Zubov's result (1956)

ẋ = f (x) (1)
f (0) = 0 , x * = 0 is asymptotically stable.

(2)

Theorem

A set A containing 0 in its interior is the domain of attraction of (1) if and only if there exist continuous functions V , h such that

◮ V (0) = h(0) = 0, 0 < V (x) < 1 for x ∈ A \ {0}, h > 0 on R n \ {0} ◮ V (x n ) → 1 for x n → ∂A or x n → ∞,
Zubov's result ( 1956)

ẋ = f (x) (1)
f (0) = 0 , x * = 0 is asymptotically stable.

(2)

Theorem

A set A containing 0 in its interior is the domain of attraction of (1) if and only if there exist continuous functions V , h such that

◮ V (0) = h(0) = 0, 0 < V (x) < 1 for x ∈ A \ {0}, h > 0 on R n \ {0} ◮ V (x n ) → 1 for x n → ∂A or x n → ∞, ◮ DV (x) • f (x) = -h(x)(1 -V (x)) 1 + f (x) 2

Robust domains of attraction

Consider systems

ẋ(t) = f (x(t), d(t)) , t ∈ R
with a perturbation term d. We are interested in robust stability properties. In particular, the robust domain of attraction.

Robust domains of attraction

Consider systems

ẋ(t) = f (x(t), d(t)) , t ∈ R
with a perturbation term d. We are interested in robust stability properties. In particular, the robust domain of attraction.

Robust domains of attraction

ẋ(t) = f (x(t), d(t)) , t ∈ R Assumptions: • f : R n × D → R n continuous, locally Lipschitz continuous in x, uniformly in d • D ⊂ R m compact, convex, d(t) ∈ D a.e.
Robust domains of attraction Robust domains of attraction

ẋ(t) = f (x(t), d(t)) , t ∈ R Assumptions: • f : R n × D → R n continuous, locally Lipschitz continuous in x, uniformly in d • D ⊂ R m compact, convex, d(t) ∈ D a.
ẋ(t) = f (x(t), d(t)) , t ∈ R Assumptions: • f : R n × D → R n continuous, locally Lipschitz continuous in x, uniformly in d • D ⊂ R m compact, convex, d(t) ∈ D a.

A robust version of Zubov's theorem

Theorem:[Zubov's theorem for perturbed systems, SICON 2001] Under suitable growth conditions on g there is a unique viscosity solution of

inf d∈D {-Dv (x)f (x, d) -(1 -v (x))g (x, d)} = 0 v (0) = 0
The robust domain of attraction satisfies

A D (0) = v -1 ([0, 1)) . Example ẋ1 = -x 1 + x 2 ẋ2 = 0.1x 1 -2x 2 -x 2 1 -0.1x 3 1
Fixed points: [0, 0], unstable [-2.5505, -2.5505], asymptotically stable [-7.4495, -7.4495], asymptotically stable.

Example ◮ Curse of dimensionality: For systems of dimension greater than 4 the method is not really applicable.

ẋ1 = -x 1 + x 2 ẋ2 = 0.1x 1 -2x 2 -x 2 1 -(0.1 + d(t)) x 3 1 D = {0} Example ẋ1 = -x 1 + x 2 ẋ2 = 0.1x 1 -2x 2 -x 2 1 -(0.1 + d(t)) x 3 1 D = [-0.

Interconnection of two systems

Consider

Σ 1 : ẋ1 = f 1 (x 1 , x 2 ) Σ 2 : ẋ2 = f 2 (x 1 , x 2 ) f i : R N 1 +N 2 +Nu → R N i Σ 2 Σ 1 ✛ ✲ x 2 x 1
Input-to-state stability (ISS) -Lyapunov version Input-to-state stability (ISS) -Lyapunov version

V > γ( u ) =⇒ V < 0
Input-to-state stability (ISS) -Lyapunov version

V > γ( u ) =⇒ V < 0 {x : V (x) ≤ c}
Input-to-state stability (ISS) -Lyapunov version

V > γ( u ) =⇒ V < 0 {x : V (x) ≤ c} 0 {x : V (x) ≤ γ( u )} ẋ(t)
Input-to-state stability (ISS) -Lyapunov version

{x : V (x) ≤ c} 0 {x : V (x) ≤ γ( u )} V > γ( u ) =⇒ V < 0 Interconnection of two systems Consider Σ 1 : ẋ1 = f 1 (x 1 , x 2 ) Σ 2 : ẋ2 = f 2 (x 1 , x 2 ) f i : R N 1 +N 2 +Nu → R N i Σ 2 Σ 1 ✛ ✲ x 2 x 1

