanishes when z belongs to the theta divisor. Fix x 0 , t 0 ∈ R and denote by z 0 and y 0 the corresponding values of z and y. Assume that z 0 is a zero of the theta function of order n ≥ 1. Then by (4.1.30), the function x y (y) has a zero at y 0 of order 2n. It follows that function x(y)x(y 0 ) has a zero of order 2n + 1 at y 0 , and then

y(x) -y 0 = O (x -x 0 ) 1 2n+1 . (4.1.36)
On the other hand, it can be seen from (4.1.4) that

u y (y) = p 2 Θ(z) Θ z + r 2 Θ z -r 2 [Θ(z) ψ(z) -2 D b Θ(z)] , (4.1.37)
where

ψ(z) = D b ln Θ(z + r 2 ) Θ(z -r 2 )
. Identity (4.1.37) implies that function u y (y) has a zero at y 0 of order 2n -1, namely, u(y)u(y 0 ) = O (yy 0 ) 2n .

Solutions of the Dym-type equation

This part deals with the Dym-type equation

u xxt + 2 u x u xx + uu xxx -2k u x = 0, (4.2.1)
where k ∈ C. Analogously to the Camassa-Holm equation, we rst prove identities satised by the theta function and derive from them solutions of equation (4.2.1). The study of real-valued and smooth solutions follows the line of the CH equation.

Theta-functional identities

Here we give theta-functional identities useful to construct algebro-geometric solutions of the Dym-type equation. Recall that R g denotes the hyperelliptic curve (4.0.1).

g(z) = (D e D f ln Θ(z) -q 1 ) -1 , (4.2.2)
where q 1 = q 1 (e, f ) is given in (2.3.11), and denote by r = f e ω, where ω is the vector of normalized holomorphic dierentials. Then there exist n, m ∈ Z g such that r = 1 2 (2iπn + Bm).

D e D f ln Θ(z + r ) = κ 2 g(z) + q 1 , (4.2.4
)

D e D 2 f ln Θ(z + r ) = -κ 2 g(z) 2 D e D 2 f ln Θ(z), (4.2.5)
for any z ∈ C g , where κ = q 2 exp{-1 2 Bm, m } with q i = q i (e, f

), i = 1,
D e D f ln Θ(z) -q 1 = q 2 e -1 2 Bm,m Θ(z + r ) 2 Θ(z) 2 e φ(z) , (4.2.6)
where φ(z) = z + r , m . Moreover, the change of variable z → z + r in (2.3.10) leads to

D e D f ln Θ(z + r ) -q 1 = q 2 e -1 2 Bm,m Θ(z) 2 Θ(z + r ) 2 e -φ(z) . (4.2.7)
Combining (4.2.6) and (4.2.7), we deduce that

κ -1 D e D f ln Θ(z + r ) -q 1 = κ -1 (D e D f ln Θ(z) -q 1 ) -1 = κ g(z), (4.2.8)
where κ = q 2 exp{-

x + α 1 y + α 2 t + ζ = D f ln Θ(Z -d), (4.2.9)
where α 1 = q 1 (e, f ) and the vector Z is given by Z(x, t) = V e y + V f t.

(4.2.10)

Here the scalar q 1 (e, f ) and the vector V e (resp. V f ) are dened in (2.3.11) and (2.3.3); D f denotes the directional derivative along the vector V f . Then the following function

u(x, t) = D 2 f ln Θ(Z -d) -α 2 , (4.2.11)
is solution of the Dym-type equation

u xxt + 2 u x u xx + uu xxx + 4 u x = 0. (4.2.12)
Notice that if u is solution of (4.2.12), the function

v(x, t) = - k 2 u(x, -k 2 t) (4.2.
u(x, t) = β D 2 f ln Θ(z) + δ, (4.2.15)
where z = Zd. By (4.2.9), the derivative with respect to x of the implicit function y(x, t) is given by

y x = (D e D f ln Θ(z) -α 1 ) -1 ;
assuming α 1 = q 1 (e, f ), the function y x becomes

y x = g(z), (4.2.16)
where the function g is dened in (4.2.2). Analogously, it can be checked that

y t = -g(z) u β - δ β -α 2 . ( 4
u x = β g(z) D e D 2 f ln Θ(z) = β D f ln (D e D f ln Θ(z) -α 1 ) , (4.2.18)
which by (4.2.6) leads to

u x = 2β D f ln Θ(z + r ) Θ(z) -β D f φ(z). (4.2.19)
The second derivative of u with respect to x reads

u xx = 2β g(z) D e D f ln Θ(z + r ) Θ(z) ,
where we used (4.2.16) and the fact that φ is a linear function of x. Moreover, from (4.2.8) we deduce that

D e D f ln Θ(z + r ) Θ(z) = κ 2 g(z) -g(z) -1 , (4.2.20)
which yields

u xx = 2β κ 2 g(z) 2 -1 . ( 4
u xt = -2 g(z) (u -δ -β α 2 ) D e D f ln Θ(z + r ) Θ(z) + 2β D 2 f ln Θ(z + r ) Θ(z) , (4.2.22)
which by (4.2.20) implies

u xt = -2 (u -δ -β α 2 ) κ 2 g(z) 2 -1 + 2β D 2 f ln Θ(z + r ) Θ(z) . ( 4
u xt + uu xx = 2 (κ 2 g(z) 2 -1) [u (β -1) + δ + β α 2 ] + 2β D 2 f ln Θ(z + r ) Θ(z) . (4.2.24)
Now let us choose β = 1 and α 2 =δ. Thus (4.2.24) takes the simple form:

u xt + uu xx = 2 D 2 f ln Θ(z + r ) Θ(z) . ( 4
(u xt + uu xx ) x = -2 g(z) κ 2 g(z) 2 + 1 D e D 2 f ln Θ(z). ( 4

Real-valued solutions and smoothness conditions

In this part, we are interested in real-valued and smooth algebro-geometric solutions obtained in the previous section. These solutions can be investigated analogously to the CH equation, therefore some details will be omitted. Assume that R g is a real hyperelliptic curve which admits real ovals with respect to an anti-holomorphic involution τ for some n ∈ N \ {0} and x 0 ∈ R, which correspond to cusps.

Proof. Real-valuedness of the solutions can be proved analogously to the CH equation in the case where points a and b are stable under τ (see Proposition 4.1.3). Fix t 0 ∈ R. Let us study smoothness conditions, with respect to the variable x, of the function u(x, t 0 ). Put z = Zd where Z is dened in (4.2.10). From (4.2.9) and (4.2.6), it can be seen that

x y (y) = κ Θ(z + r ) 2 Θ(z) 2 e -φ(z) , (4.2.27)
where r = f e ω = 1 2 (2iπn + Bm) for some n, m ∈ Z g . Let us consider both cases: m = 0 and m = 0.

-If m = 0, then z + r ∈ iR g and the function x y (y) does not vanish since the curve is an M-curve. By similar statements used to prove smoothness of the function u(x, t 0 ) where u is solution of the CH equation, we conclude that the function u(x, t 0 ) where u (4.2.11) is solution of the Dym-type equation (4.2.12) is smooth.

-If m = 0, the function x y (y) vanishes each time that z + r belongs to the theta divisor. Fix x 0 , t 0 ∈ R and denote by z 0 , y 0 the corresponding value of z, y respectively. Assume that z 0 + r is a zero of the theta function of order n. Then by (4.2.27), y 0 is a zero of the function x y (y) of order 2n, which leads to y(x) -

y 0 = O (x -x 0 ) 1 2n+1 . (4.2.28)
The derivative of function u (4.2.11) with respect to the variable y can be checked applying the operator D e to identity (4.2.6), thus one gets

u y (y) = -κ Θ(z + r ) Θ(z) 2 e -φ(z) Θ(z + r ) ψ(z) -2 D f Θ(z + r ) , (4.2
k c (τ p) = -k c (p),
for any p ∈ R g lying in a neighbourhood of c ∈ {e, f }, one gets real-valued solutions for the choices d ∈ R g and α 2 ∈ R. In this case, solutions (4.2.11) are smooth if r = r , that is n = 0 in (4.2.3), otherwise they admit innite number of cusps.

Chapter 5 Solitonic limit

It is well known that solitonic solutions of integrable equations arise from their algebro-geometric solutions when the associated spectral curve degenerates to a curve of genus zero. In this chapter, we study the solitonic limit of the theta-functional solutions to the Camassa-Holm and Dym-type equations investigated in Chapter 4. The obtained solitary waves maintain their shape and speed after interacting nonlinearly with other solitary waves. This method to nd solitonic solutions is eective for many integrable systems, such as the focusing and defocusing nonlinear Schrödinger equations and the sine-Gordon equation, see, for instance, [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF].

Here we show that solutions obtained after degeneration of the hyperelliptic spectral curve depend on the choice of the ramication points in Proposition 4.1.2 and Proposition 4.2.2. Different choices lead to dierent solutions such as solitons, cuspons or peakons, presented here in a simple parametric form.

Degeneracy of hyperelliptic Riemann surfaces

Degeneration of the hyperelliptic curve (4.0.1) of genus g into a genus zero algebraic curve can be done by pinching all A-cycles of the associated Riemann surface (see [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF] for more details). It consists in colliding 2g branch points of the curve, namely, we consider the limit λ 2j+1 , λ 2j+2 -→ β j , j = 1, . . . , g,

where β i = β j for i = j. In this limit, putting λ 1 =α and λ 2 = α with α > 0, the curve (4.0.1) can be written as

µ 2 = (λ 2 -α 2 ) g j=1 (λ -β j ) 2 , (5.1.2)
where β j = α. Desingularization of the curve (5.1.2) yields a Riemann surface of genus zero given by the polynomial equation

µ 2 = λ 2 -α 2 , (5.1.3)
where g pairs of points on the surface each cover a β j in the base.

Degeneracy of hyperelliptic normalized Abelian dierentials

Quantities entering theta-functional solutions of the CH and Dym-type equations, obtained in Degenerate holomorphic dierentials. It is well known that a basis of normalized holomorphic dierentials (ω 1 , . . . , ω g ) on the hyperelliptic curve (4.0.1) is given by:

ω k (λ) = ϕ k µ dλ, (5.1.4)
for k = 1, . . . , g, where the function ϕ k reads

ϕ k (λ) = c k1 λ g-1 + c k2 λ g-2 + . . . + c kg ,
with c kj ∈ C; moreover, ϕ k is chosen such that (2.1.9) holds. In the limit (5.1.1), the normalized holomorphic dierential ω k (5.1.4) becomes

ω 0 k (λ) = ϕ 0 k √ λ 2 -α 2 g j=1 (λ -β j ) dλ, for k = 1, . . . , g, where ϕ 0 k (λ) = c 0 k1 λ g-1 + c 0 k2 λ g-2 + . . . + c 0 kg .
The functions ϕ 0 k are determined by the normalization conditions:

2iπδ j,k = A j ω 0 k = -2iπ Res(ω 0 k , β j ),
for j, k = 1, . . . , g, which yields

ϕ 0 k (λ) = c 0 k1 i =k (λ -β i ), c 0 k1 = -β 2 k -α 2 , k = 1, . . . , g.
Therefore, in the limit considered here, the normalized holomorphic dierential ω k (5.1.4) becomes a meromorphic dierential ω 0 k on (5.1.3) having simple poles with residue 1 and -1 at the points over β k :

ω 0 k (λ) = c 0 k1 √ λ 2 -α 2 (λ -β k ) dλ.
(

p → ∞ ± ⇐⇒ µ λ → ± 1 when λ → ∞ , from (5.1.6) one has Ω 0 b-a (λ) = -k ∞ (λ) λ µ - dk ∞ k ∞ (λ) 2 , which leads to Ω 0 b-a (λ) ≈ ± 1 k ∞ (λ) dk ∞ (λ), λ ≈ ∞.
Now let us consider the limit form of the normalized meromorphic dierential Ω a-e in both cases: e = (λ 2 , 0) and e = (λ i 0 , 0) with i 0 = 1, 2. In the rst case, it can be checked that the dierential Ω a-e becomes

Ω 0 a-e = 1 µ - 1 λ -α dλ 2 , (5.1.7)
and in the second case,

Ω 0 a-e = 1 µ - µ + µ i 0 λ -β i 0 dλ 2 ,
(5.1.8)

where the limit point e = (β i 0 , µ i 0 ) lies on the curve (5.1.3). The dierential Ω e-f with e = (λ 2 , 0) and f = (λ 1 , 0) becomes in the limit (5.1.1):

Ω 0 e-f = 1 λ -α - 1 λ + α dλ 2 .
(5.1.9)

Degenerate bidierential. Following [START_REF] Korotkin | Riemann-Hilbert problem for Hurwitz Frobenius manifolds: regular singularities[END_REF], for any points p = (λ p , µ p ), q = (λ q , µ q ) lying on the hyperelliptic curve (5.1.3), the bidierential Ω(p, q) is given by: Ω(p, q) = dh(p) dh(q) (h(p)h(q)) 2 , (5. 1.10) where h is the uniformization map (i.e., a biholomorphic map) between the Riemann surface associated to the curve (5.1.3) and the Riemann sphere, which is dened by

h(p) = - 1 α (λ p + µ p ).
If one of the arguments of Ω(p, q) coincides with the ramication point e = (α, 0), says q = e, one gets

Ω(p, e) = Ω(p, q) dk e q=e = α 2 dλ p (λ p -α) 3/2 (λ p + α) 1/2 , (5.1.11)
where we have chosen the local parameter k e near the ramication point e to be k e (q) = λ qα for any point q lying in a neighbourhood of e.

Degeneration of hyperelliptic theta functions

As explained in [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF] in the framework of the NLS equation, the limit (5.1.1) allows to write down the multi-dimensional theta function as a nite sum of exponential terms. The solitonic solutions of the studied equations thus arise as limiting cases of algebro-geometric solutions.

Let us rst describe the behavior of the matrix B of B-periods of the normalized holomorphic dierentials (5.1.4) in the limit (5.1.1). From (5.1.5) one has:

β 0 α ω k -→ c 0 k1 β 0 α dλ √ λ 2 -α 2 (λ -β k ) = ln γ k + γ 0 γ k -γ 0 , (5.1.12)
where

γ k = β k -α β k + α = β 2 k -α 2 β k + α , k =
0, . . . , g.

(5.1.13)

Now assume that β k are ordered according to Re(β j ) > Re(β k ), j > k.

Therefore, the limit values B 0 jk of the non-diagonal elements of the Riemann matrix B become

B 0 jk = 2 α β j ω 0 k = 2 ln γ k -γ j γ k + γ j , j > k, ( 5
Θ g (z) = m∈{0,1} g exp    1≤j<k≤g B 0 jk m j m k + g j=1 m j z j    , (5.1.17)
for any z ∈ C g . Moreover, as we shall see in Proposition 7.1.1 in a more general situation, the degenerate theta function Θ g admits the following determinantal representation:

Θ g (z) = det(T), (5.1.18)
where T is a g × g matrix with elements

(T) jk = δ j,k + 2 γ j γ j + γ k e 1 2 (z j +z k ) .

(5.1.19)

Soliton-like solutions

In this section we prove that the choice e = (λ 2 , 0) in ( 4

V b,j = (β j + α) γ j , V e,j = 2 α 1 γ j , j = 1, . . . , g, (5.2.1)
where γ j is dened in (5.1.13). From (5.1.12) we deduce that

r j = b a ω 0 j = 2 ln γ j + 1 γ j -1 , j = 1, . . . , g. (5.2.2)
Moreover, according to (4.1.13) and (4.1.25), from (5.1.6) and (5.1.7) we obtain

α 1 = 2 α , α 2 = α + k. ( 5

.2.3)

To get real solutions, assume that |β j | > α for j = 1, . . . , g. Thus it can be seen that if β j 0 > α for some j 0 ∈ {1, . . . , g}, then one has r j 0 = r j 0 + 4iπ.

(5.2.4)

In this case, as we saw in Proposition 4.1.4, one has to choose the vector d ∈ R g appearing in (4.1.14) such that

d j 0 = d j 0 + 2iπ.
(5.2.5)

Hence the degenerate solutions given in the following proposition describe interactions between soliton and cusps. It can be seen that the number of cusps which appear in this interaction equals the number of parameters β j such that β j > α. We have proved:

Proposition 5.2.1. Take α > 0 and β j ∈ R all distinct, such that |β j | > α. Let d ∈ R g such that d j satises (5.2.5) if β j > α, and choose ζ, k ∈ R. Assume that the variables x, y, t are related by the condition

x + 2 α y + (α + k) t + ζ = ln Θ g (z + r 2 ) Θ g (z -r 2 ) , (5.2.6)
where z(x, t) = V e y + V b td. Here Θ g denotes the degenerated theta function (5.1.17). Moreover, the components of the vectors V b (resp. V e ) and r are given by (5.2.1) and (5.2.2). Hence the following function

u(x, t) = D b ln Θ g (z + r 2 ) Θ g (z -r 2 ) -α -k, (5.2.7)
is a real-valued solution of the CH equation involving elementary functions only. It describes interaction between gs solitons and s cusps, where s is the number of parameters β j such that β j > α.

Soliton-like solutions of the Dym-type equation

Analogously to the CH equation, we construct soliton-like solutions of the Dym-type equation

k e = √ λ -α, k f = i √ λ + α.
(5.2.8)

From (5.1.5) one gets V e,j = 2 α 1 γ j , V f,j = - 2 α γ j , j = 1, . . . , g, (5.2.9)
where γ j is dened in (5.1.13). Moreover, it is clear that x -

r j = - α -α ω 0 j = ± iπ, j = 1, . . . ,
1 2 α y + α 2 t + ζ = D f ln Θ g (z),
(5.2.12)

where z(x, t) = V e y + V f td and vectors V e , V f are given by (5.2.9). Hence the following function

u(x, t) = D 2 f ln Θ g (z) -α 2 , (5.2.13)
is a real-valued solution of the Dym-type equation (4.2.1) for k = -2, and describes interaction between g cusps. Solutions for arbitrary k ∈ R \ {0} follow from (4.2.13).

Peakon solutions

Camassa and Holm [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF] described a class of peaked solutions for the shallow water equation (4.1.2). These solutions, called peakons, behave like solitons and dier from the cuspons study in Section 5.2 by their discontinuous derivatives at the peak points (a cuspon is a kind of peaked soliton whose left and right derivatives equal innity at the cusp).

Here we explain how the choice e = (λ i 0 , 0) in ( 4 In what follows we assume that β k ∈ R.

Peakon solutions of the CH equation

Let us rst consider the solutions for xed t 0 ∈ R. Put a = ∞ -, b = ∞ + , choose e = (λ 2g+2 , 0) and consider the limit (5.1.1). In this limit e = (β g , µ g ) on the desingularized curve (5.1.3).

Hence by (5.1.5), one has the following asymptotic estimate for the coordinate V e,g :

V e,g -→ ∞.

(5.3.1)

This allows to consider the following change of variable

y -→ γ V e,g y (5.3.2)
in (4.1.11) and (4.1.14), for some γ ∈ R \ {0}. Analogously to the soliton-like solutions investigated in Section 5.2.1, in the limit considered here one gets, for j = 1, . . . , g:

V b,j = (β j + α) γ j ,

(5.3.3)

r j = 2 ln γ j + 1 γ j -1 , (5.3.4)
α 2 = α + k.

(5.3.5)

Therefore, taking into account the change of variable (5.3.2), for xed t 0 ∈ R, identity (4. 1.11) becomes in the limit:

x(y, t 0 ) = ln

Θ g (z + r 2 ) Θ g (z -r 2 ) -α 2 t 0 + d 0 , (5.3.6)
where

z g = γ y + V b,g t 0 + d g , z j = V b,j t 0 + d j , j = 1, . . . , g -1, (5.3.7)
for some d k ∈ R, k = 0, . . . , g. Moreover, it can be seen that the g-dimensional theta function Θ g dened in (5.1.17) admits a decomposition with respect to the (g -1)-dimensional theta function Θ g-1 , given by: Θ g (z) = e zg Θ g-1 (z + Bg ) + Θ g-1 (z), (5.3.8) where z = (z 1 , . . . , z g-1 ) and Bg = (B 1 g , . . . , B g-1 g ). Hence (5.3.6) can be written as

x(y, z) = ln e zg+ rg 2 Θ g-1 (z + r 2 + Bg ) + Θ g-1 (z + r 2 ) e zg-rg 2 Θ g-1 (z -r 2 + Bg ) + Θ g-1 (z -r 2 ) -α 2 t 0 + d 0 , (5.3.9)
and the solutions obtained in (4.1.14) take the following form in the limit considered here:

u(y(x), z) = D b ln e zg+ rg 2 Θ g-1 (z + r 2 + Bg ) + Θ g-1 (z + r 2 ) e zg-rg 2 Θ g-1 (z -r 2 + Bg ) + Θ g-1 (z -r 2 ) -α 2 .
(5.3.10)

Now assume that β j <α, j = 1, . . . , g, (5.3.11) in such a way that the function x y (y, z) does not vanish, as we saw for soliton-like solutions. This yields strict monotonicity, with respect to the real variable y, of the function x(y, z) and its inverse. Now by (5.3.9), in the limit y → ±∞ it can be seen that the function x y (y, z) tends to zero, whereas functions x(y, z) and u(y, z) have nite values. This means that the set x(R, z) is an interval and that the associated function u(x, z) is dened only on this interval, where it takes nite values. Hence, expression (5.3.10) together with (5.3.9) describes a nite piece of a solution to the CH equation for xed t 0 ∈ R, where u(x, z) is dened on the interval x(R, z). We denote this solution by

U 0 (x, t 0 ) = {u(x, z), x ∈ x(R, z)} .
(5.3.12)

Now let us construct globally dened solutions for xed t 0 ∈ R. To this end, note that

x(+∞, z) = x(-∞, z + Bg ) + r g , (5.3.13)
and u(+∞, z) = u(-∞, z + Bg ).

(

This means that the function U 1 dened by

U 1 (x, t 0 ) = u(x, z + Bg ), x ∈ x(R, z + Bg ) + r g (5.3.15)
is also a nite piece of solutions for xed t 0 ∈ R, and takes the same value as U 0 at their common endpoint. Moreover, it can be seen that the derivative of both pieces U 0 and U 1 takes nite but dierent values at this common point. Therefore, gluing these two solutions one gets an extended solution U 0 (x, t 0 ) ∪ U 1 (x, t 0 ) having a peak at this point. This construction allows to dene a global solution U for all values of x ∈ R and for xed t 0 ∈ R, namely, the function

U (x, t 0 ) = ∪ n∈Z u(x, z + n Bg ), x ∈ x(R, z + n Bg ) + n r g (5.3.16)
is dened on R and consists of an innite number of pieces U n (x, t 0 ), n ∈ Z, glued at peak points.

Finally we have proved:

Proposition 5.3.1. For xed t 0 ∈ R, peakon solutions of the CH equation consist of an innite number of pieces U n (x, t 0 ), n ∈ Z, glued at peak points, given in the following parametric form with y ∈ R:

U n (y) = D b ln Θ g (z g + rg 2 , z + r 2 + n Bg ) Θ g (z g - rg 2 , z -r 2 + n Bg ) -α 2 ,
(5.3.17)

x(y) = ln Θ g (z g + rg 2 , z + r 2 + n Bg ) Θ g (z g - rg 2 , z -r 2 + n Bg ) -α 2 t 0 + d 0 + n r g , (5.3.18)
where z g = γ y + V b,g t 0 + d g and z = Vb t 0 + d for some d ∈ R g-1 , and γ, d 0 , d g ∈ R.

Finally, let us consider t 0 as a variable parameter t. In this case, using (5.3.9) and (5.3.13), for any n ∈ Z we observe that the piece U n (x, t) is bounded by the curves C n and C n+1 dened by

C n = {x = c n (t), t ∈ R} , (5.3.19) where c n (t) = ln Θ g-1 (z + r 2 + n Bg ) Θ g-1 (z -r 2 + n Bg ) -α 2 t + d 0 + n r g .
(5.3.20)

Moreover, it can be seen that U n-1 (x, t) and U n (x, t) are glued along the curve C n , with

U n (c n (t), t) = D b ln Θ g-1 (z + r 2 + n Bg ) Θ g-1 (z -r 2 + n Bg ) -α 2 .
(5.3.21)

Since U n-1 (x, t) and U n (x, t) have dierent partial derivatives in x and t along this curve, we conclude that a global piecewise smooth solution U (x, t) has peak curves along C n , n ∈ Z.

Peakon solutions of the Dym-type equation

Following the line of Section 5.3.1, we rst construct globally dened peakon solutions of the Dym-type equation for xed t 0 ∈ R. Choose e = (λ 2g+2 , 0) and f = (λ 2 , 0), and consider the limit (5.1.1). In this limit, the point e corresponds to the point (β g , µ g ) on the curve (5.1.3), and f corresponds to the ramication point (α, 0). Moreover, we have seen that the coordinate V e,g has the behavior (5.3.1). Therefore, we consider the change of variable (5.3.2) in (4.2.9) and (4.2.11), with γ ∈ R \ {0}. Now assume that |β j | > α, j = 1, . . . , g.

(5.3.22)

Therefore, for xed t 0 ∈ R, taking into account the change of variable (5.3.2) and identity (5.3.8), the function x(y) dened in (4.2.9) takes the following form:

x(y, z) = V f,g e zg Θ g-1 (z + Bg ) e zg Θ g-1 (z + Bg ) + Θ g-1 (z) + g-1 k=1 V f,k e zg ∂ z k Θ g-1 (z + Bg ) + ∂ z k Θ g-1 (z) e zg Θ g-1 (z + Bg ) + Θ g-1 (z) -α 2 t 0 + d 0 , (5.3.23)
where

z g = γ y + V f,g t 0 + d g , z j = V f,j t 0 + d j , j = 1, . . . , g -1, (5.3.24)
for some d k ∈ R, k = 0, . . . , g. Notice that

x(+∞, z) = x(-∞, z + Bg ) + V f,g ,

K 1 (a, b) = 1 2 D a Θ[δ](0) D a Θ[δ](0) + D a ln Θ[δ] b a , ( 6
D c -D b ln Θ(z + b c ) Θ(z) + D 2 b ln Θ(z + b c ) Θ(z) + D b ln Θ(z + b c ) Θ(z) + K 1 (b, c) 2 + 2 D 2 b ln Θ(z) = 0, (6.1.4)
where the scalar K 1 (b, c) is dened in (6.1.2).

Proof of Lemma 6.1.1. Let us introduce the notation Θ ab = Θ(z + b a ω) and Θ = Θ(z). Dierentiating (2.3.1) twice with respect to the local parameter k d (p), where p lies in a neighbourhood of d, and taking the limit d → b, we obtain

D b ln Θ + D 2 b ln Θ + (D b ln Θ) 2 + p 3 p 2 D b ln Θ ca - p 3 p 2 D b ln Θ (6.1.5) = p 1 p 3 p 2 -2 D b ln Θ ca D b ln Θ cb + 2 D b ln Θ D b ln Θ cb + 2 p 1 D b ln Θ cb -p 4 -2 p 1 D b ln Θ ca + D b ln Θ ca + D 2 b ln Θ ca + (D b ln Θ ca ) 2 ,
where we took into account the relation

∂ 2 k d Θ(z + d b ) d=b = D b Θ(z) + D 2 b Θ(z).
The quantities p j = p j (a, b, c) for j = 1, 2 are dened in (2.3.6) and (2.3.7), whereas quantities p j = p j (a, b, c) for j = 3, 4 are given by

p 3 (a, b, c) = - E(a, c) E(a, b) ∂ 2 kx E(b, x) E(c, x) x=b , p 4 (a, b, c) = - E(c, b) E(a, b) ∂ 2 kx E(a, x) E(c, x) x=b . (6.1.6)
Dierentiating (6.1.5) with respect to the local parameter k a (p), where p lies in a neighbourhood of a, and taking the limit a → c, we get

D c D b ln Θ + D c D 2 b ln Θ -2 D c D b ln Θ D b ln Θ cb Θ + 2 q 1 D b ln Θ cb Θ - p 3 p 2 D c D b ln Θ + K = 0, (6.1.7)
where the scalar K depends on the points b and c, but not on the vector z ∈ C g (knowledge of the exact value of K is not needed for our purpose). Here the scalars q 1 , p

D c D b ln Θ cb -D c D 2 b ln Θ cb -2 D c D b ln Θ cb D b ln Θ cb Θ +2 q 1 D b ln Θ cb Θ - p 3 p 2 D c D b ln Θ cb +K = 0.
-D b ln Θ cb Θ + D 2 b ln Θ cb Θ + D b ln Θ cb Θ + 1 2 p 3 p 2 2 - 1 4
p 3 p 2 2 - p 1 p 3 p 2 + 2 D 2 b ln Θ = -D b ln Θ ab Θ + D 2 b ln Θ ab Θ + D b ln Θ ab Θ + 1 2
p 3 p 2 + 2 p 1 2 - 1 4
p 3 p 2 + 2 p 1 2 -p 4 + 2 D 2 b ln Θ.
By Lemma 6.1.1, the directional derivative of the left hand side of the previous equality along the vector V c equals zero. Hence for any distinct points a, b, c ∈ R g , we get

D c -D b ln Θ ab Θ + D 2 b ln Θ ab Θ + D b ln Θ ab Θ + 1 2
p 3 p 2 + 2 p 1 2 + 2 D 2 b ln Θ = 0. (6.1.9)
Moreover, from (2.3.6), (2.3.7) and (6.1.6), it can be seen that the expression 1 2 ( p 3 p 2 +2p 1 ) does not depend on the point c and equals K 1 (b, a) given by (6.1.2). Now let us introduce the following function of the variable z ∈ C g : for any z ∈ C g . Interchanging a and b, and changing the variable z by -z in (6.1.10) we get (6.1.1). The expression (6.1.3) for the scalar K 2 (a, b) follows from (6.1.10) putting z = 0.

f (b,a) (z) = -D b ln Θ ab Θ + D 2 b ln Θ ab Θ + D b ln Θ ab Θ + K 1 (b, a) 2 + 2 D 2 b ln Θ.

