C Kalla
email: caroline.kalla@u-bourgogne.fr

Breathers and solitons of generalized nonlinear Schrödinger equations as degenerations of algebro-geometric solutions

Breathers and solitons of generalized nonlinear

Schrödinger equations as degenerations of algebro-geometric solutions

Introduction

One of the signicant advances in mathematical physics at the end of the 19th century has been the discovery by Gardner, Greene, Kruskal and Miura [START_REF] Gardner | Method for Solving the Korteweg-de Vries Equation[END_REF] of the applicability of the Inverse Scattering Transform (IST) to the Korteweg-de Vries equation, and the construction of multisoliton solutions. 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 soliton solutions to the nonlinear Schrödinger equation (NLS)

i ∂ψ ∂t + ∂ 2 ψ ∂x 2 + 2ρ |ψ| 2 ψ = 0, (1.1)
where ρ = ±1, was proved by Zakharov and Shabat [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF] using a modication of the IST. The NLS equation is a famous nonlinear dispersive partial dierential equation with many applications, e.g. in hydrodynamics (deep water waves), plasma physics and nonlinear ber optics. The Nsoliton 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 relies 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 solutions satisfy the equation since dierentiation of a Wronskian is simple. On the other hand, multi-soliton solutions of (1.1) 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].

In the present paper, we construct solutions in terms of elementary functions of two generalizations of the NLS equation (1.1): the multi-component NLS equation (n-NLS), where the number of dependent variables is increased, and the Davey-Stewartson equation (DS), an integrable generalization to 2 + 1 dimensions. The solutions of n-NLS and DS presented in this paper are obtained by degenerating algebro-geometric solutions, previously investigated by the author in [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF] using Fay's identity [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF]. 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.

One way to generalize the NLS equation is to increase the number of dependent variables in (1.1). 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)
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) 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]. Multi-soliton solutions of (1.2) 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].

In this paper, we present a family of dark and bright multi-solitons, breather and rational breather solutions to the multi-component NLS equation. This 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.

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

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, (1.3)
where α = i, 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 corresponds to the case α = i, and DS2 ρ to α = 1. The DS equation (1.3) 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]. 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-anddecaying 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] Radhakrishnan | 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 DS2equation 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 to the DS1 and DS2

+ equations, as well as a family of bright multi-solitons for the DS1 and DS2equations, obtained by degener- ating 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. In this paper it is shown how the simplest solutions, the dromion and the lump solutions can be derived from algebro-geometric solutions.

The paper is organized as follows: Section 2 contains various facts from the theory of theta functions and identities due to Fay. These identities were used to construct algebro-geometric solutions of n-NLS and DS equations in [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF], and will be needed for the degeneration of the underlying theta-functional solutions. Section 3 provides technical tools dealing with 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 4 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 (1.2). In Section 5 a similar program is carried out for the DS equations; well known solutions such as multi-solitons, dromion or lump are rediscovered from an algebro-geometric approach.

2 Theta functions and Fay's identity Solutions of equations (1.2) and (1.3) in terms of the multi-dimensional theta function were discussed in [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF]. In this section we recall some facts from the construction of these solutions which will be used in the following to get particular solutions as limiting cases of algebro-geometric solutions.

Theta functions

Let R g be a compact Riemann surface of genus g > 0. Denote by {A j , B j } g j=1 a canonical homology basis, and by {ω j } g j=1 the dual basis of holomorphic dierentials normalized via

A k ω j = 2iπδ k, j k, j = 1, . . . , g. (2.1)
The matrix B of B-periods of the normalized holomorphic dierentials with entries (B) kj =

B k ω j is symmetric and has a negative denite real part. The theta function with (half integer)

characteristic δ = [δ , δ ] is dened by Θ[δ](z|B) = m∈Z g exp 1 2 B(m + δ ), m + δ + m + δ , z + 2iπδ (2.2)
for any z ∈ C g ; here δ , δ ∈ 0, 1 2 g are the vectors of characteristic and ., . denotes the scalar product u, v = i u i v i for any u, v ∈ C g . The theta function Θ[δ](z) is even if the characteristic δ is even i.e, 4 δ , δ is even, and odd if the characteristic δ is odd, i.e., 4 δ , δ 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.

Corollaries of Fay's identity

Let us rst introduce some notation. Let k a denote a local parameter near a ∈ R 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) + o (k a (p))) dk a (p), (2.3) where p lies in a neighbourhood of a, and V a, j , W 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 , (2.4)
where F : C g -→ C is an arbitrary function, and denote by D a the operator of directional derivative along the vector W a = (W a,1 , . . . , W a,g ) t . Now let δ be a non-singular odd characteristic. For any z ∈ C g and any distinct points a, b ∈ R g , the following two versions of Fay's identity [START_REF] Fay | Theta functions on Riemann surfaces[END_REF] hold (see [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF] and [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF])

D a D b ln Θ(z) = q 1 + q 2 Θ(z + r) Θ(z -r) Θ(z) 2 , (2.5) D a ln Θ(z + r) Θ(z) + D 2 a ln Θ(z + r) Θ(z) + D a ln Θ(z + r) Θ(z) -K 1 2 + 2 D 2 a ln Θ(z) + K 2 = 0, (2.6)
where the scalars q i , K i for i = 1, 2 depend on the points a, b and are given by

q 1 (a, b) = D a D b ln Θ[δ](r), (2.7
)

q 2 (a, b) = D a Θ[δ](0) D b Θ[δ](0) Θ[δ](r) 2 , (2.8) K 1 (a, b) = 1 2 D a Θ[δ](0) D a Θ[δ](0) + D a ln Θ[δ](r),
(2.9)

K 2 (a, b) = -D a ln Θ(r) -D 2 a ln (Θ(r) Θ(0)) -D a ln Θ(r) -K 1 (a, b) 2 .
(2.10)

Here we used the notation r = b a ω where ω = (ω 1 , . . . , ω g ) t is the vector of the normalized holomorphic dierentials.

2.3 Integral representation of q 2 (a, b) and K 1 (a, b)

Quantities q 2 (a, b) and K 1 (a, b) dened in (2.8) and (2.9) respectively, admit integral representation which will be more convenient for our purposes. These integral representations follow from the fact that meromorphic dierentials normalized by the condition of vanishing A-periods can be expressed in terms of theta functions.