Interconnection of two systems

Consider Σ 1 : ẋ1 = f 1 (x 1 , x 2 ) Σ 2 : ẋ2 = f 2 (x 1 , x 2 )

f i : R N 1 +N 2 +Nu → R N i Σ 2 Σ 1 ✛ ✲ x 2 x 1
with two Lyapunov functions such that V 1 (x 1 ) > γ 12 (V 2 (x 2 )) ⇒ V1 < -α 1 ( x 1 )

V 2 (x 2 ) > γ 21 (V 1 (x 1 )) ⇒ V2 < -α 2 ( x 2 )

Goal

Can we compute ISS Lyapunov functions using a Zubov approach ?

  e. • f (0, d) = 0 for all d ∈ D • 0 is locally uniformly asymptotically stable Notation: D := {d : R → D ; d Lebesgue measurable}

  e. • f (0, d) = 0 for all d ∈ D • 0 is locally uniformly asymptotically stable Notation: D := {d : R → D ; d Lebesgue measurable} A D (0) := {x ∈ R n | φ(t; x.d) → 0 ∀d ∈ D}

  Contents Zubov's Method Robust domains of attraction Interconnected Systems Zubov's Method and Interconnected Systems

02, 0.02] Examples Drawbacks ◮ Solution of the PDE is obtained by discretisation of the state space.

Drawbacks

◮ Solution of the PDE is obtained by discretisation of the state space.

◮ Curse of dimensionality: For systems of dimension greater than 4 the method is not really applicable.

Idea:

View systems in larger dimension as interconnection of low-dimensional systems.

Consider

with two Lyapunov functions such that

The ISS small gain theorem Theorem (Jiang, Mareels, Wang 1996)

If there exist K ∞ -functions ρ 1 , ρ 2 such that

See also [Jiang, Teel, Praly 1994] [Grüne 2002] and [Dashkovskiy, Rüffer, W. 2007, 2009].

Zubov and ISS Lyapunov functions

Choose a gain γ ∈ K ∞ and define fγ :

with Lyapunov gain γ -1 . Outline of procedure (i) For each of the subsystems i = 1, 2 choose γ i ∈ K ∞ and compute the robust Lyapunov function v i .

Outline of procedure (i) For each of the subsystems i = 1, 2 choose γ i ∈ K ∞ and compute the robust Lyapunov function v i . (ii) For each v γ,i compute bounds

Outline of procedure (i) For each of the subsystems i = 1, 2 choose γ i ∈ K ∞ and compute the robust Lyapunov function v i . (ii) For each v γ,i compute bounds

The gain for each of the Lyapunov functions is then given by γij

Outline of procedure (i) For each of the subsystems i = 1, 2 choose γ i ∈ K ∞ and compute the robust Lyapunov function v i . (ii) For each v γ,i compute bounds

The gain for each of the Lyapunov functions is then given by γij := ψ j,2 • γ -1 i • ψ -1 i,1 . (iv) Do the two gains γ12 , γ21 satisfy the small gain condition ?

Outline of procedure (i) For each of the subsystems i = 1, 2 choose γ i ∈ K ∞ and compute the robust Lyapunov function v i . (ii) For each v γ,i compute bounds

The gain for each of the Lyapunov functions is then given by γij

i,1 . (iv) Do the two gains γ12 , γ21 satisfy the small gain condition ? (v) If this is the case there is a constructive procedure to get a Lyapunov function for the interconnected system, (see Dashkovskiy, Rüffer, W. 2009).

Outline of procedure (i) For each of the subsystems i = 1, 2 choose γ i ∈ K ∞ and compute the robust Lyapunov function v i . (ii) For each v γ,i compute bounds

The gain for each of the Lyapunov functions is then given by γij := ψ j,2 • γ -1 i • ψ -1 i,1 . (iv) Do the two gains γ12 , γ21 satisfy the small gain condition ? (v) If this is the case there is a constructive procedure to get a Lyapunov function for the interconnected system, (see Dashkovskiy, Rüffer, W. 2009). (vi) Use this function to estimate the domain of attraction.

Conclusions

◮ An approach for the computation of lower estimates for the domain of attraction of interconnected systems has been presented.

◮ The idea still needs a lot of tuning:

◮ which γ do you choose ◮ is γ ∈ K ∞ really necessary ? (Angeli and Astolfi, 2007) says clearly that this is not the case ◮ Do state transformations help ?