6.2

Computation of the argument of the fundamental scalar q 2 (a, b)

This section is devoted to the computation of the argument arg{q 2 (a, b)}, where q 2 (a, b) is dened by (2.3.12). Let R g be a real compact Riemann surface of genus g with an anti-holomorphic involution τ . For two distinct points a, b ∈ R g connected by a contour which does not intersect the basis of cycles, let us denote by (A, B, ) the generators of the relative homology group H 1 (R g , {a, b}) and by (A, B, S b ) the generators of the homology group H 1 (R g \ {a, b}) (see Section 3.3). In the following we assume that the basis of cycles (A, B) satises (3.2.1). We compute the argument of q 2 (a, b) both in the case τ a = b, as well as in the case τ a = a, τ b = b.

Integral representation for q 2 (a, b)

The computation of the scalar arg{q 2 (a, b)} is based on the following integral representation for q 2 (a, b). Recall that Ω b-a denotes the normalized meromorphic dierential of the third kind which has residue 1 at b and residue -1 at a. Proposition 6.2.1. Let a, b be distinct points on a compact Riemann surface R g of genus g.

Denote by k a and k b local parameters in a neighbourhood of a and b respectively. Then the scalar q 2 (a, b) dened in (2.3.12) admits the following integral representation

q 2 (a, b) = -lim b→b ã→a k a (ã) k b ( b) -1 exp b ã Ω b-a (p) , (6.2.1)
where the integration contour between ã and b, which in the sequel is denoted by ˜ , does not cross any cycle from the canonical homology basis.

Proof. Notice that the scalar q 2 (a, b) does not depend on the choice of the contour ˜ , assuming that ˜ lies in the fundamental polygon of the Riemann surface.

Denote by k x a local parameter in a neighbourhood of a point x ∈ R g . To prove (6.2.1),

recall that (see (2.2.13)) b ã Ω b-a (p) = ln Θ[δ]( b b ) Θ[δ]( b a ) + ln Θ[δ]( ã a ) Θ[δ]( ã b ) . (6.2.2)
Since δ is an odd non-singular characteristic, the expression

Θ[δ]( p b ) Θ[δ]( p a )
has a simple zero at b and a simple pole at a. Therefore, if we consider ã in a neighbourhood of a, and b in a neighbourhood of b, we get (with α 1 , β 1 = 0)

Θ[δ]( b b ) Θ[δ]( b a ) = α 1 k b ( b) + o(k b ( b)), (6.2.3) Θ[δ]( ã a ) Θ[δ]( ã b ) = β 1 k a (ã) + o(k a (ã)). ( 6
k a (ã) k b ( b) -1 exp b ã Ω b-a (p) = α 1 β 1 . (6.2.5)
Moreover, using the denition (2.3.12) of q 2 (a, b), it follows from (6.2.3) and (6.2.4) that α 1 β 1 = q 2 (a, b), which by (6.2.5) completes the proof.

Argument of q 2 (a, b) when τ a = b

Here we compute the argument of the fundamental scalar q 2 (a, b) in the case where τ a = b. Proposition 6.2.2. Let (A, B, ) be the generators of the relative homology group H 1 (R g , {a, b}).

Assume that τ a = b, with local parameters satisfying the relation k b (τ p) = k a (p) for any point p in a neighbourhood of a. Then the scalar q 2 (a, b) is real, and its sign is given by: 1. if intersects the set of real ovals of R g only once, and if this intersection is transversal, then q 2 (a, b) < 0,

if

does not cross any real oval, then q 2 (a, b) > 0. By (3.3.8), the last term in the right hand side of (6.2.7) is equal to

1 2i τ ˜ + ˜ Ω b-a (p). The closed contour τ ˜ + ˜ admits the following decomposition in H 1 (R g \ {a, b}), τ ˜ + ˜ = N t A + α S b , (6.2.8)
where α = (τ ˜ + ˜ ) • and N ∈ Z g is dened in (3.3.1). Since the dierential Ω b-a has vanishing A-periods, by (6.2.8) we obtain

τ ˜ + ˜ Ω b-a (p) = 2iπα, (6.2.9)
which leads to (6.2.6). Therefore, the sign of q 2 (a, b) depends on the parity of the intersection index α = (τ ˜ + ˜ ) • .

Let us now consider cases 1. and 2. separately. Case 1. Assume that each of the contours and ˜ intersects the set of real ovals of R g transversally only once, and, moreover, this intersection point is the same for and ˜ ; we denote it by p 0 ∈ R g (R). Then the closed contour τ ˜ + ˜ can be decomposed into a sum of two closed contours c ˜ 1 and c ˜ 2 , having the common point p 0 , and such that τ sends the set of points {c ˜ 1 } into the set of points {c ˜ 2 }. Therefore, if the orientation of c ˜ 1 and c ˜ 2 is inherited from the orientation of τ ˜ + ˜ , we have τ c ˜ 1 = c ˜ 2 as elements of H 1 (R g \ {a, b}). Then,

c ˜ 1 • = -τ c ˜ 1 • τ = -c ˜ 2 • (-+ N t A) = c ˜ 2 • ,
where we used the action (3.3.1) of τ on the contour , and the fact that the intersection index between c ˜ 2 and A-cycles is zero by (6.2.8). Hence the intersection index α satises

α = (τ ˜ + ˜ ) • = (c ˜ 1 + c ˜ 2 ) • = 2,
which by (6.2.6) leads to q 2 (a, b) < 0.

Case 2. Let V be a ring neighbourhood of the path τ ˜ + ˜ , bounded by two closed paths denoted by ∂V 1 and ∂V 2 , in such way that the path lies in V and τ {∂V 1 } = {∂V 2 }. We assume that V is chosen such that no point of V is invariant under τ . Then V can be decomposed into two connected components denoted by V 1 and V 2 as follows: V 1 is bounded by ∂V 1 and τ ˜ + ˜ , and V 2 is bounded by ∂V 2 and τ ˜ + ˜ . Then τ V 1 = V 2 since the set of points {τ ˜ + ˜ } is invariant under τ . In particular, if a ∈ V 1 then b ∈ V 2 . Thus the intersection index α = (τ ˜ + ˜ ) • is odd, which leads to q 2 (a, b) > 0. Then the argument of the scalar q 2 (a, b) is given by

Argument of

arg{q 2 (a, b)} = arg{k a (ã) k b ( b)} + π(1 + α) + π 2 HM, M - 1 2i ( BM, M + 2 r, M ) , (6.2.10)
where α equals the intersection index (τ ˜ -˜ ) • . Here r = ω, and M ∈ Z g is dened in (3.3.9).

Proof. From the integral representation (6.2.1) of q 2 (a, b) we get

arg{q 2 (a, b)} = π + arg{k a (ã) k b ( b)} + Im ˜ Ω b-a (p) . ( 6
2i Im ˜ Ω b-a (p) = ˜ Ω b-a - τ ˜ Ω b-a - k M k τ ˜ ω k .
The closed contour τ ˜ -˜ ∈ H 1 (R g ) satises τ (τ ˜ -˜ ) = -(τ ˜ -˜ ); thus by (3.3.12) it has the following decomposition in H 1 (R g \ {a, b})

τ ˜ -˜ = N t A + M t B + α S b , (6.2.12)
for some α ∈ Z, where N, M ∈ Z g are dened in (3.3.9). Hence we get

2i Im ˜ Ω b-a (p) = - M t B Ω b-a + 2iπα - k M k ˜ ω k + j (B jk -iπ H jk ) M j , (6.2.13)
where we used the fact that the normalized dierential Ω b-a has vanishing A-periods, and that the integral over the small contour S b of the holomorphic dierentials is zero. Since by denition the contour does not cross any cycle of the absolute homology basis, from (2.1.11) one has where r = ˜ ω. Considering the limit when ã tends to a and b tends to b, we obtain (6.2.10).

M t B Ω b-a = M, r .

Algebro-geometric solutions of the multi-component NLS equation

In this section, we derive from the degenerated Fay identity (6.1.1) new theta-functional solutions of the multi-component nonlinear Schrödinger equation (n-NLS s )

i ∂ψ j ∂t + ∂ 2 ψ j ∂x 2 + 2 n k=1 s k |ψ k | 2 ψ j = 0, j = 1, . . . , n, (6.3.1)
where s = (s 1 , . . . , s n ), s k = ±1. Here ψ j (x, t), for j = 1, . . . , n, are complex valued functions of the real variables x and t.

Solutions of the complexied n-NLS equation

To construct algebro-geometric solutions of (6.3.1), let us rst introduce the complexied version of the n-NLS s equation, which is a system of 2n equations of 2n dependent variables ψ j , ψ * j for j = 1, . . . , n:

i ∂ψ j ∂t + ∂ 2 ψ j ∂x 2 + 2 n k=1 ψ k ψ * k ψ j = 0, -i ∂ψ * j ∂t + ∂ 2 ψ * j ∂x 2 + 2 n k=1 ψ k ψ * k ψ * j = 0, j = 1, . . . , n, (6.3.2)
where ψ j (x, t) and ψ * j (x, t) are complex valued functions of the real variables x and t. This system reduces to the n-NLS s equation (6.3.1) under the reality conditions ψ * j = s j ψ j , j = 1, . . . , n.

(6.3.3)

Theta-functional solutions of the system (6.3.2) are given by: Theorem 6.3.1. Let R g be a compact Riemann surface of genus g > 0 and let f be a meromorphic function of degree n+1 on R g . Let z a ∈ C be a non critical value of f , and consider the ber f -1 (z a ) = {a 1 , . . . , a n+1 } over z a . Choose the local parameters k a j near a j as k a j (p) = f (p)z a , for any point p ∈ R g in a neighbourhood of a j . Let d ∈ C g and A j = 0 be arbitrary constants.

Then the following functions ψ j , ψ * j , j = 1, . . . , n, give solutions of the system (6.3.2)

ψ j (x, t) = A j Θ(Z -d + r j ) Θ(Z -d) exp {-i (E j x -F j t)} , ψ * j (x, t) = q 2 (a n+1 , a j ) A j Θ(Z -d -r j ) Θ(Z -d) exp {i (E j x -F j t)} . (6.3.4)
Here r j = a j a n+1 ω, where ω is the vector of normalized holomorphic dierentials, and

Z = i V a n+1 x + i W a n+1 t. (6.3.5)
The vectors V a n+1 and W a n+1 are dened in (2.3.3), and the scalars E j , F j are given by

E j = K 1 (a n+1 , a j ), F j = K 2 (a n+1 , a j ) -2 n k=1
q 1 (a n+1 , a k ).

(6.3.6)

The scalars q 1 (a n+1 , a j ), q 2 (a n+1 , a j ), K 1 (a n+1 , a j ) and K 2 (a n+1 , a j ) are dened in (2.3.11),

(2.3.12), (6.1.2) and (6.1.3) respectively.

Proof. We start with the following technical lemma: Lemma 6.3.1. Let R g be a compact Riemann surface of genus g > 0 and let a 1 , . . . , a n+1 be distinct points on R g . Then the vectors V a j for j = 1, . . . , n + 1, are linearly dependent if and only if there exists a meromorphic function f of degree n + 1 on R g , and z a ∈ CP 1 such that f -1 (z a ) = {a 1 , . . . , a n+1 }.

Proof of Lemma 6.3.1. Assume that there exist γ 1 , . . . , γ n+1 ∈ C * such that n+1 k=1 γ k V a k = 0. By (2.1.10), the left hand side of this equality equals the vector of B-periods of the normalized dierential of the second kind Ω = n+1 k=1 γ k Ω

a k . Hence all periods of the dierential Ω vanish, which implies that the Abelian integral p -→ p p 0 Ω is a meromorphic function of degree n + 1 on R g having simple poles at a 1 , . . . , a n+1 .

Conversely, assume that there exists a meromorphic function f of degree n + 1 on R g , and z a ∈ C such that f -1 (z a ) = {a 1 , . . . , a n+1 } (the case z a = ∞ can be treated in the same way). The function h(p) = (f (p)-z a ) -1 is a meromorphic function of degree n+1 on R g having simple poles at a 1 , . . . , a n+1 only. Therefore, all periods of the dierential dh vanish. Let p 0 ∈ R g satisfy h(p 0 ) = 0. Using Riemann's bilinear identity (2.1.8) one gets Rg

ω j ∧ dh = ∂Fg ω j p p 0 dh = ∂Fg ω j h(p) = 0,
where F g denotes the simply connected domain with the boundary ∂F g = g j=1 (A j + A -1 j + B j + B -1 j ). By Cauchy's theorem, taking local parameters k a j near a j such that k a j (p) = f (p)z a for any point p ∈ R g in a neighbourhood of a j , we deduce that n+1 k=1 V a k = 0.

To prove Theorem 6.3.1, substitute the functions (6.3.4) in the rst equation of (6.3.2) to get

D a n+1 ln Θ(z + r 1 ) Θ(z) + D 2 a n+1 ln Θ(z + r 1 ) Θ(z) + D a n+1 ln Θ(z + r 1 ) Θ(z) -E 1 2 + F 1 -2 n k=1 q 2 (a n+1 , a k ) Θ(z + r k ) Θ(z -r k ) Θ(z) 2 = 0. (6.3.7)
It can be shown that equation (6.3.7) holds as follows: in (6.1.1), let us choose a = a n+1 and b = a 1 to obtain

D a n+1 ln Θ(z + r 1 ) Θ(z) + D 2 a n+1 ln Θ(z + r 1 ) Θ(z) + D a n+1 ln Θ(z + r 1 ) Θ(z) -K 1 2 (6.3.8) + K 2 + 2 D 2 a n+1 ln Θ(z) = 0, (6.3.9)
for any z ∈ C g , and in particular for z = Zd; here we used the notation K i = K i (a n+1 , a 1 ) for i = 1, 2. By Lemma 6.3.1 the sum n+1 k=1 V a k equals zero, which implies n+1 k=1 D a k = 0.

Replacing -n k=1 D a k with D a n+1 in (6.3.9) and using (2.3.10) we obtain (6.3.7), where

E 1 = K 1 , F 1 = K 2 -2 n k=1
q 1 (a n+1 , a k ).

In the same way, it can be proved that the functions in (6.3.4) satisfy the 2n -1 other equations of the system (6.3.2).

The solutions (6.3.4) of the complexied system (6.3.2) depend on the Riemann surface R g , the meromorphic function f of degree n + 1, a non critical value z a ∈ C of f , and arbitrary constants d ∈ C g , A j = 0. A transformation of the local parameters given by

k a j -→ β k a j + µ k 2 a j + O k 3 a j , (6.3.10)
where β, µ are arbitrary complex numbers (β = 0), leads to a dierent family of solutions of the complexied system (6.3.2). These new solutions are obtained via the following transformations:

ψ j (x, t) -→ ψ j β x + 2βλ t, β 2 t exp -i λ x + λ 2 t , ψ * j (x, t) -→ β 2 ψ * j β x + 2βλ t, β 2 t exp i λ x + λ 2 t , (6.3.11)
where λ = µ β -1 .

Reality conditions

Algebro-geometric solutions of the n-NLS s equation (6.3.1) are constructed from solutions (6.3.4) of the complexied system by imposing the reality conditions ψ * j = s j ψ j . Let R g be a real compact Riemann surface with an anti-holomorphic involution τ . Denote by R g (R) the set of xed points of the involution τ (see Section 3.1). Let us choose the homology basis satisfying (3.2.1). A meromorphic function f on R g is called real if f (τ p) = f (p) for any point p ∈ R g .

In the next proposition we derive theta-functional solutions of the system (6.3.1). The signs s j , which appear in the reality conditions (6.3.3), are expressed in terms of certain intersection indices on R g . These intersection indices are dened as follows: let f be a real meromorphic function of degree n + 1 on R g . Let z a ∈ R be a non critical value of f , and assume that the ber f -1 (z a ) = {a 1 , . . . , a n+1 } over z a belongs to the set R g (R). Let ãn+1 , ãj ∈ R g (R) lie in a neighbourhood of a n+1 and a j respectively such that f (ã n+1 ) = f (ã j ). Denote by ˜ j an oriented contour connecting ãn+1 and ãj , and having the following decomposition in H 1 (R g \ {a n+1 , a j })

(see Section 3.3.2): τ ˜ j = ˜ j + N t j A + M t j B + α j S a j , (6.3.12)
for some α j ∈ Z, where the vectors N j , M j ∈ Z g are the same as in (3.3.9). Then

α j = (τ ˜ j -˜ j ) • j (6.3.13)
is the intersection index between the closed contour τ ˜ j -˜ j and the contour j ; this index is computed in the relative homology group H 1 (R g , {a n+1 , a j }).

Theta-functional solutions of (6.3.1) are given by: Proposition 6.3.1. Let f be a real meromorphic function of degree n+1 on R g . Let z a ∈ R be a non critical value of f , and assume that the ber f -1 (z a ) = {a 1 , . . . , a n+1 } over z a belongs to the set R g (R). Choose the local parameters k a j near a j as k a j (p) = f (p)z a , for any point p ∈ R g in a neighbourhood of a j . Denote by (A, B, j ) the standard generators of the relative homology group H 1 (R g , {a n+1 , a j }). Let d R ∈ R g , T ∈ Z g , and dene d = d R + iπ 2 (diag(H) -2 T). Moreover, take θ ∈ R. Then the following functions ψ j for j = 1, . . . , n, give solutions of n-NLS s

ψ j (x, t) = |A j | e iθ Θ(Z -d + r j ) Θ(Z -d) exp {-i (E j x -F j t)} , (6.3.14)
where Z = i V a n+1 x + i W a n+1 t, and

|A j | = |q 2 (a n+1 , a j )| 1/2 exp 1 2 d R , M j . (6.3.15)
Here r j = j ω, the vectors V a n+1 , W a n+1 are dened in (2.3.3), and the vector M j ∈ Z g is dened by the action of τ on the relative homology group H 1 (R g , {a n+1 , a j }) (see (3.3.9)). The scalars q 2 (a n+1 , a j ) and E j , F j are introduced in (2.3.12) and (6.3.6) respectively. The signs s 1 , . . . , s n are given by s j = exp {iπ(1

+ α j ) + iπ T, M j } , (6.3.16)
where the intersection indices α j ∈ Z are dened in (6.3.13).

Proof. Let us check that under the conditions of the theorem, the functions ψ j and ψ * j (6.3.4) satisfy the reality conditions (6.3.3). First of all, invariance with respect to the anti-involution τ of the point a n+1 implies the reality of vector (6.3.5): Z = Z.

(6.3.17)

In fact, using the expansion (2.3.3) of the normalized holomorphic dierentials ω j near a n+1 we get

τ * ω j (a n+1 )(p) = V a n+1 , j + W a n+1 , j k a n+1 (p) + o(k a n+1 (p)) dk a n+1 (p),
for any point p in a neighbourhood of a n+1 . Then, by (3.2.2) the vectors V a n+1 and W a n+1 appearing in the expression (6.3.5) satisfy

V a n+1 = -V a n+1 , W a n+1 = -W a n+1 , (6.3.18)
which leads to (6.3.17). Moreover, from (3.2.2) and (3.3.9) we get

r j = -r j -2iπN j -BM j , (6.3.19)
where N j , M j ∈ Z g are dened in (3.3.9) and satisfy 2 N j + HM j = 0.

(6.3.20)

From (6.1.1), it is straightforward to see that the scalars K 1 (a n+1 , a j ) and K 2 (a n+1 , a j ) dened by (6.1.2) and (6.1.3) satisfy

K 1 (a n+1 , a j ) = K 1 (a n+1 , a j ) -V a n+1 , M j , K 2 (a n+1 , a j ) = K 2 (a n+1 , a j ) + W a n+1 , M j .
Moreover, one can directly see from (2.3.10) that q 1 (a n+1 , a j ) is real. Hence we get

E j = E j -V a n+1 , M j , F j = F j + W a n+1 , M j . (6.3.21)
Under the assumptions of the theorem, and by (6.2.10), the argument of q 2 (a n+1 , a j ) is given by arg{q 2 (a n+1 , a j )} = π(1

+ α j ) + π 2 HM j , M j - 1 2i ( BM j , M j + 2 r j , M j ) . (6.3.22)
Therefore, the reality conditions (6.3.3) together with (6.3.4) lead to

|A j | 2 = s j |q 2 (a n+1 , a j )| Θ(Z -d -r j ) Θ(Z -d + iπ diag(H)) Θ(Z -d -r j + iπ diag(H)) Θ(Z -d) × exp iπ(1 + α j ) + iπ 2 HM j , M j + d -iπ diag(H), M j , (
d = d -iπ diag(H) + 2iπT, (6.3.24)
for some T ∈ Z g , where we have used (3.2.3) and the fact that B has a non-degenerate real part. It follows that the vector d can be written as

d = d R + iπ 2 (diag(H) -2 T), (6.3.25)
for some d R ∈ R g and T ∈ Z g . Therefore, (6.3.23) becomes

|A j | 2 = s j |q 2 (a n+1 , a j )| exp iπ(1 + α j ) + iπ 2 HM j , M j + d, M j . (6.3.26)
We deduce from (6.3.25) and (6.3.26) that s j = exp iπ(1 + α j ) + iπ 2 HM j + diag(H), M jiπ T, M j .

From (6.3.20) and the denition of the matrix H (see Section 3.2), it can be deduced that the quantity 1 2 HM j + diag(H), M j is even in each case, which yields (6.3.16) and (6.3.15).

Functions ψ j for j = 1, . . . , n, given in (6.3.14) describe a family of algebro-geometric solutions of (6.3.1) depending on: a real Riemann surface (R g , τ ), a real meromorphic function f on R g of degree n + 1, a non critical value z a ∈ R of f such that the ber over z a belongs to the set R g (R), and arbitrary constants d R ∈ R g , T ∈ Z g , θ ∈ R. Note that the periodicity properties of the theta function imply without loss of generality that the vector T can be chosen in the set {0, 1} g . The case where the Riemann surface R g is dividing and T = 0 is of special importance, because the related solutions are smooth, as explained in the next proposition. In this case, the sign s j (6.3.16) is given by s j = exp{iπ(1 + α j )}.

Since the theta function is entire, singularities of the functions ψ j can appear only at the zeros of their denominator. Following Vinnikov's result [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF] and if the argument Zd of the theta function in the denominator is real, which by (6.3.17) leads to the choice d ∈ R g (and then T = 0 in Proposition 6.3.1).

The following assertions were proved in [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF]: let R g (R) = ∅; if R g is non dividing, then T v ∩ (Θ) = ∅ for all v, where (Θ) denotes the set of zeros of the theta function; if R g is dividing, then T v ∩ (Θ) = ∅ if and only if v = 0. It follows that if solutions are smooth for any vector d in a component T v (3.4.4) of the Jacobian, then the curve is dividing and v = 0. Hence d ∈ T 0 where T 0 = R g .

Solutions of n-NLS + and n-NLS -

Here, we consider the two most physically signicant situations: the completely focusing multicomponent system n-NLS + (which corresponds to s = (1, . . . , 1)), and the completely defocusing system n-NLS -(which corresponds to s = (-1, . . . , -1)).

Starting from a pair (R g , f ), where R g is a Riemann surface of genus g, and where f is a meromorphic function of degree n + 1 on R g , which has n + 1 simple poles, we construct an (n + 1)-sheeted branched covering of CP 1 , which we denote by R g,n+1 . The ramication points of the covering correspond to critical points of f ; we assume that all of them are simple. We denote by x 1 , . . . , x 2g+2n ∈ R g,n+1 the critical points of the meromorphic function f , and by z j = f (x j ) ∈ C the critical values.

For any point a ∈ R g,n+1 which is not a critical point or a pole of the meromorphic function f , we use the local parameter k a (p) = f (p)f (a), for any point p in a neighbourhood of a.

According to [START_REF] Eisenbud | On the Hurwitz scheme and its monodromy[END_REF], by an appropriate choice of the set of generators {γ j } 2g+2n j=1 of the fundamental group π 1 (CP 1 \ {z 1 , . . . , z 2g+2n }, z 0 ) of the base, which satisfy γ 1 . . . γ 2g+2n = id, the covering R g,n+1 can be represented as follows: consider the hyperelliptic covering of genus g and attach to it n -1 spheres as shown in Figure 6.1. More precisely, the generators γ j can be chosen in such way that the loop γ j encircles only the point z j ; the corresponding elements σ j ∈ S n+1 (where S n+1 denotes the symmetric group of order n + 1) of the monodromy group of the covering are given by

σ j = (n + 1, n), j = 1, . . . , 2g + 2, σ 2g+2k+3 = σ 2g+2k+4 = (n -k, n -k -1), k = 0, . . . , n -2.
Assume that the branch points {z j } 2g+2n j=1 are real or pairwise conjugate, and order them as follows:

Re(z 1 ) ≤ . . . ≤ Re(z 2g+2n ).

Let us introduce an anti-holomorphic involution τ on R g,n+1 , which acts as the complex conjugation on each sheet.

. . . Solutions of n-NLS + .

x 1 x 2 x 3 x 4 x 2g+1 x 2g+2n x 2g+2 x 2g+3 x 2g+4 x 2g+2n-1 a 1 a 2 a n+1 a n a n-1 . . . . . . 1 2 n -1 n n + 1
Here we construct solutions of the n-NLS

+ system i ∂ψ j ∂t + ∂ 2 ψ j ∂x 2 + 2 n k=1 |ψ k | 2 ψ j = 0, j = 1, . . . , n. (6.3.27)
Let us rst describe the covering and the homology basis used in the construction of the solutions.

Assume that all branch points of the covering R g,n+1 are pairwise conjugate. Denote this covering by R + g,n+1 , referring to the focusing system (6.3.27). The covering R + g,n+1 admits two real ovals if the genus g is odd, and only one if g is even. Each of them consists of a closed contour on the covering having a real projection into the base. It is straightforward to see that the covering R + g,n+1 is dividing (see Section 3.1): two points which have respectively a positive and a negative imaginary projection onto C, cannot be connected by a contour which does not cross a real oval. Hence the set of xed points of the anti-holomorphic involution τ separates the covering into two connected components. Now let us choose the canonical homology basis such that all basic cycles belong to sheets n + 1 and n, and such that the anti-holomorphic involution τ acts on them as in (3.2.1). By the previous topological description of R + g,n+1 , the matrix H in (3.2.1) reads:

H =          0 1 1 0 . . . 0 1 1 0 0          if g is odd, H =        0 1 1 0 . . . 0 1 1 0        if g is even.
The canonical homology basis is described explicitly in Figure 6.2 for odd genus, and in Figure 6.3 for even genus.

As proved in the following theorem, among all coverings having a monodromy group described in Figure 6.1, only the covering R + g,n+1 leads to algebro-geometric solutions of the focusing system (6.3.27).

< < > > A 1 A 2 B 1 B 2 z 1 z 2 z 3 z 4 . . . A g B g > > < B g-1 A g-1 < z 2g+2 z 2g+1 z 2g z 2g-1 z 2g-2 z 2g-3 z 5 z 6 < > A g-2 B g-2 < > > < > > Figure 6
.2: Homology basis on the covering R + g,n+1 when the genus g is odd. The solid line indicates the sheet n + 1, and the dashed line sheet n.

< < > > A 1 A 2 B 1 B 2 z 1 z 2 z 3 z 4 . . . A g B g > > < B g-1 A g-1 < z 2g+2 z 2g+1 z 2g z 2g-1 z 2g-2 z 2g-3 < > > < > > Figure 6
.3: Homology basis on the covering R + g,n+1 when the genus g is even. The solid line indicates the sheet n + 1, and the dashed line sheet n.

z 1 z 2 z 3 z 4 . . . > < < z 2g+2 z 2g+1 z 2g z 2g-1 z 2g-2 z 2g-3 > z a z 2g+3 z 2g+4 z 2g+2n z 2g+2n-1 z ã . . . > < n τ ˜ n -˜ n Figure 6.4: The closed contour τ ˜ n -˜ n ∈ H 1 R + g,n+1 \ {a n+1 , a n } is homologous to a closed contour which encircles the vertical cut [z 2g+1 , z 2g+2 ], then α n = (τ ˜ n -˜ n ) • n = 1.
Theorem 6.3.2. Consider the covering R + g,n+1 and the canonical homology basis discussed above.

Fix z a ∈ R such that z a > Re(z j ) for j = 1, . . . , 2g + 2n. Consider the ber f -1 (z a ) = {a 1 , . . . , a n+1 } over z a , where a j ∈ R + g,n+1 (R) belongs to sheet j (each of the a j is invariant under the involution τ ). Let d R ∈ R g and θ ∈ R. Then the following functions ψ j for j = 1, . . . , n, give smooth solutions of n-NLS + : 6.3.28) where Z = i V a n+1 x + i W a n+1 t, and the remaining notation is as in Proposition 6.3.1.

ψ j (x, t) = |A j | e iθ Θ(Z -d R + r j ) Θ(Z -d R ) exp {-i (E j x -F j t)} , (
Proof. Let us check that the conditions of the theorem imply that the functions ψ j , j = 1, . . . , n, in (6.3.14) give solutions of n-NLS s for s = (1, . . . , 1). Since the matrix H associated to the covering R + g,n+1 satises diag(H) = 0, and d R ∈ R g (i.e., T = 0 in Proposition 6.3.1), the quantity s j dened in (6.3.16) becomes

s j = exp {iπ(1 + α j )} . (6.3.29)
Let us rst compute the intersection index α n . Let ãn+1 , ãn ∈ R + g,n+1 (R) lie in a neighbourhood of a n+1 and a n respectively such that f (ã n+1 ) = f (ã n ) = z ã. Denote by ˜ n an oriented contour connecting ãn+1 and ãn . Then the intersection index α n between the closed contour τ ˜ n -˜ n and the contour n satises (see Figure 6.4): (6.3.30) which leads to s n = 1. Intersection indices α j for j = 1, . . . , n -1 can be computed in the same way. Therefore

α n = (τ ˜ n -˜ n ) • n ≡ 1 (mod 2),
α 1 ≡ α 2 ≡ . . . ≡ α n ≡ 1 (mod 2),
which implies s j = 1. By Proposition 3.4.1, smoothness of the solutions is ensured by the reality of the vector Zd R and the fact that the curve is dividing.