Let a, b ∈ R g be two distinct points connected by a contour which does not intersect A and B-cycles. Hence we can dene the normalized meromorphic dierential of the third kind Ω b-a which has residue 1 at b and residue -1 at a. Now let a ∈ R g , and N ∈ N with N > 1. The normalized meromorphic dierential of the second kind Ω (N ) a has only one singularity at the point a and is of the form

Ω (N ) a (p) = 1 k a (p) N + O(1) dk a (p), p ∈ R g , (2.11)
where k a is a local parameter in a neighbourhood of a.

Proposition 2.1. Let a, b ∈ R g be distinct points. Denote by k a and k b local parameters in a neighbourhood of a and b respectively. The quantities q 2 (a, b) and K 1 (a, b) dened in (2.8) and (2.9) respectively admit the following integral representations:

q 2 (a, b) = -lim b→b ã→a k a (ã) k b ( b) -1 exp b ã Ω b-a (p)
, (2.12) where the integration contour does not cross any cycle of canonical basis, and

K 1 (a, b) = lim ã→a ã c Ω (2) a (p) + 1 k a (ã) - b c Ω (2) a (p), (2.13)
where c is an arbitrary point on R g .

Proof of (2.12) can be found in [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF], where similar statements lead to (2.13).

Uniformization map 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 give explicitly the constants (2.7)-(2.10) in this limit. As illustrated in Section 4 and 5, 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.

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 -, (3.1)
where > 0 is a small parameter, such that the vanishing cycle coincides with the homology class of a closed contour 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. (3.2)
• Dierentials of the second kind:

Ω (2) a = 1 k a (w a ) dw (w -w a ) 2 , (3.3)
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 (2.11) 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, (3.4)
(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) 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 (1.2) and (1.3) 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 )    . (3.5)
Here we use the same notation for the quantities (B) ik on the degenerated surface. The expression in the right hand side of (3.5) can be put into a determinantal form (see Proposition 3.1) which will be used in the whole paper. This determinantal form can be obtained from the following representation of the components (B) ik after degeneration, obtained from (3.2) and (3.4),

(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 . (3.6)
Hence, following [START_REF] Matsuno | Multiperiodic and multisoliton solutions of a nonlocal nonlinear Schrödinger equation for envelope waves[END_REF] one gets Proposition 3.1. 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), (3.7)
where T is a g × g matrix with entries

(T) ik = δ i,k + w v i -w u i w v i -w u k e 1 2 (z i +z k ) .
(3.8)

Degenerate constants

The next step is to give explicitly the quantities (independent of the vector z) appearing in (2.5) and (2.6), i.e., V a , W a , r, 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) and (3.2) that V a,k = 1 k a (w a ) 1 w a -w v k - 1 w a -w u k , (3.9)
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 , (3.10) r k = ln w b -w v k w b -w u k w a -w u k w a -w v k , (3.11)
for k = 1, . . . , g. Moreover, from the integral representation of q 2 (a, b) and K 1 (a, b) (see (2.12) and (2.13)), using (3.2) and (3.3) one gets

q 2 (a, b) = 1 k (w a )k (w b )(w a -w b ) 2 , (3.12) K 1 (a, b) = 1 k (w a )(w b -w a ) - 1 2
k (w a ) k 2 (w a ) .

(3.13)

Putting z = 0 in (2.5) and taking the limit → 0 leads to

q 1 (a, b) = -q 2 (a, b), (3.14)
due to the fact that the theta function tends to one and that its partial derivatives tend to zero.

In the same way, taking the limit

→ 0 in (2.6) one gets K 2 (a, b) = -K 1 (a, b) 2 . (3.15)
4 Degenerate algebro-geometric solutions of n-NLS One way to construct solutions of (1.2) is rst to solve its complexied version, a system of 2n

equations of 2n dependent variables ψ j , ψ * j n j=1 , 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, (4.1)
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) under the reality conditions

ψ * j = s j ψ j , j = 1, . . . , n. (4.2)
Algebro-geometric solutions of the system (4.1) were obtained in [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF] by the use of the degenerated versions (2.5) and (2.6) of Fay's identity; these solutions are given by: Theorem 4.1. 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 are solutions of the system (4.1)

f -1 (z a ) = {a 1 , . . . ,
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)} . (4.3)
Here Θ denotes the theta function (2.2) with zero characteristic, and

Z = i V a n+1 x + i W a n+1 t,
where vectors V a n+1 and W a n+1 are dened in (2.3). Moreover, r j = a j a n+1 ω, where ω is the vector of normalized holomorphic dierentials, 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 ). (4.4)
The scalars q i , K i for i = 1, 2 are dened in (2.7)-(2.10).

The proof of this theorem is based on the following identity:

n+1 k=1 V a k = 0, (4.5)
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 .

We shall use this relation to construct solutions of (1.2) in terms of elementary functions.

Remark 4.1. The relationship between solutions of the Kadomtesv-Petviashvili (KP1) equation (generalization of the KdV equation to two spatial variables, see, for instance, [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]) and solutions of the multi-component NLS equation was investigated in [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF]. This relationship implies that all solutions of equation (1.2) constructed in this paper provide also solutions of the KP1 equation as explained in [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF].

In the next section, solutions of (4.1) in terms of elementary functions are derived from solutions (4.3) by degenerating the associated Riemann surface R g into a Riemann surface of genus zero. Imposing reality conditions (4.2), by an appropriate choice of the parameters one gets special solutions of (1.2) such as multi-solitons and breathers. To the best of our 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 (4.3) in the limit when the genus tends to zero, we use the uniformization map between the degenerate Riemann surface and the sphere. Details of such a degeneration were presented in Section 3.