Reachable Sets

Let t 0 < t f , U = ∅ convex, compact, and X 0 ⊆ R n be given.

Control problem

For a given u

∈ L ∞ ([t 0 , t f ], R m ) find x ∈ W 1,∞ ([t 0 , t f ], R n ) with x (t) = f (t, x(t), u(t))
a.e. in [t 0 , t f ] Approximation on grid with stepsize h = (t f -t 0 )/N:

x(t 0 ) ∈ X 0 ψ(x(t f )) = 0 s(t,
grid functions u h (t i ) = u i , x h (t i ) = x i , i = 0, . . . , N Discrete Control Problem (Euler) For a discretized control function u h (•) find a solution x h (•) with x h (t i+1 ) = x h (t i ) + hf (t i , x h (t i ), u h (t i )), i = 0, 1, . . . , N -1 x h (t 0 ) ∈ X 0 ψ(x h (t N )) = 0 s(t i , x h (t i )) ≤ 0, i = 0, 1, . . . , N u h (•) ∈ U h Discrete reachable set at t i : R h (t i , t 0 , X 0 ) := y ∈ R n | ∃u h (•)
d H (R(T , t 0 , x 0 ), R h (T , t 0 , x 0 )) ≤ C 1 h.
Hausdorff distance: Approximation of Discrete Reachable Set I Strategy 1: collect all grid points with distance

√ n 2 ρ R 1 ρ := gρ∈Gρ dist(gρ,R h )≤ √ n 2 •ρ {g ρ }
To be computed:

dist(g ρ , R h ) = inf r ∈R h g ρ -r M.
{ŝ ρ }
To be computed:

ŝρ ∈ Π R h (g ρ ) = {r ∈ R h : g ρ -r = dist(g ρ , R h )} M. Gerdts
Optimal control techniques for reachable set computations SADCO, Kickoff, Mar 3-4, 2011

Approximation of Discrete Reachable Set III Strategy 3: outer approximation by complement of open balls

R 3 ρ := R n \ gρ∈ρZ n int B dist(gρ,R h ) (g ρ )
To be computed: Computing Distance and Best Approximations Algorithm: Choose a region G ⊆ R n and cover G by a grid G h with step-size h (or h p ) For every g h ∈ G h solve (discretized) optimal control problem:

dist(g ρ , R h ) = inf r ∈R h g ρ -r
(OCP g h ) Min 1 2 x(t f ) -g h 2 2 s.t. x (t) = f (t, x(t), u(t)) x(t 0 ) ∈ X 0 ψ(x(t f )) = 0 s(t, x(t)) ≤ 0 u(t) ∈ U Solution: x (•; g h )
d H R, R 1 ρ ≤ (C 1 + √ n 2 • C 2 ) • h
Strategy 2 (best approximations): Direct Shooting Example 3: Kenderov Example 4: Bilinear

d H R, R 2 ρ ≤ (C 1 + √ n • C 2 ) • h Strategy 3 (complement of open balls): d H R, R 3 ρ ≤ (C 1 + √ n • C 2 ) • h Assumptions: f lipschitz w.r.t. t, C 1 w.r.t. x, C 0 w.r.t. u, ∅ = U convex, compact, ρ = C 2 • h,
¡ ¢ £ ¤ ¤ ¥ ¥ ¦ § © ! " # $ $ $ $ $ $ $ $ % % % % % % % % control grid t 0 t N u 1 u N u h (B-
i ; z), u h (t i ; z)) ≤ 0, ∀i, s(x h (t i ; z)) ≤ 0, ∀i, ψ(x h (0; z), x h (1; z)) = 0,
Control problem: (t ∈ [0, 1], u(t) ∈ [-1, 1]) x (t) = 8 (a 11 x(t) + a 12 y (t) -2a 12 y (t)u(t)) , x(0) = 2 y (t) = 8 (-a 12 x(t) + a 11 y (t) + 2a 12 x(t)u(t)) , y (0) = 2 (a 11 = σ 2 -1, a 12 = σ 1 -σ 2 , σ = 0.
(t ∈ [0, 1], u(t) ∈ [0, 1]) x (t) = πy (t), x(0) = -1 y (t) = -πu(t)x(t), y (
Control problem: (t ∈ [0, 1], u(t) ∈ [0, 1]) x (t) = πy (t), x(0) = -1 y (t) = -πu(t)x(t), y (0)

Adaptivity

Idea:

Let g h be a grid point and x (t f ; g h ) an optimal solution.