The functions ψ j , j = 1, . . . , n, given in (6.3.28) describe a family of smooth algebro-geometric solutions of the focusing multi-component NLS equation depending on g +n complex parameters: z 2k-1 ∈ C \ R for k = 1, . . . , g + n; and g + 2 real parameters: z a , θ ∈ R, and d R ∈ R g . Solutions of n-NLS -. Now let us construct solutions of the system n-NLS -

i ∂ψ j ∂t + ∂ 2 ψ j ∂x 2 -2 n k=1 |ψ k | 2 ψ j = 0, j = 1, . . . , n. (6.3.31)
As for the focusing case, let us rst describe the covering and the homology basis used in our construction of the solutions of (6.3.31).

Assume that the branch points z k of the covering R g,n+1 are real for k = 1, . . . , g + 2, and that the branch points z 2k-1 , z 2k are pairwise conjugate for k = g + 2, . . . , g + n. Denote by R - g,n+1 this covering, referring to the defocusing system (6.3.31). It is straightforward to see that such a covering is an M-curve (see Section 3.1), that is it admits a maximal number of real ovals g + 1 with respect to the anti-holomorphic involution τ . On the other hand, it can be directly seen that R - g,n+1 is dividing: two points which lie on the sheet n + 1 and have respectively a positive and a negative imaginary projection onto C cannot be connected by a contour which does not cross a real oval. Now let us choose the canonical homology basis such that all basic cycles belong to sheets n + 1 and n, and which satises (3.2.1). Since the covering R - g,n+1 is an M-curve, the matrix H involved in (3.2.1) satises H = 0. Such a canonical homology basis is shown in Figure 6.5.

B 1 A 1 < < > > z 1 z 2 z 3 z 4 z 2g+1 z 2g+2
A g B g . . . In the following theorem, we construct algebro-geometric solutions of the defocusing system (6.3.31) associated to the covering R - g,n+1 .

Theorem 6.3.3. Consider the covering R - g,n+1 and the canonical homology basis discussed above.

Fix z a ∈ R \ {z 1 , . . . , z 2g+2 } such that z a > Re(z j ) for j = 1, . . . , 2g + 2n. Consider the ber f -1 (z a ) = {a 1 , . . . , a n+1 } over z a , where a j ∈ R - g,n+1 (R) belongs to sheet j (each of the a j is invariant under the involution τ ). Let d R ∈ R g and θ ∈ R. Then the functions ψ j dened in (6.3.28) for j = 1, . . . , n give smooth solutions of n-NLS -.

Proof. Analogously to the focusing case, one has to check that all s j dened in (6.3.16) satisfy s j = -1. Since all branch points z k are real for k = 1, . . . , 2g + 2, the intersection index α n

z 1 z 2 z a z 2g+1 z 2g+2 z 2g+3 z 2g+4 z 2g+2n z 2g+2n-1 . . . . . . z ã < n < τ ˜ n -˜ n > Figure 6.6: The closed contour τ ˜ n -˜ n ∈ H 1 R - g,n+1 \ {a n+1 , a n } is homologous to zero, then α n = (τ ˜ n -˜ n ) • n = 0.
between the closed contour τ ˜ n -˜ n and the contour n satises (see Figure 6.6) (6.3.32) which leads to s n = -1. Intersection indices α j for j = 1, . . . , n -1 can be computed in the same way, and we get α 1 ≡ α 2 ≡ . . . ≡ α n ≡ 0 (mod 2), which implies s j = -1. Smoothness of the solutions is ensured by the reality of the vector Z-d R and the fact that the curve is dividing. Solutions ψ j , j = 1, . . . , n, constructed here describe a family of smooth algebro-geometric solutions of the defocusing multi-component NLS equation depending on n -1 complex parameters: z 2g+2k+1 ∈ C for k = 1, . . . , n -1; and 3g + 4 real parameters: z k ∈ R for k = 1, . . . , 2g + 2, z a , θ ∈ R, and d R ∈ R g . Remark 6.3.1. Smooth solutions of n-NLS s for a vector s with mixed signs can be constructed in the same way.

α n = (τ ˜ n -˜ n ) • n ≡ 0 (mod 2),

Stationary solutions of n-NLS

It is well-known that the algebro-geometric solutions (6.3.14) on an elliptic surface describe travelling waves, i.e., the modulus of the corresponding solutions depends only on xct, where c is a constant. Due to the Galilei invariance of the multi-component NLS equation (see (6.3.11)), the invariance under transformations of the form

ψ j (x, t) -→ ψ j (x + 2λ t, t) exp -i λ x + λ 2 t , where λ = -1 2 W a n+1 (V a n+1 ) -1
, leads to stationary solutions (t-independent) in the transformed coordinates.

For arbitrary genus of the spectral curve, stationary solutions of the multi-component NLS equation are obtained from solutions (6.3.14) under the vanishing condition:

W a n+1 = 0.

( 6.3.33) This condition is equivalent to the existence of a meromorphic function h of degree two on R g , such that the point a n+1 is a critical point of h (this can be proved analogously to Lemma 6.3.1).

Therefore, stationary solutions of the multi-component NLS equation can be constructed from the algebro-geometric data {R g , f, h, z a }, where:

• R g is a real Riemann surface of genus g, and f is a real meromorphic function of degree n + 1

on R g , • z a ∈ R is a non critical value of f such that f -1 (z a ) = {a 1 , . . . , a n+1 },
• h is a real meromorphic function of degree two on R g , and a n+1 is a critical point of h,

• for j = 1, . . . , n, local parameters k a j near a j are chosen to be k a j (p) = h(p)h(a j ) for any point p in a neighbourhood of a j , and k a n+1 (p) = (h(p)h(a n+1 )) 1/2 for any point p in a neighbourhood of a n+1 . With this choice of local parameters, we get f (p)z a = β j k a j (p) + µ j k a j (p) 2 + O k a j (p) 3 , for any point p ∈ R g which lies in a neighbourhood of a j , where β j , µ j ∈ R. Hence solutions (6.3.14) can be rewritten using this choice of local parameters and then are expressed by the use of the scalars β j and µ j .

Moreover, choosing a n+1 as a critical point of h, we get (6.3.33). In this case, the modulus of solutions (6.3.14) does not depend on the variable t.

Reduction of n-NLS to (n -1)-NLS

It is natural to wonder if starting from solutions of n-NLS we can obtain solutions of (n -1)-NLS for n > 2. Such a reduction is possible if one of the functions ψ j solutions of n-NLS vanishes identically.

Let R + g,n+1 be the (n + 1)-sheeted covering introduced in Section 6.3.3; to obtain solutions of (n -1)-NLS + from solutions of n-NLS + , we consider the following degeneration of the covering R + g,n+1 : let the branch points z 2g+2n and z 2g+2n-1 coalesce, in such way that the rst sheet gets disconnected from the other sheets (see Figure 6.1); denote by R + g,n the covering obtained in this limit.

Then the normalized holomorphic dierentials on R + g,n+1 tend to normalized holomorphic dierentials on R + g,n ; on the rst sheet, all holomorphic dierentials tend to zero. Therefore, in this limit, each component of the vector V a 1 tends to 0.

Hence by (2.3.12) and (6.3.15), the function ψ 1 tends to zero as z 2g+2n and z 2g+2n-1 coalesce. Functions ψ j for j = 2, . . . , n obtained in this limit give solutions of (n -1)-NLS + associated to the covering R + g,n .

Remark 6.3.2. A similar degeneration produces a solution of (n -1)-NLS -from a solution of n-NLS -. Repeating this degeneration n -1 times, we rediscover (see [START_REF] Its | Inversion of hyperelliptic integrals and integration of nonlinear dierential equations[END_REF]) algebro-geometric solutions of the focusing (resp. defocusing) nonlinear Schrödinger equation (1.1.2) associated to a hyperelliptic curve with pairwise conjugate branch points (resp. real branch points).

Hyperelliptic solutions of n-NLS

In this part we construct smooth hyperelliptic theta-functional solutions of n-NLS s . The following lemma gives an example where the vectors V a 1 , . . . , V a n+1 dened in (2.3.3) are linearly dependent. This property ensures by Lemma 6.3.1 and Theorem 6.3.1 that we can construct hyperelliptic solutions of n-NLS s . Let R g be a hyperelliptic curve of genus g > 0 given in a general form by (6.3.34) with the projection map π(λ, µ) -→ λ of degree two on R g . Lemma 6.3.2. Let a j = (λ a j , µ a j ) ∈ R g be n + 1 points having distinct projection, such that λ a j = λ k , and assume n ≥ g. Let γ g+1 , . . . , γ n ∈ C \ {0} be arbitrary constants and put γ n+1 = 1.

µ 2 = 2g+2 k=1 (λ -λ k ),
Then det V a 1 , . . . , V ag = 0, and there exist γ 1 , . . . , γ g ∈ C \ {0} such that n+1 k=1 γ k V a k = 0. For j = 1, . . . , g, the scalars γ j are given by

γ j = - n+1 k=g+1 γ k µ a j µ a k 1≤i≤g i =j λ a k -λ a i λ a j -λ a i . (6.3.35)
Proof. A basis of holomorphic dierentials on R g is given by ν = (ν 1 , . . . , ν g ) t = 1, λ, . . . , λ g-1 t dλ µ .

(6.3.36)

Normalized holomorphic dierentials ω = (ω 1 , . . . , ω g ) t are obtained as follows:

ω = 2iπ (A t ) -1 ν, (6.3.37) where A ij = A i ν j . From (6.3.37) and denition (2.3.3) of the vectors V a i we deduce that

     V a 1 ,1 . . . V ag,1 V a 1 ,2 . . . V ag,2 . . . . . . . . . V a 1 ,g . . . V ag,g      = 2iπ (A t ) -1        1 µa 1 . . . 1 µa g λa 1 µa 1 . . . λa g µa g . . . • • • . . . λ g-1 a 1 µa 1 . . . λ g-1 ag µa g        . (6.3.38)
Since by assumptions all λ a j are distinct, equality (6.3.38) involves the linear independence of the vectors V a 1 , . . . , V ag (the determinant of the last matrix in the right hand side in (6.3.38) is a Vandermonde determinant which does not vanish). Denote by γ (p) := (γ 1 , . . . , γ p ) t . It follows that the system V a 1 , . . . , V a n+1 γ (n+1) = 0, rewritten as

V a 1 , . . . , V ag γ (g) = - n+1 k=g+1 γ k V a k ,
can be solved using Cramer's method. Therefore, using (6.3.38) we get, for j = 1, . . . , g,

γ j = - n+1 k=g+1 γ k µ a j µ a k
Vd(λ a 1 , . . . , j λ a k , . . . , λ ag )

Vd(λ a 1 , . . . , λ ag ) ,

where Vd(λ a 1 , . . . , λ ag ) := 1≤i<k≤g (λ a kλ a i ) is the Vandermonde determinant.

Remark 6.3.3. On the other hand, if n < g, the proof of Lemma 6.3.2 shows that for any points a 1 , . . . , a n+1 ∈ R g having distinct projection, the vectors V a 1 , . . . , V a n+1 cannot be linearly dependent.

Theorem 6.3.4. Let R g be a real hyperelliptic curve (6.3.34) and denote by τ an anti-holomorphic involution. Choose the canonical homology basis which satises (3.2.1). Take n ≥ g and let a 1 , . . . , a n+1 ∈ R g (R) having distinct projection π(a j ) = λ a j into the base, such that λ a j = λ k . Denote by j an oriented contour between a n+1 and a j which does not intersect cycles of the canonical homology basis. Let d R ∈ R g , T ∈ Z g , and dene d = d R + iπ 2 (diag(H)-2 T). Take θ ∈ R and let γ g+1 , . . . , γ n ∈ R be arbitrary constants with γ n+1 = 1. Put ŝ = (sign(γ 1 ) s 1 , . . . , sign(γ n ) s n ) where s j and γ j are given in (6.3.16) and (6.3.35) respectively. Then by Lemma 6.3.2, the following functions ψ j for j = 1, . . . , n give solutions of n-NLS

ŝ ψ j (x, t) = |γ j | 1/2 |A j | e iθ Θ(Z -d + r j ) Θ(Z -d) exp{-i (E j x -F j t)}, (6.3.39)
where |A j | is given in (6.3.15). Here r j = j ω, and Z = i V a n+1 x + i W a n+1 t, where the vectors V a n+1 and W a n+1 were introduced in (2.3.3). The scalars E j , F j are given by

E j = K 1 (a n+1 , a j ), F j = K 2 (a n+1 , a j ) -2 n k=1 γ k q 1 (a n+1 , a k ),
where q 1 (a n+1 , a j ), q 2 (a n+1 , a j ), K 1 (a n+1 , a j ) and K 2 (a n+1 , a j ) are respectively dened in (2.3.11), (2.3.12), (6.1.2) and (6.1.3). If R g is dividing and d ∈ R g , functions (6.3.39) give smooth hyperelliptic solutions of n-NLS ŝ.

Example 6.3.1. (Hyperelliptic solutions of 2-NLS ŝ in genus 2.)

Let us construct smooth hyperelliptic theta-functional solutions of 2-NLS ŝ in genus 2, for any vector of signs ŝ. Consider the hyperelliptic curve (6.3.34) with g = 2 and let a 1 = (λ a 1 , µ a 1 ), a 2 = (λ a 2 , µ a 2 ) and a 3 = (λ a 3 , µ a 3 ) be three points on the curve such that λ a 1 = λ a 2 = λ a 3 and λ a j = λ k . By Lemma 6.3.2, the vectors V a j are related by

V a 3 + γ 1 V a 1 + γ 2 V a 2 = 0 with γ 1 = - µ a 1 µ a 3 λ a 3 -λ a 2 λ a 1 -λ a 2 , γ 2 = - µ a 2 µ a 3 λ a 3 -λ a 1 λ a 2 -λ a 1 . (6.3.40)
We consider two cases: i) the curve has real branch points only. Let us prove that on such a curve one can construct solutions of 2-NLS ŝ for ŝ = (1, -1), ŝ = (-1, -1), and ŝ = (-1, 1). Let λ 1 , . . . , λ 6 ∈ R satisfy λ 1 < . . . < λ 6 . First, note that in the case where a j and a n+1 lie on the same sheet, or lie on dierent sheets, one has µa j µa n+1 s j > 0 (see Section 6.3.3). Now assume that λ a 3 > λ a 2 > λ a 1 . By (6.3.40) we deduce that ŝ1 = 1 and ŝ2 = -1. Hence by Theorem 6.3.4, functions (6.3.39) give smooth hyperelliptic solutions of 2-NLS ŝ where ŝ = (1, -1). Analogously, the choice λ a 2 > λ a 3 > λ a 1 leads to solutions of 2-NLS ŝ with ŝ = (-1, -1), and the choice λ a 2 > λ a 1 > λ a 3 provides solutions of 2-NLS ŝ for ŝ = (-1, 1).

ii) the curve has all its branch points pairwise conjugate. As explained below, on such a curve one gets solutions of 2-NLS ŝ for any vector of signs ŝ. Assume that λ 2k-1 = λ 2k for k = 1, 2, 3. First, note that if a j and a n+1 lie on the same sheet, µa j µa n+1 > 0 and s j > 0, otherwise µa j µa n+1 < 0 and s j > 0. Following the reasoning of case i), if a 1 , a 2 and a 3 lie on the same sheet, the functions (6.3.39) give smooth hyperelliptic solutions of 2-NLS ŝ with ŝ = (1, -1), ŝ = (-1, -1) or ŝ = (-1, 1), depending on the projections λ a j . Now assume that a 1 and a 3 belong to the same sheet, and take a 2 on another sheet. Then, for the choice λ a 3 > λ a 2 > λ a 1 , the vector of signs satises ŝ = (1, 1).

Relationship between solutions of KP1 and solutions of n-NLS

Historically, the KP equation is the rst example of a system with two space variables for which it has been possible to completely solve the problem of reality of the algebro-geometric solutions.

Here we show that starting from our solutions of the multi-component NLS equation and its complexication, we can construct a subclass of complex and real solutions of the Kadomtsev-Petviashvili equation (KP1)

3 4 u yy = u t - 1 4 (6 uu x -u xxx ) x . (6.3.41)
Let R g be an arbitrary compact Riemann surface with marked point a, and let k a be an arbitrary local parameter near a. Dene vectors V a , W a , U a as in (2.3.3) and let d ∈ C g . Then, according to Krichever's theorem [START_REF] Krichever | Algebro-geometric construction of the Zakharov-Shabat equations and their periodic solutions[END_REF] (see also [START_REF] Krichever | Algebraic curves and nonlinear dierence equations[END_REF][START_REF] Krichever | Nonlinear equations and elliptic curves[END_REF]), the function

u(x, y, t) = 2 D 2 a ln Θ(i V a x + i W a y + i U a t -d) + 2 β (6.3.42)
is a solution of KP1; here the constant β is dened by the expansion near a of the normalized meromorphic dierential Ω

a having a pole of order two at a only:

Ω (2) a (p) = k a (p) -2 + β k a (p) + o(k a (p)),
where p lies in a neighbourhood of a.

Let us check that if the local parameter k a is dened by the meromorphic function f of degree n + 1 on R g as k a (p) = f (p)f (a), then formula (6.3.42) naturally arises from our construction of solutions of the n-NLS s system. Namely, identify a with a n+1 . Then, due to the fact that n+1 j=1 V a j = 0 where {a 1 , . . . , a n+1 } = f -1 (z a ) (see Lemma 6.3.1), the solution (6.3.42) of KP1 can be rewritten as

u(x, y, t) = -2 n j=1 D a n+1 D a j ln Θ(z) + 2 β, where z = i V a n+1 x + i W a n+1 y + i U a n+1 t -d. Using Corollary (2.3.10) of Fay's identity, we get u(x, y, t) = -2 n j=1 q 1 (a n+1 , a j ) + q 2 (a n+1 , a j ) Θ(z + r j ) Θ(z -r j ) Θ(z) 2 + 2 β. (6.3.43)
Now let us consider solutions ψ j , ψ * j (6.3.4) of the complexied multi-component NLS equa- tion, and make the change of variables (x, t) → (x, y) and d →i U a n+1 t + d. Then by (6.3.43), the complex-valued solutions u (6.3.42) of KP1 and solutions ψ j , ψ * j (6.3.4) of the complexied n-NLS system are related by u(x, y, t) = γ -2 n j=1 ψ j (x, y, t) ψ * j (x, y, t), (6.3.44) where

γ = -2 n j=1
q 1 (a n+1 , a j ) + 2 β.

If we impose the reality conditions (6.3.3), we obtain real solutions of KP1 from our solutions

D 1 = ∂ xx -α 2 ∂ yy , D 2 = ∂ xx + α 2 ∂ yy .
Introduce also the characteristic coordinates

ξ = 1 2 (x -iα y), η = 1 2 (x + iα y), α = i or 1.
In these coordinates, the Davey-Stewartson equations (1.2.9) become

i ψ t + D 1 ψ + 2 φ ψ = 0, D 2 φ + ρ D 1 |ψ| 2 = 0, (6.4.1)
where the dierential operators D 1 and D 2 are given by

D 1 = 1 2 (∂ 2 ξ + ∂ 2 η ), D 2 = ∂ ξ ∂ η .
In what follows, DS1

ρ denotes the Davey-Stewartson equation when α = i (in this case ξ and η are both real), and DS2

ρ the Davey-Stewartson equation when α = 1 (in this case ξ and η are pairwise conjugate).

Solutions of the complexied Davey-Stewartson equations

Consider rst the complexied version of the Davey-Stewartson equations:

i ψ t + 1 2 (ψ ξξ + ψ ηη ) + 2 ϕ ψ = 0, -i ψ * t + 1 2 (ψ * ξξ + ψ * ηη ) + 2 ϕ ψ * = 0, (6.4.2)
ϕ ξη + 1 2 ((ψψ * ) ξξ + (ψψ * ) ηη ) = 0,
where ϕ := Φ + ψψ * . This system reduces to (6.4.1) under the reality condition :

ψ * = ρ ψ, (6.4.3)
which leads to ϕ = φ. Theta-functional solutions of system (6.4.2) are given by: Theorem 6.4.1. Let R g be a compact Riemann surface of genus g > 0, and let a, b ∈ R g be distinct points. Take arbitrary constants d ∈ C g and A, κ 1 , κ 2 ∈ C \ {0} , h ∈ C. Denote by an oriented contour connecting a and b which does not intersect cycles of the canonical homology basis. Then for any ξ, η, t ∈ C, the following functions ψ, ψ * and ϕ are solutions of system (

= A Θ(Z -d + r) Θ(Z -d) exp -i G 1 ξ + G 2 η -G 3 t 2 , ψ * (ξ, η, t) = - κ 1 κ 2 q 2 (a, b) A Θ(Z -d -r) Θ(Z -d) exp i G 1 ξ + G 2 η -G 3 t 2 , 6.4.2) ψ(ξ, η, t)
ϕ(ξ, η, t) = 1 2 (ln Θ(Z -d)) ξξ + 1 2 (ln Θ(Z -d)) ηη + h 4 . (6.4.4)
Here r = ω, where ω is the vector of normalized holomorphic dierentials, and

Z = i κ 1 V a ξ -κ 2 V b η + (κ 2 1 W a -κ 2 2 W b ) t 2 , (6.4.5)
where the vectors V a , V b and W a , W b were introduced in (2.3.3). The scalars G 1 , G 2 , G 3 are given by

G 1 = κ 1 K 1 (a, b), G 2 = κ 2 K 1 (b, a), (6.4.6)
G 3 = κ 2 1 K 2 (a, b) + κ 2 2 K 2 (b, a) + h, (6.4.7)
and the scalars q 2 (a, b), K 1 (a, b), K 2 (a, b) are dened in (2.3.12), (6.1.2), (6.1.3) respectively.

Proof. Substitute the functions (6.4.4) in the rst equation of system (6.4.2) to get

κ 2 1 D a ln Θ(Z -d + r) Θ(Z -d) + κ 2 1 D 2 a ln Θ(Z -d + r) Θ(Z -d) + 2 κ 2 1 D 2 a ln Θ(Z -d) + G 3 -h + κ 1 D a ln Θ(Z -d + r) Θ(Z -d) -G 1 2 + κ 2 D b ln Θ(Z -d + r) Θ(Z -d) + G 2 2 -κ 2 2 D b ln Θ(Z -d + r) Θ(Z -d) + κ 2 2 D 2 b ln Θ(Z -d + r) Θ(Z -d) + 2 κ 2 2 D 2 b ln Θ(Z -d) = 0.
By (6.1.1), the last equality holds for any Zd ∈ C g , and in particular for Z dened in (6.4.5).

In the same way, it can be checked that functions (6.4.4) satisfy the second equation of system (6.4.2). Moreover, from (2.3.10) we get

(ψψ * ) ξξ = κ 3 1 κ 2 D 3 a D b ln Θ(Z -d), (ψψ * ) ηη = κ 1 κ 3 2 D a D 3 b ln Θ(Z -d).
Therefore, taking into account that

ϕ ξη = - 1 2 κ 3 1 κ 2 D 3 a D b ln Θ(Z -d) + κ 1 κ 3 2 D a D 3 b ln Θ(Z -d) ,
the functions (6.4.4) satisfy the last equation of system (6.4.2).

The solutions (6.4.4) depend on the Riemann surface R g , the points a, b ∈ R g , the vector d ∈ C g , the constants κ 1 , κ 2 ∈ C \ {0}, h ∈ C, and the local parameters k a and k b near a and b.

The transformation of the local parameters given by

k a -→ β k a + µ 1 k 2 a + O k 3 a , k b -→ β k b + µ 2 k 2 b + O k 3 b ,
where β, µ 1 , µ 2 are arbitrary complex numbers (β = 0), leads to a dierent family of solutions of the complexied system (6.4.2). These new solutions are obtained via the following transformations:

ψ(ξ, η, t) -→ ψ β ξ + βλ 1 t, β η + βλ 2 t, β 2 t exp -i λ 1 ξ + λ 2 η + λ 2 1 + λ 2 2 -α t 2 , ψ * (ξ, η, t) -→ β 2 ψ * β ξ + βλ 1 t, β η + βλ 2 t, β 2 t exp i λ 1 ξ + λ 2 η + λ 2 1 + λ 2 2 -α t 2 , φ(ξ, η, t) -→ β 2 φ β ξ + βλ 1 t, β η + βλ 2 t, β 2 t + α 4 ,
where λ i = κ i µ i β -1 for i = 1, 2, and α = h (1β 2 ).

Reality condition and solutions of the DS1 ρ equation

Let us consider the DS1 ρ equation

i ψ t + 1 2 (∂ 2 ξ + ∂ 2 η )ψ + 2 φ ψ = 0, ∂ ξ ∂ η φ + ρ 1 2 (∂ 2 ξ + ∂ 2 η )|ψ| 2 = 0, (6.4.8)
where ρ = ±1. Here ξ, η, t are real variables. Algebro-geometric solutions of (6.4.8) are constructed from solutions ψ, ψ * (6.4.4) of the complexied system, under the reality condition ψ * = ρ ψ.

Let R g be a real compact Riemann surface with an anti-holomorphic involution τ . Denote by R g (R) the set of xed points of the involution τ . Let us choose the homology basis satisfying (3.2.1). Then the solutions of (6.4.8) are given by: Theorem 6.4.2. Let a, b ∈ R g (R) be distinct points with local parameters satisfying k a (τ p) = k a (p) for any p in a neighbourhood of a, and k b (τ p) = k b (p) for any p in a neighbourhood of b. Denote by (A, B, ) the standard generators of the relative homology group H 1 (R g , {a, b}) (see

Section 3.3). Let d R ∈ R g , T ∈ Z g , and dene d = d R + iπ 2 (diag(H) -2 T). Moreover, take θ, h ∈ R, κ1 , κ 2 ∈ R \ {0} and put κ 1 = -ρ κ2 1 κ 2 q 2 (a, b) exp 1 2 BM, M + r + d, M , (6.4.9)
where M ∈ Z g is dened in (3.3.9). Then the following functions ψ and φ are solutions of the

DS1 ρ equation ψ(ξ, η, t) = |A| e iθ Θ(Z -d + r) Θ(Z -d) exp -i G 1 ξ + G 2 η -G 3 t 2 , (6.4.10)
φ(ξ, η, t) = 1 2 (ln Θ(Z -d)) ξξ + 1 2 (ln Θ(Z -d)) ηη + h 4 , (6.4.11)
where

|A| = |κ 1 κ 2 q 2 (a, b)| exp { d R , M } ,
and the remaining notation is as in Theorem 6.4.1.

The case where V a + V b = 0 and κ 1 = κ 2 is treated at the end of this section. It corresponds to solutions of the nonlinear Schrödinger equation.

Proof. The proof follows the lines of Section 6.3.2, where similar statements were proven for the n-NLS s equation. Let us check that under the conditions of the theorem, the functions ψ and ψ * (6.4.4) satisfy the reality condition (6.4.3). Take κ 1 , κ 2 ∈ R. First of all, invariance with respect to the anti-involution τ of the points a and b implies the reality of vector (6.4.5). Moreover, from (3.2.2) and (3.3.9) we get

r = -r -2iπN -BM, (6.4.12)
where N, M ∈ Z g are dened in (3.3.9) and satisfy

2 N + HM = 0. (6.4.13)
From (6.1.1), it is straightforward to see that the scalars K 1 (a, b) and K 2 (a, b) dened by (6.1.2) and (6.1.3) satisfy

K 1 (a, b) = K 1 (a, b) -V a , M , K 2 (a, b) = K 2 (a, b) + W a , M , (6.4.14)
which implies

G 1 = G 1 -κ 1 V a , M , G 2 = G 2 -κ 2 V b , M , G 3 = G 3 + κ 2 1 W a , M + κ 2 2 W b , M .
Therefore, the reality condition (6.4.3) together with (6.4.4) leads to

|A| 2 = -ρ κ 1 κ 2 q 2 (a, b) Θ(Z -d -r) Θ(Z -d + iπ diag(H)) Θ(Z -d -r + iπ diag(H)) Θ(Z -d) × exp 1 2 BM, M + r + d -iπ diag(H), M , (6.4.15)
taking into account the action (3.2.4) of the complex conjugation on the theta function, and the quasi-periodicity (2.2.3) of the theta function. Let us choose a vector d ∈ C g such that

d ≡ d -iπ diag(H) (mod 2iπZ g + BZ g ),
which is, since dd is purely imaginary, equivalent to d = diπ diag(H) + 2iπT, for some T ∈ Z g . Here we used the action (3.2.3) of the complex conjugation on the matrix of B-periods B, and the fact that B has a negative denite real part. Hence, the vector d can be written as

d = d R + iπ 2 (diag(H) -2 T), (6.4.16)
for some d R ∈ R g and T ∈ Z g . Therefore, all theta functions in (6.4.15) cancel and (6.4.15)

becomes

|A| 2 = -ρ κ 1 κ 2 q 2 (a, b) exp 1 2 BM, M + r + d, M . (6.4.17)
The reality of the right hand side of equality (6.4.17) can be deduced from formula (6.2.10) for the argument of q 2 (a, b). Moreover, it is straightforward to see from (6.4.16) and (6.4.13) that exp{ d, M } is also real. Since κ 1 , κ 2 are arbitrary real constants, we can choose κ 1 as in (6.4.9), which leads to

|A| 2 = κ1 κ 2 q 2 (a, b) exp 1 2 BM, M + r + d, M 2 = |κ 1 κ 2 q 2 (a, b)| 2 exp {2 d R , M } .
Functions ψ and φ given in (6.4.10) and (6.4.11) describe a family of algebro-geometric solutions of (6.4.8) depending on: a real Riemann surface (R g , τ ),

two distinct points a, b ∈ R g (R), local parameters k a , k b which satisfy k a (τ p) = k a (p) and k b (τ p) = k b (p), and arbitrary constants d R ∈ R g , T ∈ Z g , θ, h ∈ R, κ1 , κ 2 ∈ R \ {0}.
Note that because of the periodicity properties of the theta function, without loss of generality, the vector T can be chosen in the set {0, 1} g . The case where the Riemann surface is dividing and T = 0 is of special importance, because the related solutions are smooth, as explained in Proposition 6.3.2.