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

f (w) = α n+1 i=1 w -w a i w -w b i (4.6)
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 lying in a neighbourhood of w a . Solutions of the complexied system (4.1) associated to the meromorphic function f (4.6) on the sphere are given by: Proposition 4.1. Let j, k ∈ N satisfy 1 ≤ j ≤ n and 1 ≤ k ≤ g. Let f be a meromorphic function (4.6) 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 Then the following functions are solutions of the complexied system (4.1)

w u k , w v k ∈ C satisfy f (w u k ) = f (w v k ).
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)}. (4.8)
For β = -1, 0, 1, T j,β denotes the g × g matrix with entries (3.8) 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 (3.9) and (3.10), and r j,k is dened in (3.11) 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 (3.12) and (3.13).

Proof. Consider solutions (4.3) associated to a Riemann surface R g of genus g, and assume

f (a i ) = 0 for any 1 ≤ i ≤ n + 1. Pinch all A-cycles of the associated Riemann surface R g into double points, as explained in Section 3. 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 (4.6). In the limit considered here, the theta function tends to the determinantal form (3.7). Quantities dened on the degenerated surface and independent of the variables x and t were constructed in Section 3.3 and are given in (3.9)- (3.15). Condition (4.7) 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 meromorphic function f . Note that equation (4.5) holds in the limit, since by (2.3) and (4.6) one has

n+1 i=1 V a i ,k = 1 f (w u k ) - 1 f (w v k ) (4.9)
which by (4.7) equals zero for k = 1, . . . , g.

w a i , w b i for 1 ≤ i ≤ n + 1, d k for 1 ≤ k ≤ g, and A j for 1 ≤ j ≤ n.
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 , (4.10)
where λ = µ β -1 for any µ ∈ C and any β = 0. Such a transformation may be useful to simplify the expressions in the obtained solutions and thus to facilitate the numerical implementation.

Multi-solitonic solutions of n-NLS

Imposing reality conditions (4.2) on the degenerate solutions (4.8) of the complexied system, one gets particular solutions of (1.2) 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, it is shown 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),

ψ j (x, t) = A j e iθ det(T j,1 )
det(T j,0 ) exp {i (-E j x + F j t)} .

(4.12)

Here A j = |q 2 (a n+1 , a j )| 1/2 , and the remaining notation is as in Proposition 4.1 with g = N .

Proof. Let us check that the functions ψ j and ψ * j dened in (4.8) satisfy reality conditions (4.2) 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 (4.8). 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 (3.12) leads to s j = sign(f (w a n+1 )f (w a j )). Moreover, with (4.11) it can be seen that condition (4.5) 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, (4.13)
for k = 1, . . . , N . Therefore, by (4.13) the quantity f (w a n+1 )f (w a j ) cannot be positive for all j, which yields s = (1, . . . , 1). The solutions are smooth since the denominator in (4.12) is a nite sum of real exponentials.

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) .

4.2.2

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 (4.8).

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.

Proposition 4.3. Let j ∈ N satisfy 1 ≤ j ≤ n. Take w a j , θ ∈ R and choose d ∈ C 2N such that d2k-1 = d2k . Moreover, let w u 2k , w v 2k-1 ∈ C satisfy

w u 2k = w v 2k-1 (4.14)
for 1 ≤ k ≤ 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) ,

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 . (4.16)
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 j -w v 2k-1 w a j w v 2k-1 α u 2k-1 .
Proof. Consider functions (4.8) obtained from (4.3) 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 lying in a neighbourhood of w a i , i = 1, . . . , n + 1, and assume g = 2N . Hence condition (4.5) becomes

n+1 i=1 γ i 1 w a i -w v k - 1 w a i -w u k = 0 (4.17)
for k = 1, . . . , N . Now put A j = |q 2 (a n+1 , a j )| 1/2 in (4.8), where q 2 (a n+1 , a j ) = γ n+1 γ j (w a n+1 -w 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 , (4.18)
for k = 1, . . . , N . Now put γ n+1 = 2 and consider in the determinant det(T j,1 ) appearing in (4.8) 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 (4.8) converge towards functions (4.15), 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 ). Moreover, in this limit condition (4.17) yields (4.16).

Remark 4.5. The bright N -soliton solutions (4.15) 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 (4.15) are free, contrary to the dark multi-solitons (4.12) where parameters w u k and w v k have to satisfy the polynomial equation (4.7). 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.

Breather and rational breather solutions of n-NLS

Solutions obtained here dier from the dark multi-solitons studied in Section 4.2.1 by the reality condition imposed on parameters w u i and w v i in solutions (4.8) of the complexied system for i = 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.

4.3.1

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 4.4. Let j, k ∈ N satisfy 1 ≤ j ≤ n and 1 ≤ k ≤ N . Let f be a real meromorphic function (4.6) 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 (4.7) and 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)} , (4.20) where To simplify the computation of the solutions, we apply transformation (4.10) to the solutions (4.20), with β and λ given by

w u 2k = w v 2k-1 , w u 2k-1 = w v 2k .
A j = |q 2 (a n+1 , a j )|
β = 1, λ = 1 2 f (w a n+1 ) f (w a n+1 ) -2 . (4.21)
Hence, the quantity f (w a n+1 )f (w a n+1 ) -2 V a n+1 ,k in the expression (3.10) 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 (3.13) for the scalar K 1 (a n+1 , a j ).

Example 4.2. Figure 1 shows a breather solution of the 4-NLS s equation with s = (-, -, +, -). 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 4.3. Figure 2 shows an elastic collision between two breather solutions of the 4-NLS s equation with s = (-, +, +, -). 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.

4.3.2

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 breather solutions (4.20) in the limit when the parameters w v 2k-1 and w u 2k-1 tend to each others, 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 (4.20) 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 (4.20), one gets a family of N -rational breather solutions of n-NLS s . Proposition 4.5. Let N, j ∈ N satisfy 1 ≤ N ≤ n and 1 ≤ j ≤ n. Let f be a real meromorphic function (4.6) 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 1 ≤ k ≤ N . Moreover, let w u 2k-1 , w v 2k ∈ C, 1 ≤ k ≤ N , be complex conjugate critical points of the meromorphic function f , i.e., they are solutions of f (w) = 0, which is equivalent to

n+1 i=1 1 f (w a i ) 1 (w -w a i ) 2 = 0. (4.22)
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)} , (4.23)
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 i -w 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 k is a linear function of the variables x and t given by

z k = i Va n+1 ,k x + i Ŵa n+1 ,k t -dk for k = 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+1 -w a j (w a n+1 -w u 2k-1 ) (w a j -w 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 symplify the expression for the obtained solutions, apply the transformation (4.10)

to functions (4.20) with β and λ as in (4.21). 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 , (4.24)
for some α v 2k-1 , α u 2k ∈ C, where k = 1, . . . , N . Note that equation number k of system (4.9) 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 . (4.25)
Hence, in the limit → 0, equation (4.25) 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 meromor- phic function f for k = 1, . . . N, i.e., they are solutions of f (w) = 0, in such way that equation (4.5) 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 (4.20). Note that parameters α v 2k-1 , α u 2k cancel in this limit, and the degenerated functions take the form (4.23).