Reality condition and solutions of the DS2 ρ equation

Let us consider the DS2 ρ equation

i ψ t + 1 2 (∂ 2 ξ + ∂ 2 η )ψ + 2 φ ψ = 0, ∂ ξ ∂ η φ + ρ 1 2 (∂ 2 ξ + ∂ 2 η )|ψ| 2 = 0, (6.4.18)
where ρ = ±1. Here t is a real variable and variables ξ, η satisfy ξ = η. Analogously to the case where ξ and η are real variables, algebro-geometric solutions of (6.4.18) are constructed from solutions ψ, ψ * (6.4.4) of the complexied system by imposing the reality condition ψ * = ρ ψ.

Let R g be a real compact Riemann surface with an anti-holomorphic involution τ . Let us choose the homology basis satisfying (3.2.1). Then the solutions of (6.4.18) are given by: Theorem 6.4.3. Let a, b ∈ R g be distinct points such that τ a = b, with local parameters satisfying k b (τ p) = k a (p) for any point p in a neighbourhood of a. Denote by (A, B, ) the standard generators of the relative homology group H 1 (R g , {a, b}) (see Section 3.3). Let T, L ∈ Z g satisfy

2 T + HL = diag(H), (6.4.19)
and dene d = 1 2 Re(B)L + id I , for some d I ∈ R g . Moreover, take θ, h ∈ R and κ 1 , κ 2 ∈ C \ {0} such that κ 1 = κ 2 . Let us consider the following functions ψ and φ:

ψ(ξ, η, t) = |A| e iθ Θ(Z -d + r) Θ(Z -d) exp -i G 1 ξ + G 2 η -G 3 t 2 , (6.4.20) φ(ξ, η, t) = 1 2 (ln Θ(Z -d)) ξξ + 1 2 (ln Θ(Z -d)) ηη + h 4 , (6.4.21)
where

|A| = |κ 1 | |q 2 (a, b)| 1/2 exp -1 2 Re(r), L . Then, 1. if
intersects the set of real ovals of R g only once, and if this intersection is transversal, functions ψ and φ are solutions of DS2 ρ with ρ = e iπ N,L , 2. if does not cross any real oval, functions ψ and φ are solutions of DS2 ρ with ρ =e iπ N,L , where N ∈ Z g is dened in (3.3.1), and the remaining notation is as in Theorem 6.4.1.

Proof. Analogously to the proof of Theorem 6.4.2, let us check that under the conditions of the theorem, the functions ψ and ψ * (6.4.4) satisfy the reality condition (6.4.3). First of all, due to the fact that points a and b are interchanged by τ , the vector Z (6.4.5) satises Z = -Z. (6.4.22) In fact, using the expansion (2.3.3) of the normalized holomorphic dierentials ω j near a we get

τ * ω j (a)(p) = V b,j + W b,j k a (p) + o(k a (p)) dk a (p),
for any point p in a neighbourhood of a. Then by (3.2.2), the vectors V a , V b and W a , W b appearing in the vector Z satisfy

V a = -V b , W a = -W b , (6.4.23)
which leads to (6.4.22). From (3.2.2) and (3.3.1) we get r = r -2iπN, (6.4.24) where N ∈ Z g is dened in (3.3.1). By Proposition 6.2.2, the scalar q 2 (a, b) is real. From (6.1.1), it is straightforward to see that the scalars K 1 (a, b) and K 2 (a, b), dened in (6.1.2) and ( 6

.1.3), satisfy K 1 (a, b) = K 1 (b, a), K 2 (a, b) = K 2 (b, a),
which leads to G 1 = G 2 and G 3 ∈ R. Therefore, the reality condition (6.4.3) together with (6.4.4) lead to for some d I ∈ R g , where 2 T + HL = diag(H). With this choice of vector d, (6.4.25) becomes

|A| 2 = -ρ |κ 1 | 2 q 2 (a, b) Θ(Z -d -r) Θ(Z + d + iπ diag(H)) Θ(Z + d -r + iπ diag(H)) Θ(Z -d) , (6.4
|A| 2 = -ρ |κ 1 | 2 q 2 (a, b) e -r,L . (6.4.27)
Moreover, from (6.4.24) we deduce that equality (6.4.27) holds only if

ρ = -sign(q 2 (a, b)) e -iπ N,L .
The sign of q 2 (a, b) in the case where τ a = b is given in Proposition 6.2.2, which completes the proof.

Corollary 6.4.1. From Theorem 6.4.3 we deduce that:

1. if R g is dividing and each component of the vector L is even, functions (6.4.20) and (6.4.21) are solutions of DS2 + , 2. if R g does not have real oval and each component of the vector L is even, functions (

and (6.4.21) are solutions of DS2 -.

Remark 6.4.1. To construct solutions associated to non-dividing Riemann surfaces, we rst observe from (6.4.19) that all components of the vector L cannot be even, since for non-dividing Riemann surfaces the vector diag(H) contains odd coecients (see Section 3.2). In this case, the vector N has to be computed to determine the sign ρ in Theorem 6.4.3. This vector N is dened by the action of τ on the relative homology group H 1 (R g , {a, b}) (see (3.3.1)). It follows that we do not have a general expression for this vector.

To ensure the smoothness of solutions (6.4.20) and (6.4.21) for all complex conjugate ξ, η, and t ∈ R, the function Θ(Zd) of the variables ξ, η, t must not vanish. Following the work by Dubrovin and Natanzon [START_REF] Dubrovin | Real theta function solutions of the Kadomtsev-Petviashvili equation[END_REF] on smoothness of algebro-geometric solutions of the Kadomtsev-Petviashvili equation (KP1), in the case where R g admits real ovals we get Proposition 6.4.1. Functions (6.4.20) and (6.4.21) are smooth solutions of DS2 + if the curve is an M-curve and d ∈ iR g . Assume that the curve admits real ovals and functions (6.4.20), (6.4.21) are smooth solutions of DS2 ρ for any vector d in a component Tv (3.4.6) of the Jacobian, then the curve is an M-curve, d ∈ iR g and ρ = +1.

Proof. By (6.4.22) and (6.4.26) the vector Zd belongs to the set S 2 introduced in (3.4.3). Hence by Proposition 3.4.2, the solutions are smooth if the curve is an M-curve and Zd ∈ iR g which implies d ∈ iR g by (6.4.22) (and therefore L = T = 0). Remark 6.4.2. Smoothness of the solutions to the DS2 -equation was investigated in [START_REF] Malanyuk | Finite-gap solutions of the Davey-Stewartson equations[END_REF]. It is proved that the obtained solutions are smooth if and only if the associated Riemann surface does not have real oval, and if there are no pseudo-real functions of degree g -1 on it (i.e., a function which satises f (τ p) = -f (p) -1 ).

i ψ t + ψ ξξ + 2 ρ |ψ| 2 + q 1 (a, b) + h 4 ψ = 0, (6.4.28)
which can be transformed to the NLS ρ equation

i ψt + ψξξ + 2ρ | ψ| 2 ψ = 0, by the substitution ψ(ξ, t) = ψ(ξ, t) exp -2i q 1 (a, b) + h 4 t . If all branch points of R g are real, ψ is a smooth solution of NLS -. If they are all pairwise conjugate, ψ is a smooth solution of NLS + . Proof. If a, b ∈ R g are such that σa = b and if the local parameters satisfy k a (p) = k b (σp), one has V a + V b = 0, W a + W b = 0. (6.4.29)
To verify (6.4.29), we use the action σA k = -A k of the involution σ on the A-cycles of the homology basis. Hence by (2.1.9) we have

2iπδ jk = σA k σ * ω j = - A k σ * ω j ,
for j, k = 1, . . . , g. It follows that the holomorphic dierentialσ * ω j satises the normalization condition (2.1.9), which implies, by virtue of uniqueness of the normalized holomorphic dierentials,

σ * ω j = -ω j , (6.4.30)
for j = 1, . . . , g. Using (2.3.3) we obtain

σ * ω j (a)(p) = (V b,j + W b,j k b (σp) + o(k b (σp))) dk b (σp) = (V b,j + W b,j k a (p) + o(k a (p))) dk a (p),
for any p in a neighbourhood of a, which by (6.4.30) implies (6.4.29).

Therefore, when the Riemann surface associated to solutions of DS1 ρ is hyperelliptic, assum- ing that a and b satisfy σa = b, and κ 1 = κ 2 = 1, by (6.4.29) and (2.3.10), under the reality condition ψ * = ρ ψ, the function φ in (6.4.11) satises

φ(ξ, η, t) = ρ |ψ| 2 + q 1 (a, b) + h 4 .
Hence the function ψ (6.4.10) becomes solution of equation (6.4.28), with ρ = ±1 depending on the reality of the branch points, as explained in Section 6.3.3.

Solutions of the NLS equation obtained in this way coincide with those in [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF].

Chapter 7

Degeneracy of algebro-geometric solutions

We present new solutions in terms of elementary functions of the multi-component nonlinear Schrödinger equations (6.3.1) and known solutions of the Davey-Stewartson equations (6.4.1)

as multi-soliton, breather, dromion and lump solutions. These solutions are given in a simple determinantal form and are obtained as limiting cases in suitable degenerations of previously derived algebro-geometric solutions. In particular we present for the rst time breather and rational breather solutions of the multi-component nonlinear Schrödinger equations.

The chapter is organized as follows: Section 7.1 provides technical tools for the degeneration of Riemann surfaces. We present a method which allows to degenerate algebro-geometric solutions associated to an arbitrary Riemann surface that can be applied to general integrable equations.

In Section 7.2 solutions in terms of elementary functions to the complexied n-NLS equation are derived by degenerating algebro-geometric solutions; for an appropriate choice of the parameters one gets multi-solitonic solutions, and for the rst time breather and rational breather solutions to the multi-component NLS equation. In Section 7.3 a similar program is carried out for the DS equations; well known solutions as multi-solitons, dromion or lump are rediscovered from an algebro-geometric approach.

Fay's identity and degenerate Riemann surfaces

It is well known that solutions in terms of theta functions are almost periodic due to the periodicity properties of the theta functions. In the limit when the Riemann surface degenerates to a surface of genus zero, periods of the surface diverge, and the theta series breaks down to elementary functions. Whereas this procedure is well-known in the case of a hyperelliptic surface, i.e., a two-sheeted branched covering of the Riemann sphere, where such a degeneration consists in colliding branch points pairwise, it has not been applied so far to theta-functional solutions on non-hyperelliptic surfaces.

We present here a method to treat this case based on the uniformization theorem for Riemann surfaces. In particular, we show that the theta function tends to a nite sum of exponentials in the limit when the arithmetic genus of the associated Riemann surface drops to zero, and explicitly provide the quantities (independent of the vector z) appearing in identities (2.3.10) and (6.1.1) in this limit. As illustrated in Section 7.2 and 7.3, particular solutions of n-NLS and DS such as multi-solitons, well known in the theory of soliton equations, arise from such degenerations of algebro-geometric solutions.

Uniformization map and degeneration to genus zero

Let us rst recall some techniques used for degenerating Riemann surfaces (see [START_REF] Fay | Theta functions on Riemann surfaces[END_REF] for more details). There exist basically two ways for degenerating a Riemann surface by pinching a cycle: a cycle homologous to zero in the rst case, and a cycle non-homologous to zero in the second case. The rst degeneration leads to two Riemann surfaces whose genera add up to the genus of the pinched surface, whereas the limiting situation for the second degeneration is one Riemann surface of genus g-1 with two points identied, g being the genus of the non-degenerated surface.

In both cases, locally one can identify the pinched region to a hyperboloid

y 2 = x 2 -, (7.1.1)
where > 0 is a small parameter, such that the vanishing cycle coincides with the homology class of the closed countour around the cut [-√ , √ ] in the x-plane. In what follows, we deal with the degeneration of the second type and make consecutive pinches until the surface degenerates to genus zero.

To degenerate the Riemann surface R g of genus g into a Riemann surface R 0 of genus zero, we pinch all A i -cycles into double points. After desingularization one gets R 0 , and each double point corresponds to two dierent points on R 0 , denoted by u i and v i for i = 1, . . . , g. In this limit, holomorphic normalized dierentials ω i become normalized dierentials of the third kind with poles at u i and v i . Note that the normalized dierential of the second kind Ω (N ) a with a pole of order N > 1 at a remains a dierential of the second kind with the same order of the pole after degeneration to genus zero. We keep the same notation for the dierential of the second kind on the degenerated surface.

The compact Riemann surface R 0 of genus zero is conformally equivalent to the Riemann sphere with the coordinate w. This mapping between R 0 and the w-sphere is called the uniformization map and we denote it by w(p) = w for any p ∈ R 0 . Therefore, in what follows we let R 0 stand also for the Riemann sphere with the coordinate w.

Meromorphic dierentials on R 0 can be constructed using the fact that in genus zero, such dierentials are entirely dened by their behaviors near their singularities. This leads to the following third and second kind dierentials on R 0 .

• Dierentials of the third kind:

Ω v i -u i = 1 w -w v i - 1 w -w u i dw. (7.1.2)
• Dierentials of the second kind:

Ω (2) a = 1 k a (w a ) dw (w -w a ) 2 , (7.1.3)
where k a is a local parameter in a neighbourhood of w a ∈ R 0 and the prime denotes the derivative with respect to the argument. This is the dierential on R 0 , obtained from Ω (2) a (2.1.6) dened on R g , in the limit as the surface R g degenerates to R 0 . The factor (k a (w a )) -1 ensures that the biresidue of Ω (2) a with respect to the local parameter k a is 1 as before the degeneration.

Degenerate theta function

To study the theta function with zero characteristic in the limit when the genus tends to zero, let us rst analyse the behavior of the matrix B of B-periods of the normalized holomorphic dierentials. Since holomorphic normalized dierentials ω i become dierentials of the third kind with poles at u i and v i , for a small parameter > 0, elements (B) ik of the matrix B have the following behavior:

(B) ik = v i u i Ω v k -u k + O( ), i = k, (7.1.4)
(B) kk = ln + O(1).

Therefore, the real parts of diagonal terms of the Riemann matrix tend to -∞ when tends to zero, that is when the Riemann surface degenerates into the Riemann surface R 0 . It follows that the theta function (2.2.1) with zero characteristic tends to one, since only the term corresponding to the vector m = 0 in the series may give a non-zero contribution.

To get non constant solutions of (6.3.1) and (6.4.1) after the degeneration of the Riemann surface, let us write the argument of the theta-function in the form Z -D, where D is a vector with components

D k = (1/2) (B) kk +d k , for some d k ∈ C independent of . Hence for any Z ∈ C g one gets lim →0 Θ(Z -D) = m∈{0,1} g exp    1≤i<k≤g (B) ik m i m k + g k=1 m k (Z k -d k )    . (7.1.5)
Here we use the same notation for the quantities (B) ik on the degenerated surface. The expression in the right hand side of (7.1.5) can be put into a determinantal form (see Proposition 7.1.1 below) which will be used in the whole chapter. This determinantal form can be obtained from the following representation of the components (B) ik after degeneration, obtained from (7.1.2) and (7.1.4),

(B) ik = ln w v i -w v k w v i -w u k w u i -w u k w u i -w v k . (7.1.6)
Hence, following [START_REF] Matsuno | Multiperiodic and multisoliton solutions of a nonlocal nonlinear Schrödinger equation for envelope waves[END_REF] one gets: Proposition 7.1.1. For any z ∈ C g , the following holds:

m∈{0,1} g exp    1≤i<k≤g (B) ik m i m k + g k=1 m k z k    = det(T), (7.1.7)
where T is a g × g matrix with entries given by On the other hand, let R 2 be the corresponding coecient on the right-hand side of (7.1.7).

(T) ik = δ i,k + w v i -w u i w v i -w u k e 1 2 (z i +z k ) .
Since

det(T) = e g j=1 z j det δ i,k e -z i + w v i -w u i w v i -w u k 1≤i,k≤g , we obtain R 2 = det w v i -w u i w v i -w u k 1≤i,k≤n = n j=1 (w v j -w u j ) det 1 w v i -w u k 1≤i,k≤n
.

By virtue of the formula for the Cauchy determinant:

det 1 w v i -w u k 1≤i,k≤n = (-1) n(n-1) 2 1≤i<k≤n (w v k -w v i )(w u k -w u i ) n i,k=1 (w v i -w u k ) , R 2 becomes R 2 = 1≤i<k≤n (w v k -w v i )(w u k -w u i ) (w u k -w v i )(w v k -w u i ) = exp    1≤i<k≤n (B) ik    = R 1 ,
which completes the proof.

Constants in the limit

The next step is to give explicitly the quantities (independent of the vector z) appearing in (2.3.10) and (6.1.1), i.e., V a , W a , r = b a ω, q 2 , etc, after the degeneration to genus zero. We use the same notation for these quantities on the degenerated surface. For any distinct points a, b ∈ R 0 , it follows from (2.3.3) and (7.1.2) that: .11) for k = 1, . . . , g. Moreover, from the integral representation (6.2.1) of q 2 (a, b), using (7.1.2) one gets

V a,k = 1 k a (w a ) 1 w a -w v k - 1 w a -w u k , (7.1.9) W a,k = 1 k a (w a ) 2 - 1 (w a -w v k ) 2 + 1 (w a -w u k ) 2 - k a (w a ) k a (w a ) 2 V a,k , (7.1.10) r k = ln w b -w v k w b -w u k w a -w u k w a -w v k , (7.1
q 2 (a, b) = 1 k (w a )k (w b )(w a -w b ) 2 .
(7.1.12) Putting z = 0 in (2.3.10) and taking the limit → 0 leads to q 1 (a, b) =q 2 (a, b), (7.1.13) due to the fact that the theta function tends to one and that its partial derivatives tend to zero.

To compute the scalar K 1 (a, b) in the limit, let us rst prove the following lemma: Lemma 7.1.1. Let a, b be distinct points on a compact Riemann surface R g of genus g > 0.

Denote by k a and k b local parameters in a neighbourhood of a and b respectively. Then the scalar K 1 (a, b) dened in (6.1.2) admits the following integral representation:

K 1 (a, b) = lim ã→a ã c Ω (2) a (p) + 1 k a (ã) - b c Ω (2)
a (p), (7.1.14) where c is an arbitrary point on R g .

Proof. Let us denote by k x a local parameter in a neighbourhood of a point x ∈ R g . To prove q=a . Since δ is a non-singular odd characteristic, one has

Θ[δ] ã q = γ 1 (k a (ã) -k a (q)) + γ 2 2 (k a (ã) -k a (q)) 2 + o (k a (ã) -k a (q)) 2 (7.1.16)
for any ã and q in a neighbourhood of a, where γ 1 = 0. From (7.1.15) and (7.1.16) we deduce that

ã c Ω (2) a (p) = - 1 k a (ã) + 1 2 γ 2 γ 1 -γ + O(k a (ã)
).

(7.1.17)

Moreover, by (7.1.16) it can be seen that

γ 1 = -∂ kq Θ[δ] ã q q=ã = D ã Θ[δ](0), (7.1.18) γ 2 = ∂ 2 kq Θ[δ] ã q q=ã = D ã Θ[δ](0), (7.1.19)
where we used the fact that D

D a Θ[δ](0) D a Θ[δ](0) + D a ln Θ[δ] b a .
The proof is completed taking into account the denition (6.1.2) of K 1 (a, b).

Let us discuss solutions of n-NLS in genus zero. Consider the following meromorphic function f on the sphere:

f (w) = α n+1 i=1 w -w a i w -w b i (7.2.2)
where w a j = w b k for all j, k, w a j = w a k for j = k, and α ∈ C. Without loss of generality, put α = 1. This function is of degree n + 1 on the sphere, hence it represents a genus zero (n + 1)-sheeted branched covering of CP 1 . Recall that a meromorphic function f on the sphere is called real if its zeros as well as its poles are real or pairwise conjugate.

If not stated otherwise, the local parameter in a neighbourhood of a regular point w a (i.e., f (w a ) = 0) is chosen to be k a (w) = f (w)f (w a ) for any w in a neighbourhood of w a . Solutions of the complexied system (6.3.2) associated to the meromorphic function f (7.2.2) on the sphere are given by: Proposition 7.2.1. Let j, k ∈ N satisfy 1 ≤ j ≤ n and 1 ≤ k ≤ g. Let f be a meromorphic function (7.2.2) of degree n + 1 on the sphere, with complex zeros {w a i } n+1 i=1 and complex poles {w b i } n+1 i=1 . Let d ∈ C g and A j = 0 be arbitrary constants. Moreover, assume that w u k , w

v k ∈ C satisfy f (w u k ) = f (w v k ). (7.2.3)
Then the following functions are solutions of the complexied system (6.3.2)

ψ j (x, t) = A j det(T j,1 ) det(T j,0 ) exp{-i (E j x -F j t)}, ψ * j (x, t) =
q 2 (a n+1 , a j ) A j det(T j,-1 ) det(T j,0 ) exp{i (E j x -F j t)}.

(7.2.4)

For β = -1, 0, 1, T j,β denotes the g × g matrix with entries (7.1.8) where z

(j) k = Z k -d k + β r j,k . Here Z k = i V a n+1 ,k x + i W a n+1 ,k t,
where the scalars V a n+1 ,k and W a n+1 ,k are dened in (7.1.9) and (7.1.10), and r j,k is dened in (7.1.11) with w a := w a n+1 and w b := w a j . The scalars E j and F j are given by

E j = K 1 (a n+1 , a j ), F j = -(K 1 (a n+1 , a j )) 2 + 2 n k=1 q 2 (a n+1 , a k ),
where q 2 (a n+1 , a j ) and K 1 (a n+1 , a j ) are dened in (7.1.12) and (7.1.20).

Proof. Consider solutions (6.3.4) associated to a Riemann surface R g of genus g > 0, and assume f (a i ) = 0 for i = 1, . . . , n+1. Pinch all A-cycles of the associated Riemann surface R g into double points, as explained in Section 7.1. After desingularization, the meromorphic function f of degree n + 1 on R g becomes a meromorphic function of degree n + 1 on the sphere, given in general form by (7.2.2). In the limit considered here, the theta function tends to the determinantal form (7.1.7). Quantities dened on the degenerated surface independent of the variables x and t were constructed in Section 7.1.3 and are given in (7.1.9)-(7.1.13) and (7.1.20), (7.1.21). Condition (7.2.3) follows from the fact that double points appearing after degeneration of R g are desingularized into two distinct points w u k and w v k having the same projection under the Proof. Let us check that the functions ψ j and ψ * j dened in (7.2.4) satisfy the reality conditions (6.3.3) with s j = sign(f (w a n+1 )f (w a j )). Put A j = |q 2 (a n+1 , a j )| 1/2 in (7.2.4). Then with the above assumptions, it is straightforward to see that ψ * j = s j ψ j where s j = sign(q 2 (a n+1 , a j )), which by (7.1.12) leads to s j = sign(f (w a n+1 )f (w a j )). Moreover, with (7.2.6) it can be seen that condition (7.2.1) is equivalent to

1 |w a n+1 -w v k | 2 + n j=1 f (w a n+1 ) f (w a j ) 1 |w a j -w v k | 2 = 0, (7.2.8)
for k = 1, . . . , N . Therefore, by (7.2.8) the quantity f (w a n+1 )f (w a j ) cannot be positive for all j, which yields s = (1, . . . , 1).

ψ j (x, t) = A j 1 + e Z 1 -d 1 +r j,1 1 + e Z 1 -d 1 e -i (E j x-F j t) .
Example 7.2.2. Bright multi-solitons of n-NLS s .

Bright multi-solitons of the NLS equation presented in [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF] were obtained by collapsing all branch cuts of the underlying hyperelliptic curve of the algebro-geometric solutions. This way they get solutions expressed as the quotient of a nite sum of exponentials similar to dark multi-solitons, except that the modulus of the solutions tends to zero instead of a non-zero constant when the spatial variable tends to innity. Following this approach, a family of bright multi-solitons of n-NLS s is obtained here by further degeneration of (7.2.4).

For the multi-component case there exist two sorts of bright soliton interactions: elastic or inelastic. Inelastic collisions between bright solitons were investigated in [START_REF] Radhakrishnan | Inelastic Collision and Switching of Coupled Bright Solitons in Optical Fibers[END_REF] for the two component case and in [START_REF] Kanna | Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations[END_REF] for the multi-component case. The family of bright multi-solitons of n-NLS s obtained here describes the standard elastic collision with phase shift. Notice that there exist various ways to degenerate algebro-geometric solutions. Therefore, it appears possible that bright solitons with inelastic collision can be obtained by dierent degenerations. for k = 1, . . . , N . Choose γ j ∈ R and put s j = sign(γ j ). Then the following functions give bright N -soliton solutions of n-NLS s ψ j (x, t) = A j e iθ det(K j ) det(M) , (7.2.10) where A j = |γ j | 1/2 |w a j | -1 . Here K j and M are 2N × 2N matrices with entries (K j ) ik and (M) ik given by:

-for i and k even:

(K j ) ik = δ 2,i w u i w u k e 1 2 (z 2 +z k +r j,2 +r j,k ) + δ i,k -δ 2,i δ 2,k + δ 2,k (δ 2,i -1) w u i w u 2 e 1 2 (z i -z 2 +r j,i -r j,2 )
-for i even and k odd:

(K j ) ik = α 2 u k w u i α v i -α u k α v 2 -α v i α v 2 -α u k e 1 2 (z i +z k +r j,i +r j,k )
-for i odd and k even:

(K j ) ik = w v i w v i -w u k e 1 2 (z i +z k +r j,i +r j,k )
-for i and k odd: (K j ) ik = δ i,k -for i, k even, or i, k odd: (M) ik = δ i,k -for i even and k odd:

(M) ik = α u k α v i w u i α v i -α u k e 1 2 (z i +z k )
-for i odd and k even: (M) ik = -

w v i w v i -w u k e 1 2 (z i +z k ) .
Here α v 2k = α u 2k-1 where

α u 2k-1 = n j=1 γ j 1 w a j - 1 w a j -w v 2k-1 . (7.2.11)
Moreover, z k is a linear function of the variables x and t satisfying z 2k = z 2k-1 , given by

z 2k-1 = i α u 2k-1 x + i α 2 u 2k-1 t -d2k-1 .
The scalars rj,k satisfy rj,2k =rj,2k-1 where rj,2k-1 = ln w a jw v 2k-1 w a j w v 2k-1 α u 2k-1 .

Proof. Consider functions (7.2.4) obtained from (6.3.4) for the choice of local parameters k a i :

k a i (w) = (γ i f (w a i )) -1 f (w)
for any w in a neighbourhood of w a i , i = 1, . . . , n + 1, and assume g = 2N . Hence condition (7.2.1) becomes

n+1 i=1 γ i 1 w a i -w v k - 1 w a i -w u k = 0 (7.2.12)
for k = 1, . . . , 2N . Now put A j = |q 2 (a n+1 , a j )| 1/2 in (7.2.4), where q 2 (a n+1 , a j ) = γ n+1 γ j (w a n+1w a j ) -2 .

Choose a small parameter > 0 and dene d k =ln + dk , for k = 1, . . . , 2N, and

w u 2k-1 = w a n+1 + 2 α -1 u 2k-1 , w v 2k = w a n+1 + 2 α -1 v 2k , (7.2.13)
for k = 1, . . . , N . Now put γ n+1 = 2 and consider in the determinant det(T j,1 ) appearing in (7.2.4) the substitution

L 2i -→ L 2i - (T j,1 ) 2i,2 (T j,1 ) 2,2 L 2 ,
for i = 2, . . . , N, where L k denotes the line number k of the matrix T j,1 , and (T j,1 ) i,k denotes the entries of this matrix. In the limit → 0, it can be seen that the functions ψ j given in (7.2.4) converge towards functions (7.2.10), where the following change of parameters (eliminating the parameter w a n+1 ) has been made:

w a j → w a j + w a n+1 , w u 2k → w u 2k + w a n+1 , w v 2k-1 → w v 2k-1 + w a n+1 ,
for j = 1, . . . , n and k = 1, . . . , N . Analogous statements can be made for the functions ψ * j .

By assumption, it is straightforward to see that the functions ψ j and ψ * j obtained in the limit considered here satisfy the reality conditions ψ * j = s j ψ j with s j = sign(γ j ).

Breather and rational breather solutions of n-NLS

Solutions obtained here dier from the dark multi-solitons studied in Section 7.2.2 by the reality condition imposed on parameters w u k and w v k in the solutions (7.2.4) of the complexied system for k = 1, . . . , g. By an appropriate choice of parameters, one gets periodic solutions (breathers) as well as rational solutions (rational breathers). The name `breather' reects the behavior of the prole of the solution which is periodic in time (respectively, space) and localized in space (respectively, time). This appears to be the rst time that explicit breather and rational breather solutions of n-NLS s are given.

In what follows N ∈ N with N ≥ 1.

Multi-Breathers of n-NLS s .