Remark 4.7. Functions (4.23) provide a family of rational breather solutions of n-NLS s de- pending on N complex parameters d k , a real parameter θ, and a real meromorphic function f (4.6) dened by 2n + 2 real parameters, chosen such that f admits complex conjugate critical points.

Example 4.4. With the notation of Proposition 4.5 the functions ψ j (4.23) for N = 1 are given by

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

Example 4.5. Figure 3 shows a rational breather solution of the 4-NLS s equation with s = (+, +, +, +). 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 (w -w a i ) -2 = 0. We observe that 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 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 4.6. Figure 4 shows a 2-rational breather solution of the 4-NLS s equation with s = (+, +, +, +). 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 (w -w 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 4.

ξ = 1 2 (x -iα y), η = 1 2 (x + iα y), α = i, 1 in (1.
3). In these coordinates the DS equations become

i ψ t + 1 2 (∂ 2 ξ + ∂ 2 η )ψ + 2 φ ψ = 0, ∂ ξ ∂ η φ + ρ 2 (∂ 2 ξ + ∂ 2 η )|ψ| 2 = 0.
(5.1)

Recall that DS1

ρ denotes the case α = i (here ξ and η are both real), and DS2 ρ the case α = 1

(here ξ and η are pairwise conjugate).

To construct solutions of (5.1) in terms of elementary functions, let us rst introduce its complexied version:

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, (5.2)
ϕ ξη + 1 2 ((ψψ * ) ξξ + (ψψ * ) ηη ) = 0,
where ϕ := Φ + ψψ * . This system reduces to (5.1) under the reality condition:

ψ * = ρ ψ, (5.3)
which leads to ϕ = φ. Theta-functional solutions of (5.2) were studied in [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF] and can be written in the following form.

Theorem 5.1. 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 a 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 (5.2)

ψ(ξ, η, t) = 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 , (5.4) ϕ(ξ, η, t) = 1 2 (ln Θ(Z -D)) ξξ + 1 2 (ln Θ(Z -D)) ηη + h 4 .
Proof. Consider solutions (5.4) of system (5.2) in the limit when the Riemann surface degenerates to a Riemann surface of genus zero, as explained in Section 3. 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 lying in a neighbourhood of w a ∈ R 0 , k a (w) = w -w a . Therefore, quantities independent of variables ξ, η and t are obtained from (3.9)-(3.15).

Remark 5.2. Functions (5.7) 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 investigated 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 dier-

ent directions.
In what follows N ∈ N with N ≥ 1. 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, (5.11)
functions ψ and ψ * in (5.7) 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 ) , (5.12)
since u and v can be interchanged in the proof of (3.7). Therefore, functions ψ and φ in (5.7) 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 (5.7) consists of a nite sum of real exponentials (see (3.7)), since ξ, η, t are real.

Reality condition for DS2 + . Let us check that with the following choice of parameters,

w a = w b , κ 1 = κ 2 , w u k , w v k ∈ R, k = 1, . . . , N, (5.13)
the functions ψ and ψ * (5.7) satisfy the reality condition ψ * = ψ. With (5.13), it is straightforward to see that (5.12) is also satised. Moreover, since G 1 = G 2 , G 3 ∈ R and (w a -w b ) 2 < 0, the functions ψ and ψ * (5.7) satisfy the reality condition ψ * = ψ. Therefore, they dene dark multi-soliton solutions of DS2 + .

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 v 1 < w u 1 < w v 2 < w u 2 < . . . < w v N < w u N ,
the scalars (B) ik (3.6) are real for any i, k ∈ {1, . . . , N }. Therefore, the functions ψ and φ (5.7)

are smooth, since their denominator does not vanish as a nite sum of real exponentials.

w u 2k-1 = w a + α -1 u 2k-1 (w a -w b ), w v 2k-1 = w b + α -1 v 2k-1 (w a -w b ), w u 2k = w b + α -1 u 2k (w a -w b ), w v 2k = w a + α -1 v 2k (w a -w b ), (5.14)
for k = 1, . . . , N . Moreover, put κ 1 = κ1 (w a -w b ), and κ 2 = κ2 (w a -w b ). Consider in the determinant det(T 1 ) appearing in (5.7) the substitution

L 2i -→ L 2i - (T 1 ) 2i,2 (T 1 ) 2,2 L 2
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 function ψ * . Now take the limit → 0 in (5.7). The function ψ obtained in this limit has the form (5.17). 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, (5.15)

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 as those in the (x, t) coordinates extended along a spatial variable y.

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 a -w b | -1 in (5. 7
). It is straightforward to see that with the following choice of parameters,

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 , (5.18)
for k = 1, . . . , N , functions ψ and ψ * (5.7) satisfy the reality condition ψ * = ρ ψ with ρ = sign(κ 1 κ 2 ). Therefore, analogously to the n-NLS equation, functions ψ and φ in (5.7) give N -breather solutions of DS1 ρ .

Remark 5.6. 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.

Example 5.1. Figure 5 shows the evolution in time of 2-breather solution of DS1 -with the following choice of parameters:

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.

5.3.2

Multi-rational breathers of DS1 ρ

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 interaction 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 (5.7).
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 (5.19)
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 (5.7). The function ψ obtained in this limit is an N -rational breather solution of DS1 ρ given by (5.21).

Reality condition. Imposing the following constraints on the parameters:

w a , w b , h ∈ R, κ 1 , κ 2 ∈ R \ {0} , d2k = d2k-1 , w u 2k-1 = w v 2k , k = 1, . . . , N, (5.20)
it can be seen that the functions ψ and ψ * (5.7) 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

solutions of DS1 ρ ψ(ξ, η, t) = A e iθ det(K 1 ) det(K 0 ) exp -i (G 1 ξ + G 2 η -G 3 t 2 ) , φ(ξ, η, t) = 1 2 (ln det(K 0 )) ξξ + 1 2 (ln det(K 0 )) ηη + h 4 , (5.21)
where K β , with β = 0, 1, is a 2N × 2N matrix with entries (K β ) ik given by -for i and k even:

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

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

z k = i κ 1 Va,k ξ -i κ 2 Vb,k η + i κ 2 1 Ŵa,k -κ 2 2 Ŵb,k t 2 -dk .
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: Vc,2k-1 = 1 (w c -w u 2k-1 ) 2 , Ŵc,2k-1 = -2 (w c -w u 2k-1 ) 3 ,

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 localized in all directions of the plane. These solutions arise by suitable degenerations of solutions (5.7) 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 thetafunctional 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 performed 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] Radhakrishnan | 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 (5. Moreover, put κ 1 = κ1 α v 1 and κ 2 = κ2 α u 3 . Now consider the limit → 0 in (5.7). The functions ψ and φ obtained in this limit are given by (5.24).

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.

(

Put ρ =sign(κ 1 κ2 ). With (5.23), it can be seen that the degenerated functions ψ and ψ * obtained in the considered limit satisfy the reality condition ψ * = ρ ψ. 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 .
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 (5.24) are smooth solutions of DS1 ρ .

Remark 4 . 2 .

 42 Functions (4.8) give a family of solutions to the complexied multi-component NLS equation (4.1) depending on 3n + g + 2 complex parameters:

Remark 4 . 3 .

 43 The following transformations leave equation (4.1) invariant

Remark 4 . 4 .

 44 The dark N -soliton solutions (4.12) depend on N + 1 real parameters d k , θ and a real meromorphic function f (4.6) 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 4 . 1 .

 41 With the notation of Proposition 4.1 and 4.2, functions ψ j (4.12) are given for N = 1 by

Figure 1 :

 1 Figure 1: Breather of 4-NLS --+-.

Figure 2 :

 2 Figure 2: 2-breather of 4-NLS -++-.

Figure 3 :

 3 Figure 3: Rational breather of 4-NLS ++++ .

Figure 4 :

 4 Figure 4: 2-rational breather of 4-NLS ++++ .

5

  Degenerate algebro-geometric solutions of the DS equations Solutions of the DS equations (1.3) in terms of elementary functions constructed here are obtained analogously to the solutions of the n-NLS equation, therefore some details will be omitted. Let us introduce the function φ := Φ + ρ|ψ| 2 , where ρ = ±1, and the characteristic coordinates

5. 2 . 1

 21 Dark multi-soliton of DS1 ρ and DS2 + Here dark multi-solitons of the DS1 ρ and DS2 + equations are derived from functions (5.7) for an appropriate choice of the parameters. They were investigated in [41]. Put g = N and A = |κ 1 κ 2 | 1/2 |w a -w b | -1 in (5.7). Moreover, assume h ∈ R and d ∈ R N .

Remark 5 . 4 .

 54 One gets a family of smooth dark multi-soliton of the DS2 + equation, depending on 3N + 1 real parameters h, w u k , w v k , d k , a phase θ, and 2 complex parameters w a , κ 1 . 5.2.2 Bright multi-soliton of DS1 ρ and DS2 -In this part we construct bright multi-soliton 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 (5.7) of the complexied system. Put g = 2N and A = |κ 1 κ 2 | 1/2 |w a -w b | -1 in (5.7), and take h ∈ R. Degeneration. Choose a small parameter > 0 and dene d k = -ln + dk , for k = 1, . . . , 2N , and

Figure 5 :

 5 Figure 5: 2-breather of DS1 -at a) t = 0, b) t = 45.

r2k- 1 =

 1 w a -w b (w a -w u 2k-1 ) (w b -w u 2k-1 ) , for k = 1, . . . , N . Constants G 1 , G 2 , G 3 are given in (5.10). Remark 5.7. Functions (5.21) give a family of rational solutions of DS1 ρ depending on 2N complex parameters d 2k-1 , w u 2k-1 and 6 real parameters w a , w b , h, θ, κ 1 , κ 2 . Example 5.2. Figure 6 shows the evolution in time of the 2-rational breather solution of DS1 - with the following choice of parameters: w a = 2, w b = 1, w u 1 = 2i, w u 3 = 2 + i, κ 1 = κ 2 = 1, d k = h = 0.

Figure 6 :

 6 Figure 6: 2-rational breather of DS1 -at a) t = -5, b) t = 0, c) t = 5.

Figure 7 :

 7 Figure 7: Interaction between a line rational breather and a rational breather of DS1 at 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 8 :

 8 Figure 8: Interaction between a line rational breather and a rational breather of DS1 at 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.

and w u 1 = α u 1 , w u 2 = w a + α u 2 , w u 3 = w b + α u 3 , w u 4 = α u 4 , w v 1 = w a + α v 1 , w v 2 = α v 2 , w v 3 = α v 3 , w v 4 =

 112233441122334 [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]. Assume g = 4 and putA = |κ 1 κ 2 | 1/2 |w a -w b | -1 .Degeneration. Choose a small parameter > 0 and dene d k = -ln( ) + dk for k = 1, . . . , 4, w b + α v 4 .(5.22)

  a n+1 } over z a . Choose the local parameters k a j (p) = f (p) -z a , for any point p ∈ R g lying in a neighbourhood of a j . Let D ∈ C g and A j = 0 be arbitrary constants. Then the following functions {ψ j } n j=1 and ψ *

		n
	j	j=1

  Proposition 4.2. Let j, k ∈ N satisfy 1 ≤ j ≤ n and 1 ≤ k ≤ N . Let f be a real meromorphic function (4.6) of degree n + 1 on the sphere, having n + 1 real zeros {w a i } n+1 i=1 . Choose θ ∈ R and d ∈ R N . Moreover, assume that w u k , w v k ∈ C satisfy (4.7) andw u k = w v k .