Multi-breather solutions of n-NLS s are given in the following proposition. The N -breather solution corresponds to an elastic interaction between N breathers. Proposition 7.2.4. Let j, k ∈ N satisfy 1 ≤ j ≤ n and 1 ≤ k ≤ N . Let f be a real meromorphic function (7.2.2) of degree n + 1 on the sphere, having n + 1 real zeros {w a i } n+1 i=1 . Choose θ ∈ R and take d ∈ C 2N such that d2k-1 = d2k . Let w u 2k , w u 2k-1 , w v 2k , w v 2k-1 ∈ C satisfy (7.2.3) and

w u 2k = w v 2k-1 , w u 2k-1 = w v 2k . (7.2.14)
Put s j = sign(f (w a n+1 )f (w a j )). Then the following functions dene N -breather solutions of n-NLS s

ψ j (x, t) = A j e iθ det(T j,1 ) det(T j,0 ) exp {-i (E j x -F j t)} , (7.2.15)
where A j = |q 2 (a n+1 , a j )| To simplify the computation of the solutions, we apply transformation (6.3.11) to the solutions (7.2.4), with β and λ given by

β = 1, λ = 1 2 f (w a n+1 ) f (w a n+1 ) -2 . (7.2.16)
Hence, the quantity f (w a n+1 )f (w a n+1 ) -2 V a n+1 ,k in the expression (7.1.10) for the scalar W a n+1 ,k disappears, as well as the quantity 1 2 f (w a n+1 )f (w a n+1 ) -2 in the expression (7.1.20) for the scalar K 1 (a n+1 , a j ).

Example 7.2.5. Figure 7.3 shows a breather solution of the 4-NLS s equation for the vector of signs s = (-1, -1, 1, -1). It corresponds to the following choice of parameters: w a 1 = 10, w a 2 = -5, w a 3 = -1/3, w a 4 = 1/4, w a 5 = 1/2, and w u 1 ≈ 0.55 -0.11i with f (w u 1 ) = 2i, w u 2 ≈ -0.35 + 0.07i with f (w u 2 ) = -2i.

Example 7.2.6. N -rational breathers of n-NLS s , for 1 ≤ N ≤ n.

Here we are interested in solutions of n-NLS s that can be expressed in the form of a ratio of two polynomials (modulo an exponential factor). These solutions, called rational breathers, are neither periodic in time nor in space, but are isolated in time and space. They are obtained from the breather solutions (7.2.15) in the limit when the parameters w v 2k-1 and w u 2k-1 coincide, as well as the parameters w v 2k and w u 2k , for k = 1, . . . N . An appropriate choice of the parameters d i in (7.2.15) for i = 1, . . . , 2N , leads to limits of the form `0/0' in the expression for the breather solutions. Thus, by performing a Taylor expansion of the numerator and denominator in (7.2.15), one gets a family of N -rational breather solutions of n-NLS s . Proposition 7.2.5. Let N, j ∈ N satisfy 1 ≤ N ≤ n and 1 ≤ j ≤ n. Let f be a real meromorphic function (7.2.2) of degree n + 1 on the sphere, having n + 1 real zeros {w a i } n+1 i=1 . Choose θ ∈ R and take d ∈ C 2N such that d2k = d2k-1 for k = 1, . . . , N . Moreover, let w u 2k-1 , w v 2k ∈ C, k = 1, . . . , N, be complex conjugate critical points of the meromorphic function f , i.e., they are solutions of f (w) = 0, which is equivalent to the condition

n+1 i=1 1 f (w a i ) 1 (w -w a i ) 2 = 0.
(7.2.17) Put s j = sign(f (w a n+1 )f (w a j )). Then the following functions give N -rational breathers of n-

NLS s ψ j (x, t) = A j e iθ det(K j,1 ) det(K j,0 ) exp {-i (E j x -F j t)} , (7.2.18)
where A j = |q 2 (a n+1 , a j )| 1/2 . For β = 0, 1, K j,β denotes a 2N × 2N matrix with entries (K j,β ) i,k

given by:

-for i and k even:

(K j,β ) ik = (1 -δ i,k ) 1 w v i -w v k -δ i,k (z k + β rj,k )
-for i even and k odd: (K j,β ) ik = 1 w v iw u k -for i odd and k even:

(K j,β ) ik = - 1 w u i -w v k -for i and k odd: (K j,β ) ik = -(1 -δ i,k ) 1 w u i -w u k -δ i,k (z k + β rj,k ).
Here z i is a linear function of the variables x and t given by

z i = i Va n+1 ,i x + i Ŵa n+1 ,i t -di for i = 1, . . . , 2N, where Va n+1 ,2k = -Va n+1 ,2k-1 and Ŵa n+1 ,2k = -Ŵa n+1 ,2k-1 with Va n+1 ,2k-1 = 1 f (w a n+1 ) 1 (w a n+1 -w u 2k-1 ) 2 , Ŵa n+1 ,2k-1 = - 1 f (w a n+1 ) 2 2 (w a n+1 -w u 2k-1 ) 3
for k = 1, . . . , N . Scalars rj,k satisfy rj,2k =rj,2k-1 and are given by rj,2k-1 = -w a n+1w a j (w a n+1w u 2k-1 ) (w a jw u 2k-1 )

.

Scalars E j , F j are dened by

E j = 1 f (w a n+1 ) (w a j -w a n+1 ) , F j = -E 2 j + 2 n k=1
q 2 (a n+1 , a k ).

Proof. To simplify the expression for the obtained solutions, apply the transformation (6.3.11) to functions (7.2.15) with β and λ as in (7.2.16). Let > 0 be a small parameter and dene d k = dk + iπ, for k = 1, . . . , 2N . Moreover, assume

w v 2k-1 = w u 2k-1 + α v 2k-1 , w u 2k = w v 2k + α u 2k , (7.2.19)
for some α v 2k-1 , α u 2k ∈ C, where k = 1, . . . , N . Note that equation number k of system (7.2.5) can be written as

n+1 j=1 1 f (w a j ) f (w v k ) f (w u k ) (w a j -w v k ) (w a j -w u k ) = - f (w v k ) -f (w u k ) w v k -w u k . (7.2.20)
Hence, in the limit → 0, equation (7.2.20) becomes

n+1 j=1 1 f (w a j ) f (w v 2k-1 ) 2 (w a j -w v 2k-1 ) 2 = -f (w v 2k-1 ), and n+1 j=1 1 f (w a j ) f (w u 2k ) 2 (w a j -w u 2k ) 2 = -f (w u 2k ),
for k = 1, . . . , N . Therefore, choose w v 2k-1 and w u 2k to be distinct critical points of the meromorphic function f for k = 1, . . . N, i.e., they are solutions of f (w) = 0, in such way that equation (7.2.1) holds in the limit considered here. Since the condition f (w) = 0 is equivalent to solve a polynomial equation of degree 2n, it follows that 1 ≤ N ≤ n. Now take the limit → 0 in (7.2.15). Note that parameters α v 2k-1 , α u 2k cancel in this limit, and the degenerated functions take the form (7.2.18). Remark 7.2.7. Functions (7.2.18) provide a family of rational breather solutions of n-NLS s depending on N complex parameters dk , a real parameter θ, and a real meromorphic function f (7.2.2) dened by 2n + 2 real parameters, chosen such that f has complex conjugate critical points.

Example 7.2.7. With the notation of Proposition 7.2.5 the functions ψ j (7.2.18) for N = 1 are given by ψ j (x, t) = A j e iθ B + (z 1 + rj,1 )(z 1rj,1 )

B + |z 1 | 2 exp {-i (E j x -N j t)} ,
where B = (2 Im(w u 1 )) -2 .

Example 7.2.8. Figure 7.5 shows a rational breather solution of the 4-NLS s equation with s = (1, 1, 1, 1). It corresponds to the following choice of parameters: k a k (w) = f (w a k )f (w) for k = 1, . . . , n + 1, with w a 1 = 3, w a 2 = 5, w a 3 = 7, w a 4 = 0, w a 5 = 4, and w u 1 ≈ 4.53 + 0.56i being a solution of n+1 i=1 (ww a i ) -2 = 0. We observe that the functions ψ 2 and ψ 3 coincide with the Peregrine breather well known in the scalar case [START_REF] Peregrine | Water waves, nonlinear Schrödinger equations and their solutions[END_REF], whereas the functions ψ 1 , ψ 4 belong to a new class of rational breathers which does not exist in the scalar case. This new type of rational breathers emerges due to the higher degree of the meromorphic function associated to the solutions of n-NLS s for n > 1.

Example 7.2.9. This appears to be the rst time that such solutions of DS are derived from algebro-geometric solutions.

Determinantal solutions of the complexied DS equations

Here solutions of the complexied system (6.4.2) are given as a quotient of two determinants. In the next subsections, this particular form will be more convenient to produce special solutions of the DS equations (6.4.1).

ψ(ξ, η, t) = A det(T 1 ) det(T 0 ) exp -i (G 1 ξ + G 2 η -G 3 t 2 ) , ψ * (ξ, η, t) = - κ 1 κ 2 A (w a -w b ) 2 det(T -1 ) det(T 0 ) exp i (G 1 ξ + G 2 η -G 3 t 2 ) , (7.3.1)
ϕ(ξ, η, t) = 1 2 (ln det(T 0 )) ξξ + 1 2 (ln det(T 0 )) ηη + h 4 .

For β = -1, 0, 1, T β denotes the g × g matrix with entries (7.1.8) with z k = Z kd k + β r k . Here the scalars r k are given in (7.1.11) and

Z = i κ 1 V a ξ -i κ 2 V b η + i (κ 2 1 W a -κ 2 2 W b ) t 2 (7.3.2) with V c,k = 1 w c -w v k - 1 w c -w u k , W c,k = - 1 (w c -w v k ) 2 + 1 (w c -w u k ) 2 , (7.3.3) for c ∈ {a, b}. The scalars G 1 , G 2 , G 3 are given by G 1 = κ 1 w b -w a , G 2 = κ 2 w a -w b , G 3 = -G 2 1 -G 2 2 + h. (7.3.4)
Proof. Consider solutions (6.4.4) of system (6.4.2) in the limit when the Riemann surface degenerates to a Riemann surface of genus zero, as explained in Section 7.1.1. In this limit, choose the local parameters k a and k b near a ∈ R 0 and b ∈ R 0 to be the uniformization map between the degenerate Riemann surface R 0 and the w-sphere. Hence, for any w ∈ R 0 in a neighbourhood of w a ∈ R 0 , k a (w) = ww a . Therefore, quantities independent of variables ξ, η and t are obtained from (7.1.9)-(7.1.13) and (7.1.20), (7.1.21).

Remark 7.3.1. Functions (7.3.1) give a family of solutions of the complexied system, involving elementary functions only.

These solutions depend on 3g + 6 complex parameters w a , w b , h, A, κ 1 , κ 2 and w u k , w v k , d k . Varying these parameters we will obtain dierent types of physically interesting solutions in the next subsections.

Multi-solitonic solutions of the DS equations

Soliton solutions of the DS equations were shown to be representable in terms of Wronskian determinants in [START_REF] Anker | On the Soliton Solutions of the Davey-Stewartson Equation for Long Waves[END_REF]. Single soliton and multi-soliton solutions corresponding to the known onedimensional solutions can be obtained from this representation. These solitons are pseudo-onedimensional in the sense that in the (x, y)-plane, they have the same form as one-dimensional solitons in the (x, t)-plane, but that they move with an angle with respect to the axes. The multi-soliton solution describes the interaction of many such solitons each propagating in dierent directions.

In what follows N ∈ N with N ≥ 1.

Dark multi-soliton of DS1 ρ and DS2 + .

Here dark multi-solitons of the DS1 ρ and DS2 + equations are derived from functions (7.3.1) for an appropriate choice of the parameters. They were investigated in [START_REF] Yoshida | A new type of soliton behavior of the Davey-Stewartson equations in a plasma system[END_REF].

Put g = N and A = |κ 1 κ 2 | 1/2 |w a -w b | -1 in (7.3.1). Moreover, assume h ∈ R and d ∈ R N .
Reality condition for DS1 ρ . Let us check that with the following choice of parameters,

w a , w b ∈ R, κ 1 , κ 2 ∈ R \ {0} , w v k = w u k , k = 1, . . . , N, (7.3.5)
functions ψ and ψ * in (7.3.1) satisfy the reality condition ψ * = ρ ψ with ρ =sign(κ 1 κ 2 ). Indeed, this can be deduced from the fact that G 1 , G 2 , G 3 ∈ R, and

det (T β ) = det T β = det (T -β ) , (7.3.6)
since u and v can be interchanged in the proof of (7.1.7). Therefore, functions ψ and φ in (7.3.1) dene dark multi-soliton solutions of DS1 ρ .

Smoothness. The dark multi-soliton solutions obtained here are smooth because the denominator det(T 0 ) of functions ψ and φ (7.3.1) consists of a nite sum of real exponentials (see (7.1.7)), since ξ, η, t are real.

Remark 7.3.2. One gets a family of smooth dark multi-soliton of the DS1 ρ equation, depending on N + 6 real parameters w a , w b , h, κ 1 , κ 2 , d k , a phase θ, and N complex parameters w u k .

Reality condition for DS2 + . Let us check that with the following choice of parameters, Smoothness. To get smooth solutions, additional conditions are needed to ensure that det(T 0 ) does not vanish for all complex conjugate ξ = η. For instance, if

w a = w b , κ 1 = κ 2 , w u k , w v k ∈ R, k = 1, . . . , N, (7.3
w v 1 < w u 1 < w v 2 < w u 2 < . . . < w v N < w u N ,
function ψ * . Now take the limit → 0 in (7.3.1). The function ψ obtained in this limit has the form (7.3.11). Notice that in this limit, the dependence on the parameters w a and w b disappears.

Reality condition for DS1 ρ . It is straightforward to see that, with the following choice of parameters,

κ1 , κ2 ∈ R \ {0} , d2k-1 = d2k , α u 2k-1 = α v 2k , α u 2k = α v 2k-1 , k = 1, . . . , N, (7.3.9)
the functions ψ and ψ * obtained in the limit considered here satisfy the reality condition ψ * = ρ ψ with ρ =sign(κ 1 κ2 ).

Reality condition for DS2 -. In the same way, with the following choice of parameters,

κ1 = κ2 , d2k-1 = d2k , α u 2k-1 = α u 2k , α v 2k-1 = α v 2k , k = 1, . . . , N, (7.3.10)
the functions ψ and ψ * obtained in the considered limit satisfy the reality condition ψ * =ψ.

The solutions. Let θ ∈ R. With (7.3.9), the following functions of the variables ξ, η, t obtained in the considered limit, give bright N -soliton solutions of DS1 ρ where ρ =sign(κ 1 κ2 ) and γ = 0; because of (7.3.10) these functions dene bright N -soliton solutions of DS2 -where γ = 1:

ψ(ξ, η, t) = Â e iθ det(K) det(M) , φ(ξ, η, t) = 1 2 (ln det(M)) ξξ + 1 2 (ln det(M)) ηη + h 4 , (7.3.11)
where  = |κ 1 κ2 | 1/2 . Here K and M are 2N × 2N matrices with entries (K) ik and (M) ik given by:

-for i and k even:

(K) ik = δ i,k -δ 2,i δ 2,k + δ 2,i e 1 2 (z 2 +z k +r 2 +r k ) + δ 2,k (δ 2,i -1) e 1 2 (z i -z 2 +r i -r 2 )
-for i even and k odd:

(K) ik = - α 2 u k α v i -α u k α v 2 -α v i α v 2 -α u k e 1 2 (z i +z k +r i +r k )
-for i odd and k even:

(K) ik = - α v i α u k α v i -α u k e 1 2 (z i +z k +r i +r k ) -for i and k odd: (K) ik = δ i,k , -for i, k even, or i, k odd: (M) ik = δ i,k -otherwise: (M) ik = (-1) i+1 α v i α u k α v i -α u k e 1 2 (z i +z k ) .
Here z i , i = 1, . . . , 2N, is a linear function of the variables ξ, η and t given by:

z 2k-1 = i κ1 α u 2k-1 ξ + i κ2 α v 2k-1 η + i κ2 1 α 2 u 2k-1 + κ2 2 α 2 v 2k-1 t 2 -d2k-1 + γ iπ 2 , z 2k = -i κ1 α v 2k ξ -i κ2 α u 2k η -i κ2 1 α 2 v 2k + κ2 2 α 2 u 2k t 2 -d2k + γ iπ 2 ,

Breather and rational breather solutions of the DS equations

The breather solutions of the DS equation were found in [START_REF] Tajiri | Periodic soliton solutions to the Davey-Stewartson equation[END_REF]. Here a family of breather solutions and rational breather solutions of the DS1 equation are derived from algebro-geometric solutions.

These solutions resemble their 1 + 1 dimensional analogues. In particular, the proles of the corresponding solutions of the DS equation in the (x, y, t) coordinates look in the (x, y) plane as those in the (x, t) coordinates.

Multi-Breathers of DS1 ρ .

The N -breather solution obtained here corresponds to an elastic interaction between N breathers. Put g = 2N and A = |κ 1 κ 2 | 1/2 |w aw b | -1 in (7.3.1). It is straightforward to see that with the following choice of parameters, (7.3.12) for k = 1, . . . , N , functions ψ and ψ * (7.3.1) satisfy the reality condition ψ * = ρ ψ with ρ = sign(κ 1 κ 2 ). Therefore, analogously to the n-NLS equation, the functions ψ and φ in (7.3.1) give N -breather solutions of DS1 ρ . Remark 7.3.5. One gets a family of breather solutions of DS1 ρ depending on 3N complex parameters d 2k-1 , w u 2k-1 , w u 2k and 6 real parameters w a , w b , h, κ 1 , κ 2 and a phase θ.

w a , w b , h ∈ R, κ 1 , κ 2 ∈ R \ {0} , d 2k-1 = d 2k , w v 2k = w u 2k-1 , w v 2k-1 = w u 2k ,
Example 7.3.3. Figure 7.9 shows the evolution in time of the 2-breather solution of DS1 - with the following choice of parameters: Multi-rational breathers of DS1 ρ .

w a = 8, w b = -1, w u 1 = 5 -2i, w u 2 = 2 + i, w u 3 = 3 -i, w u 4 = 1 + 4i, κ 1 = κ 2 = 1, d k = h = 0.
In this part, we deal with rational solutions (modulo an exponential factor) of the DS1 ρ equation.

These solutions are obtained as limiting cases of the breather solutions. The N -rational solutions describe elastic interactions between N rational breathers, and are expressed as a quotient of two polynomials of degree N in the variables ξ, η, t.

Assume g = 2N and put

A = |κ 1 κ 2 | 1/2 |w a -w b | -1 in (7.3.1).
Degeneration. Let > 0 be a small parameter and dene d k = dk + iπ, for k = 1, . . . , 2N, and

w v 2k-1 = w u 2k-1 + α v 2k-1 , w u 2k = w v 2k + α u 2k , (7.3.13)
for k = 1, . . . , N . It is straightforward to see that det(T β ) ≈ 2N P β , where P β is a polynomial of degree 2N with respect to the variables ξ, η and t. Now take the limit → 0 in (7.3.1). The function ψ obtained in this limit is an N -rational breather solution of DS1 ρ given by (7.3.15). Reality condition. Choose w a , w b , h, θ ∈ R and κ1 , κ2 ∈ R \ {0}. Moreover, assume

d2k = d2k-1 , α v 2k-1 = α u 2k , α v 2k = α u 2k-1 , k = 1, 2.
(7.3.17)

Put ρ =sign(κ 1 κ2 ). With (7.3.17), it can be seen that the degenerated functions ψ and ψ * obtained in the considered limit satisfy the reality condition ψ * = ρ ψ. Therefore, the following degenerated functions give the dromion solution of DS1

ρ ψ(ξ, η, t) = Â e iθ e z 1 +z 3 ϕ(ξ, η, t) , φ(ξ, η, t) = 1 2 ∂ ξξ ln {ϕ(ξ, η, t)} + 1 2 ∂ ηη ln {ϕ(ξ, η, t)} + h 4 , (7.3.18)
where ϕ(ξ, η, t) = 1 + A 1 e 2 Re(z 1 ) + A 2 e 2 Re(z 3 ) + A 3 e 2 Re(z 1 )+2 Re(z 3 ) .

Here z k is a linear function of the variables ξ, η, t given by

z 1 = -i κ1 α v 1 ξ -i κ2 1 α 2 v 1 t 2 -d1 , z 3 = -i κ2 α u 3 η -i κ2 2 α 2 u 3 t 2 -d3 .
The constants Â, A 1 , A 2 and A 3 are given by

 = |κ 1 κ2 | 1/2 w a w b (α v 3 -α u 1 ) α v 1 α u 3 , A 1 = w a 4 Im(α v 1 ) Im(α u 1 ) , A 2 = w b 4 Im(α v 3 ) Im(α u 3 ) , A 3 = A 1 A 2 + w a w b 4 Im(α v 1 ) Im(α u 3 ) 1 |α u 1 -α v 3 | 2 .
Moreover, in the case where A 1 > 0, A 2 > 0 and A 3 > 0, functions (7.3.18) are smooth solutions of DS1 ρ .

Remark 7.3.7. i) The functions (7.3.18) dene a family of dromion solutions of DS1 ρ depending on 6 complex parameters d1 , d3 , α u 1 , α v 1 , α u 3 , α v 3 and 6 real parameters w a , w b , κ1 , κ2 , h, θ. ii) In the case where α u 1 , α v 3 ∈ R, one gets localized breathers, namely, the solution oscillates with respect to the time variable (modulus of ψ is constant with respect to t).

Example 7.3.6. Figure 7.13 shows the evolution in time of the dromion solution of DS1 -for the following choice of parameters:

w a = 1, w b = 2, κ1 = κ2 = 1, α u 1 = 2 + i, α v 1 = -1 + i, α u 3 = 1 + 3i, α v 3 = 2i, d1 = d3 = h = 0.
Dierent degenerations can be investigated for larger values of g. The performed functions lead to particular solutions such as dromions which move along sets of straight and curved trajectories, as well as oscillating dromion solutions. We do not discuss these solutions here. Lump of DS2 -.

The lump solutions were discovered in [START_REF] Manakov | Two Dimensional Solitons of the Kadomtsev-Petviashvili Equation and Their Interaction[END_REF] for the KP1 equation, and have been extensively studied. Arkadiev et al. [START_REF] Arkadiev | Inverse scattering transform and soliton solutions for Davey-Stewartson II equation[END_REF] have constructed a family of travelling waves (the lump solutions) of DS2

that we rediscover here.

Let us consider functions ψ, ψ * , φ given in (7.3.1), assume g = 2 and put A = |κ 1 κ 2 | 1/2 |w aw b | -1 . Moreover, consider the following transformation which leaves the system (6.4.2) invariant:

ψ(ξ, η, t) → ψ (ξ + β 1 t, η + β 2 t, t) exp -i β 1 ξ + β 2 η + β 2 1 + β 2 2 t 2 , ψ * (ξ, η, t) → ψ * (ξ + β 1 t, η + β 2 t, t) exp i β 1 ξ + β 2 η + β 2 1 + β 2 2 t 2
, φ(ξ, η, t) → φ (ξ + β 1 t, η + β 2 t, t) , (7.3.19) where

β i = µ i κ -1 i for some µ i ∈ C, i = 1, 2.
Degeneration. Choose a small parameter > 0 and dene d k = iπ + dk , for k = 1, 2, and

w v 1 = w a + α v 1 , w u 1 = w a + α u 1 , w v 2 = w b + α v 2 , w u 2 = w b + α u 2 .
Moreover, put κ k = 2 κk , and µ k = 2 μk for k = 1, 2. Now take the limit → 0 in (7.3.19). The functions ψ and φ obtained in this limit are given in (7.3.20).

Reality condition. Choose w a , w b ∈ C such that w a =w b or w a , w b ∈ R. Take h, θ ∈ R and assume

κ1 = κ2 , μ1 = μ2 , d1 = d2 , α v 1 = α v 2 , α u 1 = α u 2 .
With this choice of parameters, it can be seen that the functions ψ and ψ * obtained in the limit considered here satisfy the reality condition ψ * =ψ.

Outlook

Various classes of solutions to the multi-component NLS equation and the DS equations in terms of elementary functions have been presented as limiting cases of algebro-geometric solutions discussed in Chapter 6. We did not construct all families of solutions that can be found in the literature, but we believe that dierent degenerations will lead to interesting new or known solutions that are not presented here.

In particular, future investigations might address bright multi-solitons of n-NLS with inelastic collision. This novel type of inelastic collision, which is not observed in 1 + 1 dimensional soliton systems, follows from a family of bright soliton solutions having more independent parameters than the ones presented here with standard elastic collision. We also believe that this kind of solutions arises from algebro-geometric solutions after suitable degenerations.

Chapter 8

On the numerical evaluation of algebro-geometric solutions

In the case of hyperelliptic surfaces, ecient algorithms exist even for almost degenerate surfaces.

This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis not related to automorphisms of the curve, we study symplectic transformations to an adapted basis and give explicit formulae for M-curves.

As examples we discuss solutions of the Davey-Stewartson equations ( 6

Hyperelliptic case

In this section we consider concrete examples for the solutions, in terms of multi-dimensional theta functions, to the DS and n-NLS equations on hyperelliptic Riemann surfaces. We rst review the numerical methods to visualize the solutions and discuss how to test the accuracy.

Computation on real hyperelliptic curves

The simplest example of algebraic curves are hyperelliptic curves,

µ 2 = 2g+2 i=1 (λ -λ i ), without branching at innity 2g+1 i=1 (λ -λ i ), with branching at innity
, where g is the genus of the Riemann surface, and where we have for the branch points λ i ∈ C the relations λ i = λ j for i = j. If the number of nite branch points is odd, the curve is branched at innity. Recall that all Riemann surfaces of genus 2 are hyperelliptic, and that the involution σ which interchanges the sheets, σ(λ, µ) = (λ, -µ), is an automorphism on any hyperelliptic curve. A vector of holomorphic dierentials for these surfaces is given by (1, λ, . . . , λ g-1 ) t dλ/µ.

For a real hyperelliptic curve the branch points are either real or pairwise conjugate. As we saw in Example 3.1.1, if all branch points λ i are real and ordered such that λ 1 < . . . < λ 2g+2 , the hyperelliptic curve is an M-curve with respect to both anti-holomorphic involutions τ 1 and τ 2 dened in the example. The other case of interest in the context of smooth solutions to n-NLS s and DS are real curves without real branch point. For the involution τ 1 , a curve given by µ 2 = g+1 i=1 (λλ i )(λλ i ), with λ i ∈ C \ R, i = 1, . . . , g + 1, in this case is dividing (two points whose projections onto C have respectively a positive and a negative imaginary part cannot be connected by a contour which does not cross a real oval), whereas a curve given by µ 2 = -g+1 i=1 (λλ i )(λλ i ) has no real oval, and vice versa for the involution τ 2 .

In the following, we will only consider real hyperelliptic curves without branching at innity and write the dening equation in the form

µ 2 = (λ -ξ)(λ -η) g i=1 (λ -E i )(λ -F i ).
It is possible to introduce a convenient homology basis on the related surfaces, see Figure 8.1 for the case η = ξ. The simple form of the algebraic relation between µ and λ for hyperelliptic curves makes the generation of very ecient numerical codes possible, see, for instance, [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods[END_REF][START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods: KdV and KP solutions[END_REF] for details. These codes allow the treatment of almost degenerate Riemann surfaces, i.e., the case where the branch points almost collide pairwise, where the distance of the branch points is of the order of machine precision: |E i -F i | ∼ 10 -14 . The homology basis Figure 8.1 is adapted to this kind of degeneration. The Abel map b a ω between two points a and b is computed in the following way: the sheet identied at the point a = (λ(a), µ(a)) (where we take for µ the root computed by Matlab) is labeled sheet 1, and at the point (λ(a), -µ(a)), sheet 2. Then the ramication point whose projection to the λ-sphere has the minimal distance to λ(a) is determined. For simplicity we assume always that this is the point ξ in Figure 8.1 (for another branch point, this leads to the addition of half-periods, see e.g. [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]). This means we compute with branch points -2, -1, 0, , 2, 2 + and a = (-1.9) (1) , b = (-1.1) (2) for = 1 on the left and = 10 -10 , the almost solitonic limit, on the right.

N .. E 1 F N+1 E g F g E N+1 F 1 E N F
(see, for instance, [START_REF] Trefethen | Spectral Methods in Matlab[END_REF]) for each of the physical variables. This can be used for an expansion of the computed solution in terms of Chebyshev polynomials, a so-called spectral method having in practice exponential convergence for analytic functions as the ones considered here. Since the derivatives of the Chebyshev polynomials can be expressed linearly in terms of Chebyshev polynomials, a derivative acts on the space of polynomials via a so called dierentiation matrix.

With these standard Chebyshev dierentiation matrices (see [START_REF] Trefethen | Spectral Methods in Matlab[END_REF]), the solution can be numerically dierentiated. The computed derivatives allow to check with which numerical precision the PDE is satised by a numerical solution. With these two independent tests, we ensure that the shown solutions are correct to much better than plotting accuracy (the code reports a warning if the above tests are not satised to better than 10 -6 ).

Solutions to the DS equations

The elliptic solutions are the well known travelling wave solutions and will not be discussed here.

The simplest examples we will consider for the DS solutions are given on hyperelliptic curves of genus 2. As we saw in Section 6.4.2, for DS1 ρ reality and smoothness conditions imply that the branch points of the curve are either all real (M-curve) or all pairwise conjugate (dividing curve). The points a and b must project to real points on the λ-sphere and must be stable under the anti-holomorphic involution τ (we use here τ = τ 1 , as dened in Example 3.1.1 except for DS2 -). For DS2 ρ , we have τ a = b where the projection of a onto the λ-sphere is the conjugate of the projection of b. For DS2 + the curve must have only real branch points (M-curve), whereas for DS2

it must have no real oval.