	(4.11)

  1/2 , and the remaining notation is the same as in Proposition 4.1 for g = 2N .

Remark 4.6. Functions (4.20) 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 (4.6) dened by 2n + 2 real parameters.

  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	,	(5.24)

Remark 5.3. 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 .

where the vectors V a , V b and W a , W b were introduced in (2.3). Moreover r = ω, where ω is the vector of normalized holomorphic dierentials, and the scalars G 1 , G 2 , G 3 are given by

(5.6)

Scalars q 2 (a, b), K 1 (a, b), K 2 (a, b) are dened in (2.8), (2.9), (2.10) respectively.

Remark 5.1. In the case where vectors V a and V b satisfy V a + V b = 0, as mentioned in [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF],

solutions of the Davey-Stewartson equation become solutions of the NLS equation (1.1) under an appropriate change of variables.

In this section, we study the behaviour of theta-functional solutions (5.4) of the complexied DS equations when the Riemann surface degenerates into a Riemann surface of genus zero.

Imposing the reality condition (5.3), for particular choices of the parameters one gets well-known solutions such as multi-soliton, breather, rational breather, dromion and lump. 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 (5.2) 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 (5.1).

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 (3.8) with

Here the scalars r k are given in (3.11) and

where c ∈ {a, b}. The scalars G 1 , G 2 , G 3 are given by

(5.10) 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,

the functions ψ and ψ * obtained in the considered limit satisfy the reality condition ψ * = -ψ.

The solutions. Let θ ∈ R. With (5.15), 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 (5.16) these functions dene bright N -soliton solutions of DS2 -where γ = 1:

where

Here K and M are 2N × 2N matrices with entries (K) ik and (M) ik given by:

-for i and k even:

-for i even and k odd:

-for i odd and k even:

-for i and k odd:

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

for k = 1, . . . , N . Moreover, the scalars rk are dened by rk = (-1)

Remark 5.5. i) With (5.15), functions (5.17) 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 (5.16), functions (5.17) 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.

Remark 5.8. i) Functions (5.24) 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).

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 (5.7), 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 (5.2) invariant:

where

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

Moreover, put κ k = 2 κk , and µ k = 2 μk for k = 1, 2. Now take the limit → 0 in (5.25). The functions ψ and φ obtained in this limit are given in (5.26).

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

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.

The solutions. Therefore, the following degenerated functions provide smooth solutions of

where ξ = x + iy and β 1 = μ1 κ-1 1 . Here z 1 = i Va,1 (κ 1 ξ + μ1 t) -d1 and

Va,1 = -

Simplications. To simplify (5.26), put

for arbitrary µ ∈ C.

where ξ = x + iy. Here λ, ν, µ are arbitrary complex constants, and θ, h ∈ R. Solutions (5.27) coincide with the lump solution previously obtained in [START_REF] Arkadiev | Inverse scattering transform and soliton solutions for Davey-Stewartson II equation[END_REF].

Outlook

In this paper, 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 a previous paper [START_REF] Kalla | New degeneration of Fay's identity and its application to integrable systems[END_REF]. We did not construct all families of solutions present 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 parameters than the ones presented here with standard elastic collision. We believe that also this kind of solutions arises from algebro-geometric solutions after suitable degenerations.


Hubert Cecotti

Bertrand Rivet

EFFECT OF THE VISUAL SIGNAL STRUCTURE ON STEADY-STATE VISUAL EVOKED POTENTIALS DETECTION

Keywords: SSVEP, EEG, BCI, Signal structure, Signal detection

The detection of Steady-State Visual Evoked Potential (SSVEP) responses in the Electroencephalogram (EEG) is a current challenge in signal processing applied on Brain-Computer Interfaces (BCI). BCI based on SSVEP requires visual stimuli. When these stimuli are displayed on an LCD screen, the number of frequencies for flickering object on the screen is limited. We propose to extend the number of frequencies by composing different visual patterns. We evaluate the relationship in the frequency domain between the visual stimuli and the recorded EEG signal. The signal detection across seven types of SSVEP responses is achieved by considering spatial filters based on the generalized Rayleigh quotient. The mean detection accuracy across three subjects is 89.58%.

INTRODUCTION

Brain-Computer Interfaces (BCI) based on non-invasive scalp electroencephalography (EEG) is a multidisciplinary research area where signal processing is one of the major aspect for processing brain signals. The main purpose of BCI is to allow the communication through direct neural activity measurements [START_REF] Sejnowski | Toward Brain-Computer Interfacing (Neural Information Processing)[END_REF]. Among the different existing paradigms for creating a BCI, several rely on external stimuli. These stimuli are often visual like for the detection of event related potential like the P300 or for steady-state visual evoked potentials (SSVEP). One of main challenges in the BCI community is to increase the information transfer rate (ITR) while keeping a convenient interface. The ITR depends on the number of available commands in the BCI and the detection accuracy for these commands. Furthermore, the ITR is directly related to visual stimuli. Indeed, the accuracy of the commands depends on the reliability of the visual stimuli [START_REF] Cecotti | Reliable visual stimuli on LCD screens for SSVEP based BCI[END_REF]. In addition, the device that presents visual stimuli can be a bottleneck for enabling a large number of different commands. The reli-This work has been supported by French National Research Agency (ANR) through TecSan program (project RoBIK ANR-09-TECS-013). ability of the visual stimuli presentation is therefore a critical parameter for BCI based on external stimuli like P300 or SSVEP-BCI. While the required precision of the flashing time is not so crucial for the detection of the P300, the detection of an SSVEP response requires reliable stimuli due to its particular caracteristics.

In this study, we focus on the detection of SSVEP responses [START_REF] Vialatte | Steadystate visually evoked potentials: focus on essential paradigms and future perspectives[END_REF]. BCIs based on SSVEP are presented in the literature as more accessible than other BCI systems: they allow a high information transfer rate (ITR) and little or no user training [START_REF] Cecotti | A self-paced and calibration-less SSVEP based brain-computer interface speller[END_REF][START_REF] Bin | An online multi-channel SSVEP-based braincomputer interface using a canonical correlation analysis method[END_REF]. SSVEP responses are also easier to detect in the EEG [START_REF] Gao | A BCI based environmental controller for the motion-disabled[END_REF][START_REF] Friman | Multiple channel detection of steady-state visual evoked potentials for braincomputer interfaces[END_REF]. The visual stimuli that are used for inducing SSVEP responses are flickering lights at different frequencies. When someone looks at a particular flickering object at a frequency f, then a response occurs in the visual cortex. This response corresponds to the frequency of the stimulus and its higher harmonics, as depicted in Figure 1, [START_REF] Müller-Putz | Steady-state visual evoked potential (SSVEP)-based communication: impact of harmonic frequency components[END_REF]. It is then possible to detect different responses for different frequencies in the EEG signal.

The amplitude and the phase that define an SSVEP response depend on three main parameters [START_REF] Wu | Stimulator selection in SSVEP-based BCI[END_REF]: the frequency, the intensity of the flickering light, and the structure of the repetitive visual pattern (phase, duty cycle, ...). For the frequencies, SSVEP responses are usually obtained with frequencies between 5 and 50Hz [START_REF] Regan | Human brain electrophysiology: evoked potentials and evoked magnetic fields in science and medicine[END_REF]. If it is not possible to detect a Dirac delta function at a frequency f and its harmonics on the visual stimuli, then it will become harder to detect a Dirac delta function for f and its harmonics in the EEG signal [START_REF] Cecotti | Reliable visual stimuli on LCD screens for SSVEP based BCI[END_REF]. Therefore, the visual stimuli presented to the user has a direct impact on the efficiency of an SSVEP response detection method. This statement implies some problems for the choice of the display device. It is possible to use old CRT monitors, LEDs, or to adjust to some extend the frequency of some LCD screens in relation to the resolution of the screen. The usual refresh rate of LCD screens in the mass market is currently limited to 60Hz. Due to the limit of the vertical refresh rate in current LCD screens, the number of frequencies for displaying flickering objects is limited. Such limitation can represent a drawback for commercial and clinical applications using SSVEP. Monitors possess advantages for pre- senting together visual stimuli, monitoring the EEG activity and other information to the user. We propose a new strategy for enabling different frequencies on a screen. We evaluate the relationship between the amplitudes in the visual stimuli and the EEG signal. Then, the produced SSVEP responses are detected by considering spatial filters obtained through the generalized Rayleigh quotient. The paper is organized as follows. The visual stimuli are described in the second section. The method for signal detection is presented in the third section. Then, the experimental protocol is given in section four. Finally, the accuracy of the signal detection and its relationship with the visual stimuli are presented in the last section.

VISUAL STIMULI

Visual stimuli for generating an SSVEP response are usually presented on a set of LEDs or on an LCD screen [START_REF] Zhu | A survey of stimulationmethods used in SSVEP-based BCIs[END_REF]. We consider here visual stimulation on a classical LCD screen with a vertical refresh rate of 60Hz. The repetitive visual pattern is composed of n frames with n ≥ 2 (black+white). Hence, the frequency of a stimulus based on n frames is 60/n Hz. Besides, we are limited to a maximum of 30Hz due to the Shannon theorem. A visual stimulus is represented on the screen by a flickering box (white/black). We define a basic flickering pattern P b by i white frames followed by j black frames where i + j ≥ 2. Combined patterns P can be obtained by concatening different patterns P b . We denote by N P and S P , the number of undecomposable (basic) patterns and the size (the number of frames on the screen) of the combined pattern P . Thus, a repetitive pattern P will have peaks at 60/S P Hz and its harmonics. It will have a higher peak especially at the frequency 60 * N P /S P Hz. Table 1 presents different frequencies and the corresponding structure of the flickering pattern. The frequencies are given for the higher peak. These frequencies are obtained by concatening basic patterns representing 8.671 and 7.500Hz.

SIGNAL DETECTION

The detection of several SSVEP responses corresponds to a multi-class classification problem. We consider N f classes.

Each class corresponds to an SSVEP response, i.e. a particular frequency, a visual stimulus. We consider a visual stimulation flickering at f Hz. We consider the following description for the signal y i (t) as the voltage between the electrode i and a reference electrode at a time t:

y i (t) = N h k=1 a i,k sin(2πkft + Φ i,k ) + b i,t (1)
where N h is the number of considered harmonics. The signal is decomposed into two parts: the SSVEP response and the remaining EEG activity, which is considered as noise. The first part corresponds to the evoked SSVEP response signal, which is composed of a number of sinusoids with frequencies in relation to the stimulus frequency and a number of N h harmonic frequencies. Each sinusoid is defined by its amplitude and phase: a i,k and Φ i,k . b i,t corresponds to the background EEG activity. The online detection of an SSVEP response on an EEG signal requires a time segment for the signal analysis. We consider a time segment of N t samples of the signals, with a sampling frequency of F s Hz:

y i = X f a i,f + B i (2)
where y i = [y i (1), . . . , y i (N t )] T contains the EEG signal for the i th electrode in one time segment. The SSVEP model of the frequency f, X f , is contained in a matrix N t × 2N h defined by

X f (t, 2k -1) = sin(2πkft) (3) X f (t, 2k) = cos(2πkft) (4)
with 1 ≤ k ≤ N h . For N y electrodes, the signal is defined as:

Y = X f A f + B (5)
where Y = [y 1 , . . . , y Ny ] contained the sampled EEG signals from all the electrodes. A f contains all the amplitudes for all the expected sinusoids for every electrode signal related the the expected frequency to detect. Spatial filters shall be consider to enhance the SSVEP response in the signal. A spatial filter is represented by a linear combination of the signals measured by different electrodes. We denote by s, a linear combination of y i , the EEG after a spatial filter. Its purpose is to enhance the information contained in the EEG while reducing the nuisance signals.

s = Ny i=1 w i y i = Y w (6)
where w i is the weight for the i th electrode.

Several components can be created by using several sets of weights w. We note N s the number of channels. We first estimate the background activity by removing the potential SSVEP components from the signal. It is achieved by projecting the signal onto the orthogonal complement of the SSVEP model matrix (X).