For DS we will mainly give plots for xed time since for low genus, the solution is essentially travelling in one direction. For higher genus, we show a more interesting time dependence in Figure 8.9.

We rst consider the defocusing variants, DS1 + and DS2 + on M-curves. In genus 2 we study the family of curves with the branch points -2, -1, 0, , 2, 2 + for = 1 and = 10 -10 . In the former case the solutions will be periodic in the (x, y)-plane, in the latter almost solitonic with branch points -2, -1, 0, , 2, 2 + and a = (-1.5 + 2i) (1) , b = (-1.5 -2i) (2) for = 1 on the left and = 10 -10 , the almost solitonic limit, on the right. with branch points -4, -3, -2, -2 + , 0, , 2, 2 + , 4, 4 + and a = (-3.9) (1) , b = (-3.1) (2) for = 1 on the left and = 10 -10 , the almost solitonic limit, on the right. since the Riemann surface is almost degenerate (in the limit → 0 the surface degenerates to a surface of genus 0, and the resulting solutions have been discussed in more detail in Chapter 7). To obtain nontrivial solutions in the solitonic limit, we use d =

1 2 [ 1 1 0 0 ] t in all examples.
In Figure 8.2 it can be seen that these are in fact dark solitons, i.e., the solutions tend asymptotically to a non-zero constant and the solitons thus represent `shadows' on a background of light. The well known features from soliton collisions for (1+1)-dimensional integrable equations, namely, the propagation without change of shape, and the unchanged shape and phase shift after the collision, can be seen here in the (x, y)-plane.

The corresponding solutions to DS2

+ can be seen in Figure 8. with branch points -4, -3, -2, -2+ , 0, , 2, 2+ , 4, 4+ and a = (-1.5+2i) (1) , b = (-1.5-2i) (1) for = 1 on the left and = 10 -10 , the almost solitonic limit, on the right. with branch points -2 ± i, -1 ± i, 1 ± i and a = (-4) (1) , b = (-3) (2) . The square modulus of the solution is shown on the left, real and imaginary parts on the right.

In the same way one can study on a genus 4 hyperelliptic curve the formation of the dark 4-soliton for these two equations. We consider the curve with branch points -4, -3, -2, -2 + , 0, , 2, 2 + , 4, 4 + for = 1 and = 10 -10 , and use d = Solutions to the focusing variants of these equations can be obtained on hyperelliptic curves with pairwise conjugate branch points. For such solutions the solitonic limit cannot be obtained as above since the quotient of theta functions in (6.4.10) and (6.4.20) tends to a constant in this case. To obtain the well-known bright solitons (solutions tend to zero at spatial innity) in this way, the hyperelliptic curve has to be completely degenerated (all branch points must collide pairwise to double points) which leads to limits of the form '0/0' in the expression for the solutions (6.4.10), (6.4.20) which are not convenient for a numerical treatment; see Chapter 7 with branch points -2 ± i, -1 ± i, ±i, 1 ± i, 2 ± i and a = (-4) (1) , b = (-3) (2) . Figure 8.8: Solution to the DS2 -equation at t = 0 on a hyperelliptic curve of genus 2 with branch points -2 ± i, -1 ± i, 1 ± i and a = (-1.5 + 2i) (1) , b = (-1.5 -2i) (2) .

Chapter 8. On the numerical evaluation of algebro-geometric solutions for an analytic discussion. Therefore we only consider non-degenerate hyperelliptic curves here.

To obtain smooth solutions, we use d = 0. A solution in genus 2 of the DS1 -equation is studied on the curve with the branch points -2 ± i, -1 ± i, 1 ± i in Figure 8.6.

A typical example of a DS1

solution on a hyperelliptic curve of genus 4 with branch points -2 ± i, -1 ± i, ±i, 1 ± i, 2 ± i is shown in Figure 8. -equation for several values of t on a hyperelliptic curve of genus 4 with branch points -2 ± i, -1 ± i, ±i, 1 ± i, 2 ± i and a = (-1.5 + 2i) (1) , b = (-1.5 -2i) (2) .

Solutions to the n-NLS s equation

As we saw in Section 6.3, a straightforward way to obtain solutions to the n-NLS s equation is given on an (n + 1)-sheeted branched covering of the complex plane, an approach that will be studied in more detail in the next section. As can be seen from the proof of Theorem 6.3.1, the crucial point in the construction of the solutions (6.3.14) of n-NLS s is the fact that n+1 k=1 V a k = 0. This implies that it is also possible to construct smooth theta-functional n-NLS s solutions on hyperelliptic surfaces by introducing constants γ k via n+1 k=1 γ k V a k = 0, see Theorem 6.3.4. As an example we consider, as for DS in genus 2, the family of curves with the branch points -2, -1, 0, , 2, 2 + for = 1 and = 10 -10 . In the former case the solutions will be periodic in the (x, t)-plane, in the latter almost solitonic. To obtain nontrivial solutions in the solitonic limit, we use d = In Figure 8.10 we show the case a 1 = (-1.9) (1) , a 2 = (-1.1) (1) and a 3 = (-1.8) (1) , which leads to a solution of 2-NLS ŝ with ŝ = (-1, -1). Interchanging a 2 and a 3 in the above example, we obtain a solution to 2-NLS ŝ with ŝ = (1, -1) in Figure 8.11. (6.3.39) to the 2-NLS ŝ equation with ŝ = (-1, -1) on a hyperelliptic curve of genus 2 with branch points -2, -1, 0, , 2, 2 + and a 1 = (-1.9) (1) , a 2 = (-1.1) (1) and a 3 = (-1.8) (1) for = 1 on the left and = 10 -10 , the almost solitonic limit, on the right. (6.3.39) to the 2-NLS ŝ equation with ŝ = (1, -1) on a hyperelliptic curve of genus 2 with branch points -2, -1, 0, , 2, 2 + and a 1 = (-1.9) (1) , a 2 = (-1.8) (1) and a 3 = (-1.1) (1) for = 1 on the left and = 10 -10 , the almost solitonic limit, on the right. points -4, -3, -2, -2 + , 0, , 2, 2 + , 4, 4 + and = 1 for a 1 = (-3.9) (1) , a 2 = (-3.7) (1) , a 3 = (-3.5) (1) , a 4 = (-3.3) (1) and a 5 = (-3.1) (1) , which leads to ŝ = (1, -1, 1, -1), on the left, and for a 1 = (-3.9) (1) , a 2 = (-3.7) (1) , a 3 = (-3.5) (1) , a 4 = (-3.1) (1) and a 5 = (-3.3) (1) , which leads to ŝ = (-1, 1, -1, -1) on the right.

Solutions of 4-NLS ŝ can be studied in the same way on the hyperelliptic curve of genus 4

with branch points -4, -3, -2, -2 + , 0, , 2, 2 + , 4, 4 + . We use d = 1 2 [ 1 1 1 1 0 0 0 0 ] t and the points a 1 = (-3.9) (1) , a 2 = (-3.7) (1) , a 3 = (-3.5) (1) , a 4 = (-3.3) (1) and a 5 = (-3.1) (1) . Since the vectors V a j and W a j are very similar in this case, the same is true for the functions ψ j . Therefore, we will only show the square modulus of the rst component ψ 1 in Figure 8.12 for ) on a hyperelliptic curve of genus 2 with branch points -2 ± i, -1 ± i, 1 ± i and a 1 = (-1.9) (1) , a 2 = (-1.8) (2) and a 3 = (-1.1) (1) . ŝ = (1, -1, 1, -1) on the left. Interchanging a 4 and a 5 in this case, one gets a solution to 4-NLS ŝ with ŝ = (-1, 1, -1, -1) which can be seen on the right of Figure 8.12. The almost solitonic limit = 10 -10 produces well known solitonic patterns as shown for instance for the DS equation in the previous subsection.

Hyperelliptic solutions of n-NLS ŝ with ŝj = 1, j = 1, . . . , n, can be constructed on a curve without real branch point. To get smooth solutions, we use d = 0. A solution of the 2-NLS ŝ equation is studied on the curve of genus 2 with the branch points -2±i, -1±i, 1±i in Figure 8.13.

A typical example for a hyperelliptic 4-NLS ŝ solution with ŝ = (1, 1, 1, 1) can be obtained on a curve of genus 4 with branch points -2 ± i, -1 ± i, ±i, 1 ± i, 2 ± i, as shown in Figure 8.14. ) on a hyperelliptic curve of genus 4 with branch points -2 ± i, -1 ± i, ±i, 1 ± i, 2 ± i and a 1 = (-3.9) (1) , a 2 = (-3.7) (2) , a 3 = (-3.5) (1) , a 4 = (-3.3) (2) and a 5 = (-3.1) (1) .

General real algebraic curves

The quantities entering theta-functional solutions of the DS and n-NLS s equations are related to compact Riemann surfaces. Since all compact Riemann surfaces can be dened via compactied non-singular algebraic curves, convenient computational approaches as [START_REF] Deconinck | Computing Riemann matrices of algebraic curves[END_REF][START_REF] Deconinck | Computing Riemann Theta Functions[END_REF] and [START_REF] Frauendiener | Algebraic curves and Riemann surfaces in Matlab[END_REF] are based on algebraic curves: dierentials, homology basis and periods of the Riemann surface can be obtained in an algorithmic way. We refer the reader to the cited literature for details.

In this part we are interested in non-singular real algebraic curves of the form with a mn ∈ R, on which we consider the anti-holomorphic involution given by τ (x, y) = (x, y).

The identication of the sheets of the covering dened by the algebraic curve (8.2.1) via the projection map (x, y) → x, is done, as in the hyperelliptic case, by analytic continuation of the roots y i , i = 1, . . . , N, for some non-critical point x b on the x-sphere, along a set of contours specied in [START_REF] Frauendiener | Algebraic curves and Riemann surfaces in Matlab[END_REF]. In the context of real algebraic curves for which solutions of n-NLS s and DS are discussed here, an additional problem is to establish the action of the anti-holomorphic involution τ on points on dierent sheets. A typical problem is to nd points a ∈ R g and b ∈ R g with the same projection onto the x-sphere such that τ a = b. To this end, the roots y i , i = 1, . . . , N, identied at x = x b , are analytically continued to the points projecting to x(a) on the x-sphere.

It is then established which pairs of points in the dierent sheets satisfy τ a = b.

In contrast to the hyperelliptic curves of the previous section, it is not possible for general curves to introduce a priori a basis of the homology. Thus the cited codes use an algorithm by Tretko and Tretko [148] which produces a homology basis for a given branching structure of the covering which is in general not adapted to possible automorphisms of the curve. In the context of theta-functional solutions to integrable PDEs one is often interested in real curves.

As discussed in Chapter 6, the Vinnikov basis (i.e., the canonical homology basis which satises (3.2.1)) is convenient in this context. Since solutions and smoothness conditions for n-NLS s and DS equations are formulated in this basis, a symplectic transformation relating the computed basis to the Vinnikov basis needs to be worked out. This transformation is discussed in the present section and will be applied to examples of real algebraic curves.

Symplectic transformation

Let R g be a real compact Riemann surface of genus g and τ an anti-holomorphic involution dened on it. Let (ν 1 , . . . , ν g ) be a basis of holomorphic dierentials such that τ * ν j = ν j , j = 1, . . . , g (8.2.2) (recall that τ * denotes the action of τ lifted to the space of holomorphic dierentials). For an arbitrary canonical homology basis (A, B), let us denote by P A and P B the matrices of A and B-periods of the dierentials ν j :

(P A ) ij = A i ν j , (P B ) ij = B i
ν j , i, j = 1, . . . , g. In what follows (A, B) denotes the Vinnikov basis. From (8.2.2) and (3.2.1) we deduce the action of the complex conjugation on the matrices P A and P B : According to (8.2.8), the matrix A Re P Ã + B Re P B is invertible, since the matrix P A of A-periods of a basis of holomorphic dierentials is always invertible. Moreover, since the Riemann matrix B = 2iπ P B (P A ) -1 has a (negative) denite real part, which is equal to -2π Im(P B ) Im((P A ) -1 ) for the real matrix P A here, then by (8.2.11) is satised for the Vinnikov basis, where the function ψ * (ξ, η, t) is dened in (6.4.4), it also holds for the computed basis. Therefore, putting ξ = η = 0, t = 2, d = 0, and taking the modulus of each term in (8.2.26) expressed in the computed basis, one gets:

(P A ) ij ∈ R, ( 8
Θ B[ δ]( Z -r) Θ B[ δ]( Z) exp{-i ( G3 + h)} = Θ B[ δ]( Z + r) Θ B[ δ]( Z) exp{i ( G3 + h)}
where Z = i( Wa -Wb ). It would be possible to compute the theta-functional solutions in the Vinnikov basis once the symplectic transformation between this basis and the basis determined by the code is known.

However, since this symplectic transformation is not unique, the found Vinnikov basis leads in general to a Riemann matrix for which the theta series converges only slowly, i.e., the value N θ in (8.1.1) has to be chosen very large. To avoid this problem, we compute the theta function always in the typically more convenient Tretko-Tretko basis with the characteristic of the theta functions given by (8.2.16)-(8.2.18).

Trott curve

The Trott curve [START_REF] Trott | Applying Groebner Basis to Three Problems in Geometry[END_REF] given by the algebraic equation The Trott curve has real bers and can thus be used to construct solutions to the 3-NLS equation via the projection map f : (x, y) → x, which is a real meromorphic function of degree 4 on the curve. We consider the points on the curve stable with respect to τ and projecting to the point with x = 0.1 in the x-sphere, and choose d = 0. The corresponding solution to the 3-NLS equation can be seen in Figure 8.15.

A solution to the DS1 + equation on this curve can be constructed for points a and b stable with respect to the involution τ . The solution for a = (-0.2) (1) , b = (0.2) (2) and the choice d = 0 can be seen in Figure 8. [START_REF] Anker | On the Soliton Solutions of the Davey-Stewartson Equation for Long Waves[END_REF].

Similarly, a solution to the DS2 + equation can be obtained for points a and b subject to τ a = b. For a = (0.1 + i) (1) and b = (0.1i) (1) we get Figure 8.17 + equation on the Trott curve for the points a = (0.1 + i) (1) and b = (0.1i) (1) at t = 0.

Dividing curves without real branch point

We consider the curve given by the equation 30x 4 -61x 3 y + 41y 2 x 2 -43x 2 -11y 3 x + 42xy + y 4 -11y 2 + 9 = 0 (8.2.33) which was studied in [START_REF] Dubrovin | Matrix nite-zone operators[END_REF] and [START_REF] Vinnikov | Self-adjoint determinantal representations of real plane curves[END_REF]. It is a genus 3 curve, dividing with respect to the antiholomorphic involution τ , without real branch point. This curve admits two real ovals. In this case the matrix H has the form After some calculations, one nds that the following matrices A, B, C, D provide a solution of (8. 4) (1) and b = (-3) (2) at t = 0.

From (8.2.16) and (8.2.17) one gets for the characteristic: δ = 1 2 [ 0 0 1 1 1 0 ] t .

The curve (8.2.33) has real bers and can thus be used to construct solutions to the focusing 3-NLS equation. We consider the points on the curve with x = 2.5 and stable with respect to τ , and we choose d = 0. The corresponding solution to the focusing 3-NLS equation can be seen in Figure 8.18.

A solution to the DS1

-equation can be constructed by choosing the points a = (-4) (1) and b = (-3) (2) see Figure 8.19.

Fermat curve

The Fermat curves y n + x n + 1 = 0, n > 2, n even, To construct a solution of the DS2equation on the Fermat curve, we choose the points a = (-1.5 + i) (1) and b = (-1.5i) (3) . The resulting solution for the choice d = 0 can be seen in Figure 8.20.

Outlook

In this chapter we have presented the state of the art of the numerical evaluation of solutions to integrable equations in terms of multi-dimensional theta functions associated to real Riemann surfaces by using an approach via real algebraic curves. It was shown that real hyperelliptic curves parametrized by the branch points can be treated with machine precision for a wide range of the parameters. Even almost degenerate situations where the branch points coincide pairwise can be handled as long as at least one cut stays nite. This approach to real hyperelliptic curves [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods[END_REF][START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods: KdV and KP solutions[END_REF]

9 2-++++--. 10 Solution of 2 -. 11 Solution of 2 -. 13 Solution of 2 - 1 Introduction 1 . 1

 9102112132111 Fay sur des surfaces de Riemann compactes. Cette relation généralise une identité bien connue pour la fonction birapport dans le plan complexe. Elle permet d'établir des relations entre les fonctions theta et leurs dérivées. Cela ore une approche complémentaire aux solutions algébro-géométriques des équations intégrables avec certains avantages par rapport à l'utilisation des fonctions de Baker-Akhiezer. Cette méthode a été appliquée avec succès par Mumford et al. aux équations Korteweg-de Vries, Kadomtsev-Petviashvili et sine-Gordon. Selon cette approche, nous construisons des solutions algébro-géométriques des équations de Camassa-Holm et de Dym, ainsi que des solutions de l'équation de Schrödinger non linéaire à plusieurs composantes et des équations de Davey-Stewartson. Les limites solitoniques de ces solutions sont étudiées lorsque le genre de la surface de Riemann associée tombe à zéro. De plus, nous présentons une évaluation numérique des solutions algébro-géométriques des équations intégrables lorsque la surface de Riemann associée est réelle. Introduction to the theory of integrable systems . . . . . . . . . . . . . . . . . . . 1.1.1 Dark 3-soliton of DS1 -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Bright 2-soliton of DS2 -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.breather of DS1 -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 2-rational breather of DS1 -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Interaction between a line rational breather and a rational breather of DS1 -(same direction of propagation) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 Interaction between a line rational breather and a rational breather of DS1 -(transversal propagations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13 A dromion solution of DS1 -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14 A lump solution of DS2 -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Homology basis on real hyperelliptic curves . . . . . . . . . . . . . . . . . . . . on a genus 2 hyperelliptic curve with real branch points on the left; the almost solitonic limit on the right . . . . . . . . . . . . . . . . . . . . on a genus 2 hyperelliptic curve with real branch points on the left; the almost solitonic limit on the right . . . . . . . . . . . . . . . . . . . . on a genus 4 hyperelliptic curve with real branch points on the left; the almost solitonic limit on the right . . . . . . . . . . . . . . . . . . . . on a genus 4 hyperelliptic curve with real branch points on the left; the almost solitonic limit on the right . . . . . . . . . . . . . . . . . . . . on a genus 2 hyperelliptic curve with pairwise conjugate branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.7 Solution of DS1 on a genus 4 hyperelliptic curve with pairwise conjugate branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.8 Solution of DS2 on a genus 2 hyperelliptic curve with pairwise conjugate branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . on a genus 4 hyperelliptic curve with pairwise conjugate branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8NLS -on a genus 2 hyperelliptic curve with real branch points on the left; the almost solitonic limit on the right . . . . . . . . . . . . . . . . . . . 139 8NLS +on a genus 2 hyperelliptic curve with real branch points on the left; the almost solitonic limit on the right . . . . . . . . . . . . . . . . . . . 139 8.12 Solution of 4-NLS +-+on the left, solution of 4-NLS -+-on the right, on a genus 4 hyperelliptic curve with real branch points . . . . . . . . . . . . . . . . 140 8NLS ++ on a genus 2 hyperelliptic curve with conjugate pairwise branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8.14 Solution of 4-NLS ++++ on a genus 4 hyperelliptic curve with conjugate pairwise Chapter Introduction to the theory of integrable systems

( 1 . 1 . 3 ) 3 4 (

 11334 can be written as the compatibility condition of the auxiliary linear problem [∂ y -L, ∂ t -A] = 0, where L and A are dierential operators given byL = ∂ 2 x + u, A = ∂ 3 x + u∂ x + ∂ x u) + w.

. 5 )

 5 with the potential u(x, t) = D 2 a ln Θ(Zd). When this potential is related to the function ψ by u(x, t) = ρ |ψ| 2 , with ρ = ±1, the function ψ (1.2.4) becomes a solution of the nonlinear Schrödinger equation (1.1.2). This is the starting point of our construction of algebro-geometric solutions of the multi-component nonlinear Schrödinger equation and the Davey-Stewartson equations. The nonlinear Schrödinger equation (1.1.2) is a famous nonlinear dispersive partial dierential equation with many applications, e.g. in hydrodynamics (deep water waves), plasma physics and nonlinear ber optics. Integrability of this equation was established by Zakharov and

( 2 .

 2 2.5) where K = 2iπA + B B, and where

  some n, m ∈ Z g . Consider the hyperelliptic curve as a two-sheeted covering of the Riemann sphere, denote by a path connecting points a and b on one sheet, and consider a copy ˜ of this contour connecting points b and a, which lies on the other sheet of R g . Then by(2.3.19) we deduce that r = ω = ˜ ω.

  3.33), where the vectors V a , V b are dened in (2.3.3) and d ∈ C g is arbitrary, we obtain that the function u n (x, y) is a solution of the 2D Toda lattice equation (2.3.28).

  3.1). The last intersection index in (3.3.7) equals m -N, which gives τ B = -B + HA + N S b . Finally, to prove that τ S b = S b , we use the relation S a + S b = 0, where S a is a positively oriented small contour around a, and the relation τ S b = -S a . Using the action (3.3.2) of τ on the A-cycles in the homology group H 1 (R g \ {a, b}), due to the uniqueness of the normalized dierential of the third kind Ω b-a , we get τ * Ω b-a = -Ω b-a .

2

 2 Case τ a = a and τ b = b Now let us assume that a, b ∈ R g satisfy τ a = a and τ b = b. Proposition 3.3.2. The action of the anti-holomorphic involution τ on:

  ) equals -(m + N + HM), which by (3.3.10) implies τ B = -B + HA + N S b . Finally, since the anti-holomorphic involution τ inverses orientation we have τ S b = -S b .

( 3 . 4 . 1 )

 341 for any ζ ∈ J, where n ζ ∈ Z, n ζ ≤ g, is the degree of the divisor D such that Π(D) = ζ. Now consider the following two subsets of the Jacobian:

  Proposition 3.4.2 is used in [51] to construct smooth solutions of the KP equation. In the present work, Proposition 3.4.1 and Proposition 3.4.2 are necessary to get physically meaningful smooth solutions of the studied integrable equations.

Proposition 4 . 1 . 1 .

 411 Let a, b ∈ R g such that σ(a) = b and let e ∈ R g be a ramication point,

e ln g 1 g 2 = p 1 + p 2 g 1 g 2 .

 22 3.6), (2.3.7); D b (resp. D e ) denotes the directional derivative (2.3.4). Proof. Under the changes of variables (a, b, c) → (b, e, a) and z → zr/2, identity (2.3.5) becomes D

( 4 . 1 . 7 )

 417 Here we used the fact that e a ω = b e ω = r/2 (use (2.3.19)). Applying the dierential operator D b to the previous equality one gets:

  which proves (4.1.4). To prove (4.1.5), consider the change of variables (a, b, c) → (e, b, a) in (2.3.5), which leads to

( 4 .

 4 1.9) Moreover, since D b D e ln(g 1 g 2 ) = D b D e ln Θ z + r 2 + D b D e ln Θ z -r 2 -2 D b D e ln Θ(z), using (2.3.10) one gets

( 4 .

 4 1.12) where vectors V e and V b are dened in (2.3.3). The scalars α 1 and α 2 satisfy α 1 = p 1 (b, e, a), α 2 = 2 p 1 (e, b, a) + k,(4.1.13)

( 4 .

 4 1.14) Here D b denotes the directional derivative (2.3.4) along vector V b . Proof. Let β, δ ∈ C and α 1 , α 2 ∈ C be arbitrary constants. Let us look for solutions u of CH having the following form

( 4 .

 4 1.18) By (4.1.5) and (4.1.16) we deduce the second derivative of u with respect to x:

( 4 .

 4 1.21) m t (x, t) = 4β p2 (g 1 g 2 ) 2 (y t D e ln(g 1 g 2 ) + D b ln(g 1 g 2 )) .

( 4 .

 4 1.22) Therefore, entering with the functions (4.1.15), (4.1.18), (4.1.21) and (4.1.22) into the CH equation (4.1.2) we obtain

Proposition 4 . 1 . 3 .

 413 Assume that all ramication points of R g are stable under τ and denote by e ∈ R g (R) one of them. Let a, b ∈ R g (R) such that σ(a) = b. For any c ∈ {a, b, e}, choose the local parameter k c such that k c (τ p) = k c (p) for any point p lying in a neighbourhood of c. Denote by an oriented contour between a and b containing point e, which does not intersect cycles of the canonical homology basis. Choose such that the closed path τis homologous to zero in H 1 (R g ). Take d ∈ iR g and k ∈ R. Choose ζ ∈ C in (4.1.11) such that ln Θ(d+r/2) Θ(d-r/2)ζ is real. Then solutions u(x, t) of the CH equation given in (4.1.14) are real-valued, and for xed t 0 ∈ R, the function u(x, t 0 ) is smooth with respect to the real variable x. Proof. Let us check that under the conditions of the proposition, the function u(x, t) (4.1.14) is real-valued. First of all, invariance with respect to the anti-involution τ of the points e and b implies Z = -Z,

  (4.1.23)

  3.3) of the normalized holomorphic dierentials ω j near e (resp. b) one gets τ * ω j (e)(p) = V e,j + W e,j k e (p) + o k e (p) 2 dk e (p), for any point p lying in a neighbourhood of e (resp. b). Then by (3.2.2), vectors V e and V b appearing in the vector Z are purely imaginary, which leads to (4.1.23). Moreover, since the closed contour τis homologous to zero in H 1 (R g ), from (3.2.2) one gets r =r.

( 4 .

 4 1.24) From (2.3.8), the scalars p 1 (b, e, a) and p 1 (e, b, a) appearing respectively in α 1 and α 2 (see 4.1.13) satises Ω a-b (e) = p 1 (b, e, a) dk e , Ω a-e (b) = p 1 (e, b, a) dk b .

( 4 .

 4 1.25) Therefore, from(3.3.18) it can be deduced that p 1 (b, e, a) and p 1 (e, b, a) are real, which involves

  Now let us study real-valuedness and smoothness of the solutions in the case where τ a = b. Proposition 4.1.4. Assume that all ramication points of R g are stable under τ and denote by e ∈ R g (R) one of them. Let a, b ∈ R g such that σ(a) = b and assume that τ a = b. Choose the local parameters such that k b (τ p) = k a (p) for any point p lying in a neighbourhood of a, and k e (τ p) =k e (p) for any p lying in a neighbourhood of e. Denote by an oriented contour between a and b containing point e, which does not intersect cycles of the canonical homology basis. Assume that N = 2L for some L ∈ Z g , where N ∈ Z g is dened in (3.3.1). Take k ∈ R and dene d = d R + iπ 2 N for some d R ∈ R g . Choose ζ ∈ C in (4.1.11) such that ln Θ(d+r/2) Θ(d-r/2)ζ is real. Then solutions u (4.1.14) of the CH equation are real-valued. Moreover, for xed t 0 ∈ R, the function u(x, t 0 ) is smooth with respect to the real variable x in the case where N = 0, otherwise it has innite number of singularities of the type O (xx 0 ) 2n 2n+1

( 4 .

 4 1.31) From(2.3.3) and (3.2.2) one gets V a = -V b and V e = V e . Moreover, by assumption vectors V a and V b satisfy V a + V b = 0 (use(2.3.19)); thus we deduce that V b = V b and V e = V e which proves(4.1.31). By(3.3.8) and (4.1.25) it can be deduced that α 1 and α 2 (4.1.13) satisfy

( 4 .

 4 1.38) Finally, combining (4.1.36) and (4.1.38), the function u(x, t 0 ) has an innite number of singularities of the type O (xx 0 ) 2n 2n+1 which correspond to cusps.

( 4 . 2 . 3 ) 4 . 2

 42342 Solutions of the Dym-type equation 51 Moreover, the two following identities hold:

( 4 . 1 .

 41 [START_REF] Alber | Algebraic geometrical solutions for certain evolution equations and Hamiltonian ows on nonlinear subvarieties of generalized Jacobians[END_REF]) and (4.2.11) respectively, are related to the hyperelliptic spectral curve via Abelian dierentials. Hence, before degenerating the algebro-geometric solutions constructed in Chapter 4 in the limit (5.1.1), let us rst determine the limit form of the normalized Abelian dierentials of the rst and third kind, as well as the limit form of the bidierential (2.2.14).

.1. 5 )

 5 Degenerate meromorphic dierentials. Consider the hyperelliptic curve (4.0.1) and denote by a = ∞ -, b = ∞ + the innity points. The normalized meromorphic dierential Ω b-a having a simple pole at b with residue +1 and a simple pole at a with residue -1, after degeneration reads Ω 0 b-a (λ) = -dλ µ .

( 5 . 1 . 6 )

 516 To see that, consider the local parameter k ∞ (λ) = λ -1 in the neighbourhood of points a = ∞ - and b = ∞ + respectively. Using the fact that these points on the curve (5.1.3) are distinguished by the condition

  .1.14) taking into account that B 0 jk = B 0 kj . For the diagonal elements of the matrix B, one has the following asymptotic estimate:Re(B jj ) = 2 ln |λ 2j+1λ 2j+2 | + O(1), j = 1, . . . , g, which in the limit (5.1.1) yieldsRe(B jj ) -→ -∞, j = 1, . . . , g.

( 5 . 1 . 15 )B 16 )

 511516 According to(5.1.15), in the limit considered here, the theta function (2.2.1) with zero characteristic tends to one. Therefore, to get nontrivial theta function, and hence nontrivial solutions of (4.1.1) and (4.2.1) after degeneration, we change the argument z in the theta function by z-D with D = (1/2) B jj , in such way thatΘ(z -D) = jj m j (m j -1) + j<k B jk m j m k + j m j z jNow taking the limit (5.1.1) in(5.1.16), by(5.1.15) it is straightforward to see that the degenerate theta function denoted by Θ g reads

  .1.14) and (4.2.11) leads, after degeneration (5.1.1) of the associated hyperelliptic curve, to soliton-like solutions of the CH equation (4.1.1) and the Dym-type equation (4.2.1) respectively. Here soliton-like solutions denote interactions between solitons and cuspons, namely, non-smooth soliton solutions with a cusp. In what follows we assume that β k ∈ R.5.2.1 Soliton-like solutions of the CH equationLet us rst determine quantities independent of the physical coordinates which appear in Proposition 4.1.2, when the genus of the hyperelliptic spectral curve tends to zero, as discussed in the previous section. We keep the same notation for these quantities as before the degeneration. Let us choose e = (λ 2 , 0) in (4.1.14), a = ∞ -, b = ∞ + , and consider the limit (5.1.1). With notation introduced in (5.1.1) one has e = (α, 0). From (5.1.5) one gets:

( 4 . 2 .

 42 [START_REF] Alber | The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type[END_REF]. Let us choose e = (λ 2 , 0) and f = (λ 1 , 0) in (4.2.11) and consider the limit (5.1.1). Keeping the notation introduced in (5.1.1) one has e = (α, 0) and f = (-α, 0). Let us choose the local parameters k e and k f near e and f respectively such that

  .1.14) and (4.2.11) with i 0 ∈ {3, . . . , 2g + 2}, leads to peakon solutions when the hyperelliptic spectral curve degenerates according to (5.1.1).

  , z) = u(-∞, z + Bg ).

( 5 . 3 . 26 )Theorem 6 . 1 . 1 . 2 + 2

 532661122 Moreover, it can be seen that the derivative with respect to y of the function x(y, z) does not vanish for nite values of y. Hence, by comparison with(5.3.13) and(5.3.14), similar statements used for the CH equation yield the construction of globally dened solutions of the Dym-type equation. These solutions are given by innite pieces of solutions glued at peak points:Chapter 6 Algebro-geometric solutions viaFay's identityWe prove a new degenerated version of Fay's trisecant identity (2.3.1). Following Mumford's approach, the new identity is applied to construct new algebro-geometric solutions of the multicomponent nonlinear Schrödinger (n-NLS) equation. This approach also provides an independent derivation of known algebro-geometric solutions to the Davey-Stewartson (DS) equations.The chapter is organized as follows: in Section 6.1 we derive a new degeneration of Fay's identity and in Section 6.2 we give an auxiliary computation required in the construction of algebro-geometric solutions of DS and n-NLS equations. With the degenerated version of Fay's trisecant identity obtained in Section 6.1, we construct in Section 6.3 new smooth theta-functional solutions of the multi-component NLS equation, and describe explicitly solutions in the focusing and defocusing cases. We also discuss the reduction from n-NLS to (n -1)-NLS, stationary solutions of n-NLS, hyperelliptic solutions, and the link between our solutions of n-NLS and solutions of the KP1 equation. Section 6.4 contains an independent derivation of smooth thetafunctional solutions of the Davey-Stewartson equations. This approach also provides an explicit description in terms of theta functions of the constants appearing in the solutions. 6.1 New degeneration of Fay's identity Algebro-geometric solutions of the Davey-Stewartson and the multi-component NLS equation constructed in this chapter are obtained by using the following new degenerated version of Fay's identity. Let a, b be distinct points on a compact Riemann surface R g of genus g. Fix local parameters k a and k b in a neighbourhood of a and b respectively. Denote by δ a non-singular odd characteristic. Then for any z ∈ C g , D 2 a ln Θ(z)+K 2 (a, b) = 0, (6.1.1) where the scalars K 1 (a, b) and K 2 (a, b) are given by

  .1.2) and K 2 (a, b) = -D a ln Θ start from the following lemma: Lemma 6.1.1. Let b, c ∈ R g be distinct points. Fix local parameters k b and k c in a neighbourhood of b and c respectively. Then for any z ∈ C g ,

( 6 . 1 . 8 )Now ( 6 . 1 . 4 )

 618614 is obtained by subtracting (6.1.7) and (6.1.8).To prove Theorem 6.1.1, make the change of variable z → -z+ c b in (6.1.5) and add 2 D 2 b ln Θ to each side of the equality to get

Then ( 6 . 1 . 9 ) 1 ) dk d 1 ,

 61911 can be rewritten as D c f (b,a) (z) = 0 for any z ∈ C g and for all c ∈ R g , c = b (because also D a f (b,a) (z) = 0 by Lemma 6.1.1). Since on each Riemann surface R g there exists a positive divisor d 1 + ... + d g of degree g such that the vectors ω(d Lemma 5), the function f (b,a) (z) is constant with respect to z; we denote this constant by -K 2 (b, a): f (b,a) (z) = -K 2 (b, a) (6.1.10)

  Proof. Let ã, b ∈ R g lie in a neighbourhood of a and b respectively such that τ ã = b. Denote by ˜ an oriented contour connecting ã and b. First, let us check that arg{q 2 (a, b)} = π(1 + α), (6.2.6) where α = (τ ˜ + ˜ ) • in the relative homology group H 1 (R g , {a, b}). The integral representation (6.2.1) of q 2 (a, b) leads to arg{q 2 (a, b)} = π + Im ˜ Ω b-a (p) .

  q 2 (a, b) when τ a = a and τ b = b Now let us consider the case where a and b are invariant with respect to τ . Proposition 6.2.3. Let a, b ∈ R g (R) with local parameters satisfying k a (τ p) = k a (p) for any point p in a neighbourhood of a and k b (τ p) = k b (p) for any point p in a neighbourhood of b. Let (A, B, ) be the generators of the relative homology group H 1 (R g , {a, b}). Choose ã, b ∈ R g (R) in a neighbourhood of a and b respectively, and denote by ˜ an oriented contour connecting ã and b.

  r + r + BM, M ) ,(6.2.15)

Figure 6 . 1 :

 61 Figure 6.1: Hurwitz diagram of the covering R g,n+1 . The horizontal lines indicate copies of the Riemann sphere, and the vertical lines the ramication points of the covering R g,n+1 .

Figure 6 . 5 :

 65 Figure 6.5: Homology basis on the covering R - g,n+1 . The solid line indicates the sheet n + 1, and the dashed line sheet n.

( 6 . 3 .

 63 14) of the n-NLS s equation: u(x, y, t) = γ -2 n j=1 s j |ψ j (x, y, t)| 2 . (6.3.45) Since in our construction of solutions to the multi-component NLS equation, the local parameters are dened by the meromorphic function f , complex solutions (6.3.44) and real solutions (6.3.45) of KP1 obtained in this way form only a subclass of Krichever's solutions.6.4 Algebro-geometric solutions of the Davey-Stewartson equations In this section we present another application of the degenerated Fay identity (6.1.1), which leads to algebro-geometric solutions of the Davey-Stewartson equations (1.2.9). Let us introduce the function φ := Φ + ρ |ψ| 2 , where ρ = ±1, and the dierential operators

. 25 )

 25 taking into account (3.2.4). Let us choose a vector d ∈ C g such that d =diπ diag(H) + 2iπT + BL, for some vectors T, L ∈ Z g . The reality of vector d + d together with (3.2.

6. 4 . 4

 44 Reduction of the DS1 ρ equation to the NLS equation Solutions of the nonlinear Schrödinger equation (1.1.2) can be derived from solutions of the Davey-Stewartson equations, when the associated Riemann surface is hyperelliptic. Proposition 6.4.2. Let R g be a hyperelliptic curve of genus g > 0 which admits an antiholomorphic involution τ . Denote by σ the hyperelliptic involution dened on R g . Let a, b ∈ R g (R) with local parameters satisfying k a (τ p) = k a (p) for p near a, and k b (τ p) = k b (p) for p near b. Moreover, assume that σa = b and k a (p) = k b (σp) for p near a. Then, taking κ 1 = κ 2 = 1, the function ψ dened in (6.4.10) is solution of the equation

( 7 . 1 . 8 )

 718 Proof. The proof is carried out by comparing the coecient of exp{z k 1 +. . .+z kn } in the left and right-hand side of (7.1.7), where k 1 , . . . , k n are n dierent numbers taken from 1, . . . , g. Due to the symmetry property of the left-hand side in (7.1.7) with respect to the variables z k , however, it is sucient to choose k 1 = 1, . . . , k n = n. Denote the coecient of the term exp{z k 1 +. . .+z kn } on the left-hand side of (7.1.7) by R 1 . This term arises by taking m k = 1 for k = 1, . . . , n and m k = 0 for k = n + 1, . . . , g, and has the form:

( 7 . 1 .

 71 [START_REF] Alber | Algebraic geometrical solutions for certain evolution equations and Hamiltonian ows on nonlinear subvarieties of generalized Jacobians[END_REF], recall that (see (2.2.15)) ã c Ω(2) a (p) = ∂ kq ln Θ[δ]

Figure 7 .

 7 1 shows the elastic interaction with a phase-shift between two dark soliton solutions of the 4-NLS s equation with s = (1, -1, 1, -1). It corresponds to the following choice of parameters: w a 1 = 8, w a 2 = 5, w a 3 = 3, w a 4 = -4, w a 5 = -10, w b 1 = 1, w b 2 = 2, w b 3 = 17, w b 4 = 1/4, w b 5 = -1/2, and w u 1 ≈ 1.51 + 2.43i with f (w u 1 ) = 7, w u 2 ≈ 4.41 + 3.45i with f (w u 1 ) = 1.

Proposition 7 . 2 . 3 . 1 ( 7 . 2 . 9 )Figure 7 . 1 :

 723172971 Figure 7.1: Dark 2-soliton of 4-NLS s with s = (1, -1, 1, -1).

  ψ j (x, t) = A j e iθ e z 1 -r j,1 1 + e B+2Re(z 1 ) , where B = ln -|w v 1 | 2 |α u 1 | 2 (4 Im(w v 1 ) Im(α u 1 )) -1 .Example 7.2.4.

Figure 7 .

 7 2 shows the interaction between two bright solitons solution of the 4-NLS s equation with s = (1, 1, 1, 1). It corresponds to the following choice of parameters: γ j = 1, w a 1 = -6, w a 2 = 17, w a 3 = 3, w a 4 = 7, and w v 1 = 1 + i, w v 3 = 2 -3i, dk = 0. The resulting two-soliton solution shows the standard elastic collision with a phase-shift.

Figure 7 . 2 :

 72 Figure 7.2: Bright 2-soliton of 4-NLS s with s = (1, 1, 1, 1).

Figure 7 .

 7 [START_REF] Ablowitz | Solitons and the Inverse Scattering Transform[END_REF] shows an elastic collision between two breathers, solution of the 4-NLS s equation with s = (-1, 1, 1, -1). It corresponds to the following choice of parameters:w a 1 = 1/3, w a 2 = 3, w a 3 = 1/7, w a 4 = 2, w a 5 = 1, w b 1 = -1, w b 2 = 4, w b 3 = -2, w b 4 =0, and w u 1 ≈ 0.55-0.11i with f (w u 1 ) = 2i, w u 2 ≈ -0.35+0.07i with f (w u 2 ) = -2i, w u 3 ≈ -0.91-0.52i with f (w u 3 ) = 10 -5i, and w u 4 ≈ 14.46 + 5.32i with f (w u 4 ) = 10 + 5i.

Figure 7 .

 7 6 shows a 2-rational breather solution of the 4-NLS s equation with s = (1, 1, 1, 1). It corresponds to the following choice of parameters: k a k (w) = f (w a k )f (w) for k = 1, . . . , n + 1, with w a 1 = 3, w a 2 = 5, w a 3 = 7, w a 4 = 0, w a 5 = 4, and w u 1 ≈ 4.53 + 0.56i, w u 3 ≈ 3.45 + 0.56i being solutions of n+1 i=1 (ww a i ) -2 = 0, and d k = 10. Variation of the parameters d k leads to a displacement in the (x, t)-plane of the rational breathers appearing in each of the pictures of Figure 7.6.

Figure 7 . 5 :

 75 Figure 7.5: Rational breather of 4-NLS s with s = (1, 1, 1, 1).

7 . 3

 73 Degenerate algebro-geometric solutions of the DS equationsSolutions of the Davey-Stewartson equations (6.4.1) in terms of elementary functions constructed here are obtained analogously to the solutions of the n-NLS equation, therefore some details will be omitted. In this section, we study the behaviour of theta-functional solutions (6.4.4) of the complexied DS equations (6.4.2) when the Riemann surface degenerates into a Riemann surface of genus zero. Imposing the reality condition (6.4.3), for particular choices of the parameters one gets well-known solutions such as multi-soliton, breather, rational breather, dromion and lump.

. 7 )

 7 the functions ψ and ψ * (7.3.1) satisfy the reality condition ψ * = ψ. With (7.3.7), it is straightforward to see that(7.3.6) is also satised. Moreover, since G 1 = G 2 , G 3 ∈ R and (w a -w b ) 2 < 0, the functions ψ and ψ * (7.3.1) satisfy the reality condition ψ * = ψ. Therefore, they dene dark multi-soliton solutions of DS2 + .

for k = 1 ,

 1 . . . , N . Moreover, the scalars rk are dened by rk = (-1) k ln {-α v k α u k } , k = 1, . . . , 2N. Remark 7.3.4. i) With (7.3.9), functions (7.3.11) give a family of bright multi-soliton solutions of the DS1 ρ equation depending on 3N complex parameters d2k-1 , α u 2k-1 , α u 2k and 4 real parameters h, θ, κ1 , κ2 . ii) With (7.3.10), functions (7.3.11) provide a family of bright multi-soliton solutions of the DS2equation depending on 3N +1 complex parameters d2k-1 , α u 2k-1 , α v 2k-1 , κ1 and 2 real parameters h, θ. Example 7.3.2. In the case N = 2, Figure 7.8 shows an elastic collision of two bright solitons solution of the DS2 -equation with the following choice of parameters: w a = 10 + 20i, w b = 10 -20i, α u 1 = 5 + 2i, α v 1 = 2 + i, α u 3 = 3 + i, α v 3 = 1 + 4i, κ1 = 1, dk = h = 0.

Figure 7 . 8 :

 78 Figure 7.8: Bright 2-soliton of DS2 -at a) t = 0, b) t = 50.

Figure 7 . 9 :

 79 Figure 7.9: 2-breather of DS1 -at a) t = 0, b) t = 45.

Figure 7 . 11 :

 711 Figure 7.11: Interaction between a line rational breather and a rational breather of DS1at a) t = -50, b) t = -20, c) t = -5, d) t = 0, e) t = 10, f ) t = 50. The rational breather propagates in the same direction as the line breather.

Figure 7 . 12 :

 712 Figure 7.12: Interaction between a line rational breather and a rational breather of DS1at a) t = -50, b) t = -20, c) t = -5, d) t = 0, e) t = 10, f ) t = 50. The rational breather propagates transversally to the direction of the line breather.

Figure 7 .

 7 Figure 7.13: A dromion solution of DS1 -at a) t = -5, b) t = 10.

  .4.1) and the multicomponent nonlinear Schrödinger equations (6.3.1) investigated in Chapter 6. We rst consider the hyperelliptic case and study concrete examples of low genus, also in almost degenerate situations. Finally, we consider examples of non-hyperelliptic real Riemann surfaces and discuss symplectic transformations needed to construct smooth solutions.

Figure 8 . 1 :

 81 Figure 8.1: Homology basis on the real hyperelliptic curves, contours on sheet 1 are solid, contours on sheet 2 are dashed. A-cycles are the closed contours entirely on sheet 1.

  The choice of a branch point as the base point of the Abel map has the advantage that a change of

Figure 8 . 2 :

 82 Figure 8.2: Solution (6.4.10) to the DS1 + equation at t = 0 on a hyperelliptic curve of genus 2

Figure 8 . 3 :

 83 Figure 8.3: Solution (6.4.20) to the DS2 + equation at t = 0 on a hyperelliptic curve of genus 2

Figure 8 . 4 :

 84 Figure 8.4: Solution (6.4.10) to the DS1 + equation at t = 0 on a hyperelliptic curve of genus 4

3 .

 3 We only show the square modulus of the solution here for simplicity. For the real and the imaginary part of such a solution for the DS1 --case, see Figure8.6.

Figure 8 . 5 :

 85 Figure 8.5: Solution (6.4.20) to the DS2 + equation at t = 0 on a hyperelliptic curve of genus 4

Figure 8 . 6 :

 86 Figure 8.6: Solution (6.4.10) to the DS1 -equation at t = 0 on a hyperelliptic curve of genus 2

  be seen in Figure 8.4. The corresponding solutions to DS2 + is shown in Figure 8.5.

Figure 8 . 7 :

 87 Figure 8.7: Solution (6.4.10) to the DS1 -equation at t = 0 on a hyperelliptic curve of genus 4

7 .

 7 Smooth solutions to DS2 -can be obtained on Riemann surfaces without real oval for points a and b satisfying τ a = b. As mentioned above, hyperelliptic curves of the form µ 2 = -2g+2 i=1 (λλ i ) with pairwise conjugate branch points have no real oval for the standard involution τ 1 as dened in Example 3.1.1. On the other hand, surfaces dened by the algebraic equation µ 2 = 2g+2 i=1 (λλ i ) have no real oval for the involution τ 2 (see Example 3.1.1). We will consider here τ 2 for the same curves as for DS1 -. An example for genus 2 can be seen in Figure 8.8. An example for a DS2solution of genus 4 can be seen in Figure8.9.

Figure 8 . 9 :

 89 Figure 8.9: Solution to the DS2

Figure 8 .

 8 Figure 8.10: Solution(6.3.39) to the 2-NLS ŝ equation with ŝ = (-1, -1) on a hyperelliptic curve of genus 2 with branch points -2, -1, 0, , 2, 2 + and a 1 = (-1.9)(1) , a 2 = (-1.1)(1) and a 3 = (-1.8)(1) for = 1 on the left and = 10 -10 , the almost solitonic limit, on the right.

2 Figure 8 .

 28 Figure 8.11: Solution (6.3.39) to the 2-NLS ŝ equation with ŝ = (1, -1) on a hyperelliptic curve of genus 2 with branch points -2, -1, 0, , 2, 2 + and a 1 = (-1.9)(1) , a 2 = (-1.8)(1) and a 3 = (-1.1)(1) for = 1 on the left and = 10 -10 , the almost solitonic limit, on the right.

2 Figure 8 . 12 :

 2812 Figure 8.12: Solution to the 4-NLS ŝ equation on a hyperelliptic curve of genus 4 with branch

Figure 8 . 13 :

 813 Figure 8.13: Solution to the 2-NLS ŝ equation with ŝ = (1, 1) on a hyperelliptic curve of genus 2 with branch points -2 ± i, -1 ± i, 1 ± i and a 1 = (-1.9)(1) , a 2 = (-1.8)(2) and a 3 = (-1.1)(1) .

Figure 8 . 14 :

 814 Figure 8.14: Solution to the 4-NLS ŝ equation with ŝ = (1, 1, 1, 1) on a hyperelliptic curve of genus 4 with branch points -2 ± i, -1 ± i, ±i, 1 ± i, 2 ± i and a 1 = (-3.9)(1) , a 2 = (-3.7)(2) , a 3 = (-3.5)(1) , a 4 = (-3.3)(2) and a 5 = (-3.1)(1) .

a

  mn x m y n = 0, x, y ∈ C,(8.2.1)

( 8 .

 8 2.3)

144 (x 4 + y 4 ) 1 .

 441 -225 (x 2 + y 2 ) + 350 x 2 y 2 + 81 = 0(8.2.30) is an M-curve with respect to the anti-holomorphic involution τ dened by τ (x, y) = (x, y), and is of genus 3. Moreover, this curve has real branch points only (and 28 real bitangents, namely, tangents to the curve in two places). Our computed matrices of à and B-periods read 1The Trott curve being an M-curve, the vectors of the characteristic δ satisfy (8.2.28) and (8.2.29), which leads to δ = 1 2 [ 0 0 0 1 1 0 ] t .A possible choice of a symplectic transformation bringing the computed basis to the Vinnikov basis is:Note that the matrices A, B, C, D are not unique since the action (3.2.1) of the anti-holomorphic involution on the basic cycles allows for permutations of A j -cycles for instance. These matrices can be computed as follows. Since the Trott curve is an M-curve, one has H = 0. Moreover, the matrix Im(P B) being invertible here, by (8.2.9) one gets: B = -A Im P à Im P B -and (2.1.4) it follows that A t D + C Im P à Im P B -1 = I 3 .

( 8 . 2 . 32 )

 8232 The computed matrix Im P Ã Im P B -1 being (within numerical precision) equal to Im P Ã Im P B and with C, D ∈ M 3 (Z), we get from (8.2.32) that det A = 1. Since A ∈ M 3 (Z), the condition det A = 1 implies A ∈ Gl 3 (Z). For any A ∈ Gl 3 (Z), one can see from (8.2.31), (8.2.14) and (8.2.15) that B, C, D ∈ M 3 (Z), and therefore that the matrices A, B, C, D give a solution of (8.2.8)-(8.2.11). The choice A = I 3 leads to the above matrices.

  .

Figure 8 .

 8 Figure 8.15: Solution (6.3.14) to the 3-NLS s equation on the Trott curve for the points with x = 0.1 on the x-sphere. The sheets are identied at the points projecting to x = -1.0129, (0.9582i, -0.9582i, 0.1146i, -0.1146i). The vector of signs equals s = (1, -1, -1) from top to bottom.

Figure 8 . 16 :

 816 Figure 8.16: Solution to the DS1 + equation on the Trott curve for the points a = (-0.2) (1) and b = (0.2) (2) at t = 0.

Figure 8 . 17 :

 817 Figure 8.17: Solution to the DS2

  2721 -0.0977i -0.3193 + 0.1914i -1.0668 + 0.4293i 0.2721 + 0.0977i -0.3193 -0.3341i -1.0668 -2721 -0.2932i -0.3193 + 0.3341i -1.0668 + 0.4316i 0.2721 + 0.2932i -0.3193 -0.7169i -1.0668 -

Figure 8 . 18 :

 818 Figure 8.18: Solution to the 3-NLS s equation on the dividing curve (8.2.33) of genus 3 for the points with x = 2.5 on the x-sphere. The sheets are identied at the ber over -2.1404 + 0.4404i, (-12.2492+2.0113i, -5.1634+1.3519i, -4.5915+0.9380i, -1.5405+0.5429i). The vector of signs is s = (1, 1, 1).

Figure 8 . 19 :

 819 Figure 8.19: Solution to the DS1 equation on the dividing curve (8.2.33) of genus 3 for the points a = (-4)(1) and b = (-3) (2) at t = 0.

( 8 .

 8 2.34) are real curves without real oval with respect to τ . We consider here the curve with n = 4 that has genus 3. The matrix H has the form The following matrices A, B, C, D provide a solution of (8.2.8)-(8.2.11):

  involution if their Poisson bracket {f, g} vanishes, where

	{f, g} =	n i=1	∂f ∂q i	∂g ∂p i	-	∂f ∂p i	∂g ∂q i	.

  1/2 . Moreover, if N is odd, Ĉ = C ∪ {∞} describes a compact Riemann surface with local parameter (λ, µ) → λ -1/2 near the innity point ∞; otherwise, if N is even, Ĉ = C ∪ {∞ ± } describes a compact Riemann surface with local parameters (λ, µ) → λ -1 near the innity points ∞ ± .

	For compact Riemann surfaces, a single non-negative integer yields a complete topological
	classication:
	Theorem 2.1.1. (and Denition) Any compact Riemann surface is homeomorphic to a sphere
	with handles. The number g ∈ N of handles is called the genus of the Riemann surface.
	For instance, every surface of genus zero is topologically a sphere, while a surface of positive
	genus g can be obtained topologically by identifying pairwise appropriate sides of a 4g-sided
	polygon.
	Example 2.1.2. The genus of the compactication Ĉ of the hyperelliptic curve (2.1.1) with

  Let f : M → N be a holomorphic covering. A point p ∈ M is called a branch point of f if it has no neighbourhood U (containing point p) such that f | U is injective. A covering with branch points is called ramied or branched covering (unramied otherwise). Theorem 2.1.2. Let f : M → N be a non-constant holomorphic mapping between two compact Riemann surfaces. Then there exists m ∈ N such that f takes every value c ∈ N precisely mThe number m above is called the degree of f . The covering f : M → N is called m-sheeted. For any a ∈ M, the number b f (a) = n a -1 is called the branch number of f at a, where n a ∈ N denotes the multiplicity of f at a.

	for some n ∈ N. A holomorphic mapping into C is called a holomorphic function, and a holomorphic mapping into CP 1 is called a meromorphic function. Non-constant holomorphic mappings between Riemann surfaces are also called holomorphic In particular, for a non-constant meromorphic function on a compact Riemann surface, the number m corresponds to the number of its poles (counting multiplicities). coverings. times (counting multiplicities). Example 2.1.3. Let us consider the compactication Ĉ of the hyperelliptic curve (2.1.1) together Ĉ with the meromorphic function f : (λ, µ) → λ of degree 2. The function f allows to consider as a two-sheeted branched covering of CP 1 where branch points are given by:

  .1.8) where A k , B k and A k , B k denote the A and B-periods of the dierentials ω and ω respectively. Normalized Abelian dierentials. It follows from identity (2.1.8) that the g × g matrix of A-periods of a basis of holomorphic dierentials (ω 1 , . . . , ω g ), with entries

	A kj =	ω j ,	j, k = 1, . . . , g,
	A k
	is invertible. This allows to dene normalized holomorphic dierentials as follows. Let (A, B) be a canonical basis of H 1 (R g ). The dual basis of holomorphic dierentials ω j normalized by
	ω j = 2iπδ j,k ,	j, k = 1, . . . , g,	(2.1.9)
	A k
	is uniquely dened and is called canonical.
	Abelian dierentials of the second and third kind are normalized by imposing the condition
	that all A-periods vanish. It follows that normalized Abelian dierentials of the second kind Ω (N ) a and of the third kind Ω b-a are uniquely dened.
	The B-periods of normalized Abelian dierentials Ω	(N ) a

  Abel map and divisors. Let Λ be the lattice Λ = 2iπZ g + BZ g generated by the A and B-periods of the normalized holomorphic dierentials ω 1 , . . . , ω g . The complex torus J(R g ) = C g /Λ is called the Jacobian of the Riemann surface R g . Let us denote by Π the Abel map Π : R g → J(R g ) dened by divisor D on a Riemann surface R g is a formal symbol D = N k=1 n k p k , for some n k ∈ Z and N ∈ N\{0}, where p k ∈ R g . The degree of D is dened by deg D = N k=1 n k . The Abel map can be extended linearly to all divisors on R g . A divisor D is called principal if it describes the set of zeros and poles (counting multiplicities) of a meromorphic function f on R g . In this case we use the notation D = (f ) (note that deg (f ) = 0). This allows to dene a linear equivalence relation on the set of divisors: two divisors D and D are called linearly equivalent if the divisor D -D is principal. The following Abel theorem provides necessary and sucient conditions for a divisor to be principal: Theorem 2.1.4. A divisor D is principal if and only if deg D = 0 and Π(D) ≡ 0.

	Jacobian, Π(p) =	p	ω,	(2.1.13)
		p 0
	for any p ∈ R g , where p 0 ∈ R g is the base point of the application, and ω = (ω 1 , . . . , ω g ) t is the vector of normalized holomorphic dierentials. In the whole thesis we use the notation
	b a = Π(b) -Π(a).
			g,	(2.1.12)
	is called the Riemann matrix of R g . From the Riemann's bilinear identity (2.1.8) it can be deduced that the Riemann matrix B is symmetric and has a negative denite real part (more
	generally, a Riemann matrix is a symmetric matrix with negative denite real part). These
	properties of the matrix B ensure that the so-called theta function, dened in the next section,
	is an entire function on C g .

A

  .[START_REF] Arnold | Mathematical methods of classical mechanics[END_REF]) by(2.3.3) it follows that W a,j = 0. In this case the function u(x, t) dened in (2.3.18) is solution of the KdV equation(2.3.16).

  Assume that does not cross cycles of the canonical homology basis. Choose arbitrary constants d ∈ C g and k, ζ ∈ C. Let y(x, t) be an implicit function of the variables x, t ∈ R dened by

	.1.10)
	which by (4.1.8) and (4.1.9) leads to (4.1.5).
	From identities (4.1.4) and (4.1.5) we are now able to construct theta-functional solutions of the
	CH equation:
	Proposition 4.1.2.

Let a, b ∈ R g such that σ(a) = b, and let e ∈ R g be a ramication point.

Denote by

an oriented contour between a and b which contains point e.