Yf = Y -X f (X T f X f ) -1 X T f Y (7)
Spatial filters Ŵf that maximize the Signal-to-Noise Ratio are obtained though determining the generalized Rayleigh quotient that maximizes the following expression:

Ŵf = argmax W T r(W T Y T Y W ) T r(W T Yf T Yf W ) (8)
We denote by Ŷf = X T f Y Ŵf the signal after spatial filtering. The power of the expected frequencies and their harmonics are calculated for the N s components. For each frequency, the evaluation of the SSVEP response is defined by:

= 1 N s N h Ns i=1 N h k=1 Ŷf (i, 2k -1) 2 + Ŷf (i, 2k) 2 (9)
In the next sections, N s is equal to the number of electrodes and N h = 4. The detection of an SSVEP response is simply performed by selecting the frequency with the maximum associated value R(f).

EXPERIMENTAL PROTOCOL

The EEG signal was recorded on three healthy subjects (average age= 26.3 years). Subjects were wearing an EEG cap with 10 electrodes [START_REF] Chatrian | Ten percent electrode system for topographic studies of spontaneous and evoked eeg activity[END_REF]. The location of these electrodes are depicted in Figure 2. F 7 and F 8 were dedicated to the ground and the reference, respectively. The signal was recorded on O 1 ,O 2 ,P 3 ,P 4 ,P 7 ,P 8 ,P Z and F C z . The amplifier was a FirstAmp (Brain Products GmbH) with a sampling frequency of 2kHz. The signal was bandpassed filtered (Butterworth filter of order 4) with cut-off frequencies at 4 and 45Hz. Each subject looked at a box on the screen flickering at one of the seven frequencies presented previously (Table 1). For each Fig. 2. Location of the electrodes in the system 10-20.

frequency and for each subject, we record the equivalent of 20s of EEG signal. The detection procedure of the SSVEP responses is applied every 250ms by considering a sliding window of 3s, i.e., the signal detection is only preformed on 3s of EEG data.

RESULTS

Table 2 presents the detection accuracy across three subjects for the seven types of SSVEP responses, i.e., seven different visual stimuli. The average accuracy of the SSVEP detection is 89.58%. This result is interesting as it shows that with only two basic frequencies, it is possible to create five other frequencies and to obtain a reliable accuracy for the detection. Indeed, the only difference between two SSVEP responses can correspond to only one frame on the screen, i.e. a difference of only 16.7ms in the observed visual stimulus. It is worth mentioning that the accuracy can be increased by considering only a limited number of basic patterns. In the current evaluation, up to four basic patterns were combined.

Decreasing the number of basic patterns naturally leads to the improvement of the accuracy. Table 3 presents the results in relation to the number of basic patterns that can be considered. With only two types of SSVEP responses, it is possible to reach 98.07%. The amplitudes between the harmonics is different between the visual stimuli and the estimated amplitude in the EEG. Table 4 presents the mean and standard deviation of the amplitude ratio between the main frequency (60 * N P /S P ) and its higher harmonics (H * 60 * N P /S P ), across the three subjects and seven frequencies. These results show a stable behavior between the amplitudes in the EEG, in spite of the different methods for enabling these frequencies.

CONCLUSION

With a device like an LCD screen, which has a low frequency for enabling flashing patterns at a particular frequency, it is difficult to enable a large number of different frequencies.

We have proposed an efficient way for solving this problem. The proposed detection method was sufficient for detecting the SSVEP responses at the different frequencies across three subjects. Such method for providing visual stimuli on a screen could be considered for BCI based on the detection on SSVEP. A large number of frequencies implies a large number of possible BCI commands, hence increasing the information transfer rate. While the choice of LEDs can be judicious for displaying visual stimuli, the graphical user interface will be separated from the visual stimuli. Enabling efficient visual stimuli on an LCD screen can be an advantage for rehabilitation, spelling applications, applications combining neurofeedbacks... As low frequencies (below 20Hz) can be disturbing for the user, further works will extend the proposed strategy with new materials, which are initially dedicated for nvidia 3D vision, i.e., a screen with a vertical refresh rate of 120Hz and an nvida graphic card.

Fig. 1 .

 1 Fig. 1. Frequency amplitude in the EEG (bipolar combination of electrodes O 1 and O 2 ) for a visual stimulus flickering at 7.5Hz during 13s.

Table 1 .

 1 Frequency of the structure different visual stimuli.

	f (Hz) (N P , S P ) Signal structure (P)
	8.571	(1,7)	0000111
	8.276	(4,29)	00001110000111000011100001111
	8.182	(3,22)	0000111000011100001111
	8.000	(2,15)	000011110001111
	7.826	(3,23)	00001111000011110001111
	7.742	(4,31)	0000111100001111000011110001111
	7.500	(1,8)	00001111

Table 2 .

 2 SSVEP detection accuracy (in %) across 3 subjects.

							f (Hz)	Mean SD
			Subject 8.57	8.28	8.18	8.00	7.83	7.74	7.50
			1	88.40 65.21 97.10 79.71 84.05 100.0 100.0 87.78 11.8
			2	81.16 84.06 97.10 89.85 86.96 79.71 82.60 85.92 5.57
			3	98.55 94.20 94.20 91.30 100.0 86.96 100.0 95.03 4.51
			Mean	89.37 81.16 96.13 86.95 90.34 88.89 94.2 89.58 4.52
	Table 3. SSVEP detection accuracy (in %) across 3 subjects
	(S1,S2,S3).
	N P max N f	S1	S2	S3	Mean
	4	7	87.78 85.92 95.03 89.58
	3	5	93.04 91.01 97.10 93.72
	2	3	95.17 93.72 97.58 95.49
	1	2	96.38 97.83 100.0 98.07

Table 4 .

 4 Amplitude ratio based on the main frequency.

	H	S1	S2	S3	Mean
		Mean SD Mean SD Mean SD
	2	0.53 0.13 0.39 0.12 0.34 0.03 0.42
	3	0.24 0.04 0.20 0.03 0.17 0.02 0.20
	4	0.16 0.03 0.16 0.03 0.11 0.02 0.15