  2 were introduced in (4.1.3) with z = Z(x, t)d. By (4.1.11), the derivative with respect to the variable x of the implicit function

y(x, t) is given by

  .1.17) Now let us express the function m(x, t) = uu xx + k introduced in (4.1.2) in terms of functions g 1 and g 2 . By (4.1.4) and (4.1.16), the rst derivative of the function u (4.1.15) with respect to the variable x is given by

  I ∈ R g , where the vectors T and L satisfy 2 T + HL = diag(H). For this choice of vector d, (4.1.28) becomes

.1.28) Let us choose a vector d ∈ C g such that d =diπ diag(H) + 2iπT + BL for some vectors T, L ∈ Z g . Reality of the vector d + d together with (3.2.3) imply d = 1 2 Re(B) L + i d I (4.1.29)

for some d

  Now let us check that h has a constant sign with respect to y, t ∈ R. Since Zd ± r 2 ∈ iR g , by Proposition 3.4.2 the functions Θ(Zd ± r 2 ) of the real variables y and t do not vanish if the hyperelliptic curve is an M-curve, i.e., if all branch points in (4.0.1) are real. Hence h is a real continuous non vanishing function with respect to the real variables y and t, which means it has a constant sign. Therefore, x is a real-valued function of y and t if the constant ζ in (4.1.11) is chosen such that ln(h(y, t))ζ is real. It is straightforward to see that the solution u (4.1.14) is a real-valued function of the real variables y and t, and then is real-valued with respect to the real variables x and t. Now x t 0 ∈ R and let us study smoothness conditions on the function u(x, t 0 ) with respect to the variable x. First let us check that solution u (4.1.14) is a smooth function of the real variable y. The function u is smooth with respect to y if and only if it does not have singularities.

  do not vanish if the hyperelliptic curve is dividing (in this case diag(H) = 0), and if the arguments Zd ± r 2 in the theta function are real (modulo 2iπZ g ). The vector

.35)

for some d R ∈ R g and T ∈ Z g . For this choice of vector d, by

(4.1.34)

the function h is a real-valued function of the real variables y and t. Now let us study in which cases the function h has a constant sign. The sign of the function h is constant with respect to y and t if the functions Θ(Zd ± r 2 ) do not vanish. By (4.1.31), (4.1.33) and (4.1.35), the vectors Zd ± r 2 belong to the set S 1 introduced in (3.4.2). Hence by Proposition 3.4.1, the functions Θ(Zd ± r 2 )

  satises (4.2.3) for some n, m ∈ Z g . Therefore, considering the change of variables (a, b) = (e, f ) in (2.3.10) and using the periodicity property (2.2.3) of the theta function, one gets

	2, dened in (2.3.11)
	and (2.3.12).
	Proof. As proved in (2.3.22), the vector r =

f e ω

  Note that equation (4.2.12) can be written in the form(u xt + uu xx ) x = -(4 + u xx ) u x .Let β, δ ∈ C and α 1 , α 2 ∈ C be arbitrary constants. Let us look for solutions u of equation (4.2.14) having the following form:

	(4.2.14)

[START_REF] Alber | Wave solutions of evolution equations and Hamiltonian ows on nonlinear subvarieties of generalized Jacobians[END_REF]

is solution of the Dym-type equation (4.2.1) for arbitrary k ∈ C * . Proof.

  .[START_REF] Calogero | Solvable quantum version of an integrable Hamiltonian system[END_REF] where ψ(z) = 2 D f ln Θ(z) + D f φ(z). Identity (4.2.29) implies that the function u y (y) has a zero at y 0 of order 2n -1, namely, Finally, combining (4.2.28) and (4.2.30), we conclude that the function u(x, t 0 ) has an innite number of singularities of the type O (xx 0 ) Remark 4.2.1. Analogously, if all ramication points of R g are stable under τ and if one chooses local parameters such that

	2n
	2n+1	which correspond to cusps.

u(y)u(y 0 ) = O (yy 0 ) 2n . (4.2.30)

  Contrary to the CH equation, by (5.2.10) and Remark 4.2.1, there exists only one class of real-valued solutions without pole, which represent interaction between g solitary cusps. We have proved: Proposition 5.2.2. Take α > 0 and β j ∈ R all distinct such that |β j | > α. Choose d ∈ iR g and α 2 , ζ ∈ R. Assume that variables x, y, t are related by the condition

g,

(5.2.10)

since integration of the dierential ω 0 j over a closed contour encircling the cut [-α, α] gives a non zero polar period. Finally, from (2.3.13) and (5.1.11) it can be checked that

α 1 = -

1 2 α .

(

5.2.11)

To get real solutions, assume that α 2 ∈ R and |β j | > α for j = 1, . . . , g.

  we obtain Proposition 6.3.2. Solutions(6.3.14) are smooth if the curve R g is dividing and d ∈ R g . Assume that solutions (6.3.14) are smooth for any vector d in a component T v (3.4.4) of the Jacobian, then the curve is dividing and d ∈ R g .

	Proof. By (6.3.17) and (6.3.25), the vector Z-d belongs to the set S 1 introduced in (3.4.2). Hence
	by Proposition 3.4.1, the solutions are smooth if the curve is dividing (in this case diag(H)=0),

  The solutions are smooth since the denominator in (7.2.7) is a nite sum of real exponentials. Remark 7.2.4. The dark N -soliton solutions (7.2.7) depend on N + 1 real parameters d k , θ and a real meromorphic function f (7.2.2) dened by 2n + 2 real parameters. The solitons are dark since the modulus of the ψ j tends to A j when x ∈ R tends to innity. Example 7.2.1. With the notation of Proposition 7.2.1 and 7.2.2, functions ψ j (7.2.7) are given for N = 1 by

  Moreover, in this limit condition (7.2.12) yields (7.2.11). Remark 7.2.5. The bright N -soliton solutions (7.2.10) depend on 2N complex parameters d2k-1 , w v 2k-1 , and 2n + 1 real parameters w a j , γ j , θ. Moreover, all parameters appearing in (7.2.10) are free, contrary to the dark multi-solitons (7.2.7) where parameters w u k and w v k have to satisfy the polynomial equation (7.2.3). The solitons are bright since the modulus of the ψ j tends to zero when x ∈ R tends to innity, in contrast to the dark solitons.

	Example 7.2.3. With the notation of Proposition 7.2.3, the functions ψ j (7.2.10) are given for
	N = 1 by

  1/2 , and the remaining notation is the same as in Proposition 7.2.1 for g = 2N .

	Remark 7.2.6. Functions (7.2.15) cover a family of breather solutions of n-NLS s depending on
	N complex parameters d k , a real parameter θ, and a real meromorphic function f (7.2.2) dened
	by 2n + 2 real parameters.

  Proposition 7.3.1. Let k ∈ N satisfy 1 ≤ k ≤ g. Let w a , w b , w u k , w v k , h ∈ C, and A, κ 1 , κ 2 ∈ C \ {0}. Choose d ∈ C g .Then the following functions give solutions of the system (6.4.2)

  .2.4) P B = -P B + HP A . Denote by ( Ã, B) the homology basis on R g produced by the Tretko-Tretko algorithm. As we saw in (2.1.2), there exists a symplectic matrix From the symplectic transformation (8.2.6) we obtain the following transformation law between the matrices P Ã, P B and P A , P B dened in (8.2.3):

							(8.2.5)
			A B C D	∈ Sp(2g, Z) such that
	A B C D	Ã B	=		A B	.	(8.2.6)
	A B C D	P Ã P B	=	P A P B	.
							(8.2.9)
	and by (8.2.5)
	C Re P Ã + D Re P B =	1 2	HP A	(8.2.10)

(8.2.7) Therefore, by (8.2.4) one gets A Re P Ã + B Re P B = P A (8.2.8) A Im P Ã + B Im P B = 0, C Im P Ã + D Im P B = Im (P B ) .

(8.2.11)

  the matrix C Im P Ã + D Im P B is also invertible. Lemma 8.2.1. The matrices A, B, C, D ∈ M g (Z) solving (8.2.8)-(8.2.11) satisfy:A t = Im P B C Im P Ã + D Im P B B t = -Im P Ã C Im P Ã + D Im P B Re P B A Re P Ã + B Re P BNotice that the construction of these solutions given in Chapter 6 allows to express the imaginary part of the constant h (8.2.25) in terms of the characteristic δ. Namely, since the reality condition ψ * = ρ ψ

	C t = D t =	1 2 1 2	-1 -1 A t H --1 B (8.2.26) (8.2.12) (8.2.13) (8.2.14)

t H + Re P Ã A Re P Ã + B Re P B -1 . (8.2.15)

  Remark 8.2.2. In the case where the spectral curve is an M-curve, i.e. H = 0, the vectors of characteristic (8.2.16) and (8.2.17) do not depend explicitly on the symplectic matrix appearing in the change of homology basis and are uniquely dened by: à Im P t B Re P à -Im P t à Re P B

			We deduce that
			Im( h) =	1 2	ln	Θ (8.2.27)
	δ1 =	1 2	diag Re P -1	Im P t Ã	(8.2.28)
	δ2 =	1 2	diag Re P B Im P t B Re P à -Im P t à Re P B	-1	Im P t B	.	(8.2.29)

B[ δ]( Z + r) Θ B[ δ]( Zr) -Im( G3 ).

  is being generalized to arbitrary hyperelliptic curves.

			INDEX
	gap, 7		nonlinear Schrödinger equation, 4
	genus, 18
			peakon, 61
	half-period, 132		period, 21
	Hamiltonian, 1		period lattice, 23
	Harnack's inequality, 33		polar period, 21
	holomorphic mapping, 18	Index prime form, 25
	branch number, 19
	branch point, 18		ramication point, 78
	degree, 19		real oval, 33
	ber, 73		residue (of a dierential), 21
	Abel map, 23 homology basis		Chebyshev polynomials, 134 Riemann matrix, 22
	Abel's theorem, 23 canonical, 20		collision, 104 Riemann sphere, 27
	Abelian dierential, 21 symplectic transformation, 20		elastic, 104 Riemann surface, 17
	normalized, 22 of the rst kind, 21 of the second kind, 21 of the third kind, 21 period, 21 polar period, 21 homology group rst, 19 relative, 35 hyperelliptic curve, 18 involution, 30		covering hyperelliptic, 30 m-sheeted, 19 holomorphic, 18 ramied, 18 critical point, 78 cross-ratio real, 33 real dividing, 33 with punctures, 35 Riemann theta function, 25 Riemann's bilinear identity, 21
	Abelian function, 24 algebraic curve, 17 surface, 97		on a Riemann surface, 27 on the sphere, 27 Schottky problem, 13
	dividing, 33 hyperelliptic, 18 M-curve, 33 integrable system, 1 intersection index, 20 inverse scattering transform method, 2 non-singular, 18 real, 14 analytic continuation numerical, 133 involution anti-holomorphic, 33 hyperelliptic, 30		Schottky uniformization, 13 cusp-like singularity, 46 Sine-Gordon equation, 30 cuspon, 59 soliton, 3 cycle, 19 bright, 104 dark, 104 Davey-Stewartson equations, 11 solitonic limit, 55 degenerate hyperelliptic curve, 55 almost, 135
	paths for, 142 Baker-Akhiezer function, 4 Jacobi inversion problem, 24 Jacobian, 23		Riemann surface, 98 divisor, 23 degree, 23 spectral curve, 2 spinor, 26 symplectic transformation, 20
	base point, 23 Kadomtsev-Petviashvili equation, 5 bidierential, 26 Korteweg-de Vries equation, 3 branch cuts, 105 branch number, 19 branch point, 18 Lax pair, 2 Liouville integrable system, 1 local parameter, 18 breather, 108 lump, 123 rational, 108		linearly equivalent, 23 theta divisor, 25 principal, 23 theta function theta, 25 Riemann, 25 dromion, 123 with characteristic, 23 Dym type equation, 8 Toda lattice equation, 31 elliptic Tretko-Tretko algorithm, 142 function, 25 Trott curve, 146
	Camassa-Holm equation, 8 M-curve, 33		integral, 25 uniformization map, 98
	canonical basis of cycles, 20 meromorphic function		surface, 83
	chain, 19 on a Riemann surface, 18		essential singularity, 5 Vandermonde determinant, 85
	characteristic, 23 on the sphere, 102		vector of Riemann constants, 25
	half integer, 23 real, 75		Fay's identity, 27 Vinnikov basis, 34
	non-singular, 24 multi-component nonlinear Schrödinger equa-	nite-gap potential, 4
	odd, 24 tion, 10		fundamental group, 78 Wronskian, 4

Figure 7.4: 2-breather of 4-NLS s with s = (-1, 1, 1, -1).

Figure 7.6: 2-rational breather of 4-NLS s with s = (1, 1, 1, 1).

For the ease of representation we only give 4 digits here, though at least 12 digits are known for these quantities.

Remerciements

Proposition 5.3.2. For xed t 0 ∈ R, peakon solutions of the Dym-type equation are obtained by gluing at peak points innite pieces of solutions given in parametric form, with y ∈ R, by: U n (y) = D 2 f ln Θ g (z g , z + n Bg )α 2 ,

(5.3.27)

x(y) = D f ln Θ g (z g , z + n Bg )α 2 t 0 + d 0 + n V f,g , (5.3.28) where z g = γ y + V f,g t 0 + d g and z = Vf t 0 + d for some d ∈ R g-1 and γ, d 0 , d g , α 2 ∈ R.

Now let us construct time-dependent peakon solutions. For any n ∈ Z, it can be seen that the piece U n (x, t) is bounded by the curves C n and C n+1 dened by C n = {x = c n (t), t ∈ R} , (5.3.29) where c n (t) = D f ln Θ g-1 (z + n Bg )α 2 t + d 0 + n V f,g .

(5.3.30) Moreover, we observe that functions U n-1 (x, t) and U n (x, t) are glued along the curve C n where U n (c n (t), t) = D 2 f ln Θ g-1 (z + n Bg )α 2 .

(5.3.31)

The functions U n-1 (x, t) and U n (x, t) having dierent partial derivatives in x and t along this curve, we conclude that a global piecewise smooth solution U (x, t) has peak curves along C n , with n ∈ Z.

Part III

Generalized nonlinear Schrödinger equations

Therefore, from (7.1.14) and (7.1.3) we deduce that

k (w a ) k (w a ) 2 .

(7.1.20)

Finally, taking the limit → 0 in (6.1.1) one gets

2 .

(7.1.21)

7.2

Degenerate algebro-geometric solutions of n-NLS

As mentioned in Section 6.3.1, one way to construct solutions of (6.3.1) is rst to solve its complexied version (6.3.2) and impose reality conditions (6.3.3). Algebro-geometric solutions of the complexied system were obtained in Theorem 6.3.1. Recall that the proof of this theorem is based on the following identity:

which is satised by the vectors V a k associated to the ber f -1 (z a ) = {a 1 , . . . , a n+1 } over z a ∈ CP 1 . We shall use this relation to construct solutions of (6.3.1) in terms of elementary functions.

Remark 7.2.1. The relationship between solutions of the KP1 equation and solutions of the multi-component NLS equation has been investigated in Section 6.3.7. This relationship implies that all solutions of the multi-component NLS equation constructed in this chapter provide also solutions of the KP1 equation.

In the next section, solutions of (6.3.2) in terms of elementary functions are derived from solutions (6.3.4) by degenerating the associated Riemann surface R g into a Riemann surface of genus zero. Imposing reality conditions (6.3.3), by an appropriate choice of the parameters one gets special solutions of (6.3.1) such as multi-solitons and breathers. To the best of my knowledge, such an approach to multi-solitonic solutions of n-NLS s has not been studied before.

Moreover, breather and rational breather solutions to the multi-component case are derived here for the rst time.

Determinantal solutions of the complexied n-NLS equation

Solutions of the complexied scalar NLS equation in terms of elementary functions were obtained in [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF], when the genus of the associated hyperelliptic spectral curve tends to zero. For specic choices of parameters, they get dark and bright multi-solitons of the NLS equation, as well as quasi-periodic modulations of the plane wave solutions previously constructed in [START_REF] Its | Exact integration of nonlinear Schrödinger equation[END_REF]. A direct generalization of this approach to the multi-component case is not obvious, due to the complexity of the associated spectral curve. To bypass this problem and to construct spectral data associated to algebro-geometric solutions (6.3.4) in the limit when the genus tends to zero, we use the uniformization map between the degenerate Riemann surface and the Riemann sphere.

Details of such a degeneration have been presented in Section 7.1. meromorphic function f . Note that equation (7.2.1) holds in the limit, since by (7.1.9) and (7.2.2) one has:

Multi-solitonic solutions of n-NLS

Imposing the reality conditions (6.3.3) on the degenerate solutions (7.2.4) of the complexied system, one gets particular solutions of (6.3.1) such as dark and bright multi-solitons. Dark and bright solitons dier by the fact that the modulus of the rst tends to a non zero constant and the modulus of the second tends to zero when the spatial variable tends to innity. Such solutions were obtained in [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF] for the one component case by degenerating algebro-geometric solutions, and describe elastic collisions between solitons. Elastic means that the solitons asymptotically retain their shape and speed after interaction. The interaction of vector solitons is more complex than the one of scalar solitons because inelastic collisions can appear in all components of one solution (see, for instance, [START_REF] Ablowitz | Integrable Nonlinear Schrödinger Systems and their Soliton Dynamics[END_REF]).

In what follows N ∈ N with N ≥ 1.

Dark multi-solitons of n-NLS s , s = (1, . . . , 1).

Dark multi-soliton solutions of 2-NLS s were investigated in [START_REF] Radhakrishnan | Bright and dark soliton solutions to coupled nonlinear Schrödinger equations[END_REF]. The dark N -soliton solution derived here corresponds to elastic interactions between N dark solitons. Moreover, we show that this type of solutions does not exist for the focusing multi-component nonlinear Schrödinger equation, i.e., in the case where s = (1, . . . , 1).

Put s j = sign(f (w a n+1 )f (w a j )). Then the following functions dene smooth dark N -soliton solutions of n-NLS s , where s = (s 1 , . . . , s n ) with s = (1, . . . , 1),

det(T j,0 ) exp {-i (E j x -F j t)} .

(7.2.7)

Here A j = |q 2 (a n+1 , a j )| 1/2 , and the remaining notation is as in Proposition 7.2.1 with g = N .

the scalars (B) ik (7.1.6) are real for any i, k ∈ {1, . . . , N }. Therefore, the functions ψ and φ (7.3.1) are smooth, since their denominator does not vanish as a nite sum of real exponentials. Bright multi-soliton of DS1 ρ and DS2 -.

In this part we construct bright multi-soliton solutions to the DS1 ρ and DS2equations. It is well known that such solutions can be written in terms of a quotient of sums of exponentials, for which the modulus tends to zero if the spatial variables tend to innity.

To get bright multi-soliton solutions, one degenerates once more solutions (7.3.1) of the com-

, and take h ∈ R.

Degeneration. Choose a small parameter > 0 and dene d k =ln + dk , for k = 1, . . . , 2N , and

for k = 1, . . . , N . Moreover, put κ 1 = κ1 (w aw b ), and κ 2 = κ2 (w aw b ). Consider in the determinant det(T 1 ) appearing in (7.3.1) the substitution

for i = 2, . . . , N , where L k denotes the line number k of the matrix T 1 and (T 1 ) i,k the entries of this matrix. An analogous transformation has to be considered for the matrix T -1 appearing in Reality condition. By imposing the following constraints on the parameters:

it can be seen that the functions ψ and ψ * (7.3.1) in the considered limit satisfy the reality condition ψ * = ρ ψ, with ρ =sign(κ 1 κ 2 ).

The solutions. Let θ ∈ R. Then the following degenerated functions dene N -rational breather

where K β , with β = 0, 1, is a 2N × 2N matrix with entries (K β ) ik given by -for i and k even:

-for i even and k odd:

Here z k is a linear function of the variables ξ, η and t given by

Moreover, for c ∈ {a, b}, the scalars Vc,k , Ŵc,k and rk satisfy Vc,2k = Vc,2k-1 , Ŵc,2k = Ŵc,2k-1 and r2k = r2k-1 , and are given by:

,

Example 7.3.5. Figure 7.11 (resp. Figure 7.12) shows the interaction between a line rational breather and a rational breather solution of DS1 -with the following choice of parameters:

By line rational breather we denote a growing and decaying mode localized only in one direction.

Dromion and lump solutions of the DS equations

Here we construct the dromion solution of DS1 ρ and the lump solution of DS2which correspond to solutions with exponential decrease in all directions of the plane. Such solutions are derived by suitable degenerations of solutions (7.3.1) to the complexied system, and by imposing the reality condition ψ * = ρ ψ. This appears to be the rst time that such solutions are obtained as limiting cases of theta-functional solutions.

Dromion of DS1 ρ . Boiti et al. [START_REF] Boiti | Scattering of localized solitons in the plane[END_REF] have shown that the DS1 equation has solutions that decay exponentially in all directions. The solutions they obtained can move along any direction in the plane, and the only eect of their interactions is a shift in their position, independently of their relative initial position in the plane. Later, Fokas and Santini [START_REF] Fokas | Dromions and a boundary value problem for the Davey-Stewartson 1 equation[END_REF][START_REF] Santini | Energy exchange of interacting coherent structures in multidimensions[END_REF] pointed out that by an appropriate choice of the boundary conditions, the localized solitons (called `dromions') of the DS1 equation possess properties which are dierent from the properties of one-dimensional solitons, namely, the obtained solutions do not preserve their form upon interaction. For a particular choice of their spectral parameters, they recovered solutions previously derived by Boiti et al. For details on the theory of dromion solutions the reader is referred to [START_REF] Radha | Localized Coherent Structures and Integrability in a Generalized (2 + 1)-Dimensional Nonlinear Schrödinger Equation[END_REF] and references therein. In this section we explore how the simplest dromion solution can be derived from algebro-geometric solutions.

Let us consider solutions of the complexied system obtained in (7.3.1). Assume g = 4 and put

Degeneration. Choose a small parameter > 0 and dene d k =ln( ) + dk for k = 1, . . . , 4, and

Moreover, put κ 1 = κ1 α v 1 and κ 2 = κ2 α u 3 . Now consider the limit → 0 in (7.3.1). The functions ψ and φ obtained in this limit are given by (7.3.18).

The solutions. Therefore, the following degenerated functions describe smooth solutions of DS2 -

where ξ = x + iy and β 1 = μ1 κ-1

Simplications. To simplify (7.3.20), put

for arbitrary µ ∈ C. In this way, functions (7.3.20) become To compute the integral a ξ ω, one has to analytically continue µ on the connecting line between λ(a) and ξ onto the λ-sphere. Whereas the root µ is not supposed to have any branching on the considered path, the square root in Matlab is branched on the negative real axis. To analytically continue µ on the path [λ(a), ξ], we compute the Matlab root at some λ j ∈ [λ(a), ξ], j = 0, . . . , N c and analytically continue it starting from µ(a) by demanding that

The so dened sheets will be denoted here and in the following by numbers, i.e., a point on sheet 1 with projection λ(a) into the base is denoted by (λ(a)) (1) .

Thus the computation of the Abel map is reduced to the computation of line integrals on the connecting line between λ(a) and ξ in the complex λ-plane. For the numerical computation of such integrals we use Clenshaw-Curtis integration (see, for instance, [START_REF] Trefethen | Spectral Methods in Matlab[END_REF]): to compute an integral 1 -1 h(x) dx, this algorithm samples the integrand on the N c + 1 Chebyshev collocation points x j = cos(jπ/N c ), j = 0, . . . , N c . The integral is approximated as the sum:

(see [START_REF] Trefethen | Spectral Methods in Matlab[END_REF] on how to obtain the weights w j ). It can be shown that the convergence of the integral is exponential for analytic functions h as the ones considered here. To compute the Abel map, one uses the transformation λ → λ(a)(1 + x)/2 + ξ(1x)/2, to the Clenshaw-Curtis integration variable. The same procedure is then carried out for the integral from ξ to b.

The theta functions are approximated as in [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods[END_REF] as a sum,

The periodicity properties of the theta function (2.2.3) make it possible to write z = z 0 + 2iπN + BM for some N, M ∈ Z g , where z 0 = 2iπα + Bβ with α i , β i ∈ ] -1 2 , 1 2 ] for i = 1, . . . , g. The value of N θ is determined by the condition that all terms in (2.2.1) with |m i | > N θ are smaller than machine precision, which is controlled by the largest eigenvalue of the real part of the Riemann matrix (the one with minimal absolute value since the real part is negative denite), see [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods[END_REF][START_REF] Frauendiener | Algebraic curves and Riemann surfaces in Matlab[END_REF].

To control the accuracy of the numerical solutions, we use essentially two approaches. First we check the theta identity (6.1.1), which is the underlying reason for the studied functions being solutions to n-NLS s and DS, at each point in the spacetime. This test requires the computation of theta derivatives not needed in the solution (which slightly reduces the eciency of the code since additional quantities are computed), but provides an immediate check whether the solution satises (6.1.1) with the required accuracy. Since this identity is not implemented in the code, it provides a strong test. This ensures that all quantities entering the solution are computed with the necessary precision. In addition, the solutions are computed on Chebyshev collocation points Proof. Multiplying equality (8.2.9) from the left by the matrix C t , we deduce from (2.1.3) and (2.1.4) that

and D) to determine the symplectic matrix in (8.2.7). In practice, this means that a convenient ansatz for one of the matrices has to be found. The others then follow from the relations in Lemma 8.2.1.

Thus to construct these matrices one rst checks which of the matrices Re P Ã , Re P B , Im P Ã , Im P B are invertible. This way a matrix can be identied (e.g. A) in terms of which the others can be expressed. The task is thus reduced to provide an ansatz for this matrix such that the others will have entire components. We illustrate this approach at the example of the Trott curve below. Proposition 8.2.1. Let ( Ã, B) be the canonical homology basis obtained with the Tretko-Tretko algorithm; we denote with a tilde the quantities expressed in this basis. Under the change of homology basis (8.2.6), solutions of n-NLS s and DS equations given in (6.3.14) and (6.4.10), (6.4.20) respectively, which are expressed in the basis satisfying (3.2.1), transform as follows: the vector d appearing in the solutions becomes (2iπ) -1 Kt d where K = 2iπA + B B, and the theta function Θ = Θ B with zero characteristic, transforms to the theta function

with Z = i( Wa -Wb ), and the vectors N, M dened in (3.3.9) become A t N + C t M and B t N + D t M respectively.

Proof. Under the change of the canonical homology basis (8.2.6), the vector ω = (ω 1 , . . . , ω g ) t of normalized holomorphic dierentials transforms as

where K = 2iπA + B B. According to the transformation law (2.2.5) of theta functions, it can be checked after straightforward calculations, that under this change of homology basis, quantities

(2.3.11), (2.3.12) and (6.1.2), (6.1.3) transform as:

We deduce that solutions of the n-NLS s and DS equations given in (

, respectively, transform as follows: the vector d becomes (2iπ) -1 Kt d, and the theta function Θ = Θ B with zero characteristic, transforms to the theta function Θ B[ δ] with characteristic δ.

To compute the vectors of the characteristic δ we consider the inversion of the symplectic matrix in (8.2.6), which leads to

Since the characteristic used in Chapter 6 to construct algebro-geometric solutions of n-NLS

Outlook Fay identities, Gowdy universes and Einstein-Maxwell equations

The approach based on Fay's identity to generate solutions to integrable PDEs will be applied to further interesting equations as the stationary axisymmetric Einstein-Maxwell equations which reduce to Ernst equations in the electro-vacuum case. Relevant objects in astrophysics would be disks with strong magnetic elds. An explicit solution would allow us to study the role of strong magnetic elds in astrophysically interesting objects such as neutron stars. It was shown by Korotkin [91] that the corresponding solutions are given on three-sheeted Riemann surfaces where several branch points are parametrized by the physical coordinates. Such solutions have not been studied at all so far. Ernst equations also appear in the context of so-called Gowdy universes. In contrast to the standard Ernst equation they are not studied on R 2 but on spaces with dierent topology, e.g. T 2 . This leads to dierent regularity requirements for the solutions.

The formation of singularities for regular initial data is of particular interest in this context. For this question, a formulation of the initial value problem in terms of a Riemann-Hilbert problem will be studied.

Hurwitz spaces and random matrices

The remarkable property of algebro-geometric solutions to the Ernst equations is that some of the branch points depend on the physical coordinates. Thus modular properties of Riemann surfaces will be important in this context. This is best discussed in terms of Hurwitz spaces, i.e. the space of meromorphic functions on a Riemann surface. The discoveries will also be applicable in the context of random matrix theory. This theory has a wide range of applications in physics and mathematics. Today random matrices are an important tool in physics in 2D quantum gravity, statistical physics and string theory as well as in mathematics in the problem of Laplacian growth, intersection theory on moduli spaces and number theory, see [START_REF] Forrester | Developments in random matrix theory[END_REF] for references. Hurwitz spaces occur here in the context of the partition function in the limit of diverging matrix size, see [START_REF] Eynard | 1/N 2 correction to free energy in hermitian two-matrix model[END_REF].

A better understanding of Hurwitz spaces would make a study of the little explored case of interacting two-matrix models possible. Hurwitz spaces also occur in the context of Frobenius manifolds which are used to solve the Witten-Dijkgraf-Verlinde-Verlinde equation. This equation is of central importance among others in topological conformal quantum eld theory, Gromov-Witten invariants and singularity theory.

Theta functions and Hurwitz spaces

The biggest obstacle for an extensive use of multi-dimensional theta functions in praxis has so far been the diculty in evaluating these functions and other characteristic quantities on Riemann surfaces numerically. Though at least on hyperelliptic surfaces all important quantities are explicitly known in terms of integrals on the surface, the dependence on the branch points and the number of integrals to be determined are a problem. Several approaches to the numerical treatment of theta functions have been realized, see [START_REF] Deconinck | Computing Riemann matrices of algebraic curves[END_REF][START_REF] Deconinck | Computing Riemann Theta Functions[END_REF] for references. These two codes are distributed along with Maple. Whereas these codes are an important step to provide general access to multi-dimensional theta functions, they are not ecient enough to study their modular dependence. To discuss solutions to the Ernst equation where the periods have to be calculated for each point of the spacetime, in [START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods[END_REF][START_REF] Frauendiener | Hyperelliptic theta functions and spectral methods: KdV and KP solutions[END_REF][START_REF] Klein | Ernst Equation and Riemann Surfaces[END_REF] a spectral code to treat hyperelliptic theta functions has been given. This code computes periods even on almost degenerate Riemann surfaces in fractions of a second on an average computer to the order of machine precision. In collaboration with C. Klein and V. Shramchenko, a generalization of these codes to general algebraic curves would allow among others the visualization of solutions to the electro-magnetic Ernst equations.

Abstract

Fay's identity on Riemann surfaces is a powerful tool in the context of algebro-geometric solutions to integrable equations. This relation generalizes a well-known identity for the cross-ratio function in the complex plane. It allows to establish relations between theta functions and their derivatives. This oers a complementary approach to algebro-geometric solutions of integrable