Every grid point within the ball B r (g h ) and radius r = x (t f ; g h ) -g h is not reachable and thus needs not to be projected.

  d H (S, S) = max{d(S, S), d( S, S)}, d(S, S) = sup s∈S dist(s, S) M. Gerdts Optimal control techniques for reachable set computations SADCO, Kickoff, Mar 3-4, 2011

  x h ( tj ; z) -x j = 0, ∀j Structure M = 1 (single shooting) : small & dense; z = (x 1 , u 1 , . . . , u N ) M > 1 (multiple shooting) : large-scale & sparse; z = (x 1 , . . . , x M-1 , u 1 , . . . , u N ) M. Gerdts Optimal control techniques for reachable set computations SADCO, Kickoff, Mar 3-4, 2011 Numerical Solution Software OC-ODE, OC-DAE1, SODAS [G.]: direct multiple shooting discretization SQP method (non-monotone linesearch, BFGS update, primal active-set QP solver) various integrators (Runge-Kutta, BDF methods, linearized Runge-Kutta methods) various control approximations (B-splines of order k ) gradients by sensitivity differential equation sensitivity analysis and adjoint estimation extensions to adjoint gradient computation and mixed-integer optimal control problems Large-scale problems: www.worhp.de (academic licenses available) (t ∈ [0, 1], u(t) ∈ [-π, π]) x (t) = 2gy (t) cos(u(t)), x(0) = 0 y (t) = 2gy (t) sin(u(t)), y (0) = 1 Reachable sets for N = 5, 10, 20, 40: M. Gerdts Optimal control techniques for reachable set computations SADCO, Kickoff, Mar 3-4, 2011 Example 2: Rayleigh problem Control problem: (t ∈ [0, 2.5], u(t) ∈ [-1, 1]) x (t) = y (t), x(0) = -5 y (t) = -x(t) + y (t)(1.4 -0.14y (t) 2 ) + 4u(t), y (0) = -5 Reachable sets for N = 10, 20, 40, 80, 160: M. Gerdts Optimal control techniques for reachable set computations SADCO, Kickoff, Mar 3-4, 2011 Example 3: Kenderov Control problem: (t ∈ [0, 1], u(t) ∈ [-1, 1]) x (t) = 8 (a 11 x(t) + a 12 y (t) -2a 12 y (t)u(t)) , x(0) = 2 y (t) = 8 (-a 12 x(t) + a 11 y (t) + 2a 12 x(t)u(t)) , y (0) = 2 (a 11 = σ 2 -1, a 12 = σ 1 -σ 2 , σ = 0.9) Reachable sets for N = 20, 40, 80, 160, 320: M. Gerdts Optimal control techniques for reachable set computations SADCO, Kickoff, Mar 3-4, 2011

  0) = 0 Reachable sets for N = 10, 20, 40, 80, 160: M. Gerdts Optimal control techniques for reachable set computations SADCO, Kickoff, Mar 3-4, 2011

  sets with higher order methods zooming into interesting sub-regions possible state-space grid O(h) only once and not O(h 2 ) in each Euler step as in Chahma'03, avoiding an obstacle (time to collision 0.5...2 s) Scenario 2: overtaking maneuver Questions: Can a collision be avoided at all? If yes, how? M. Gerdts Optimal control techniques for reachable set computations SADCO, Kickoff, Mar 3-4, 2011 Optimal Trajectory to a Secure Target State Objective: (t f =free final time) (y (t f ) -y target ) 2 → min Single track model: x = F x cos(ψ) -F y sin(ψ) /m y = F x sin(ψ) + F y cos(ψ) /m ψ = ( v F sv cos(δ)h F sh + v F uv sin(δ)) /I z δ = w δ Constraints: initial conditions and (a) state constraints: 1.3 ≤ y (t) ≤ 5.7 (stay on road), (F sv , F uv ) ≤ F max,v , (F sh , F uh ) ≤ F max,h (Kamm's circle) (b) boundary conditions: x(t f ) = d (d =initial distance to obstacle), y (t f ) = 0 (no velocity in y-direction when passing obstacle) (c) control constraints: w δ,min ≤ w δ ≤ w δ,max (steering velocity), F B,min ≤ F B ≤ F B,max (braking force) width 2.6 m, road width 7 m, initial y-position of car 1.75 m, distance 70 m, velocity 200 km/h, CPU: 0.05 s -0.07 s


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

New degeneration of Fay's identity and its application to integrable systems

New degeneration of Fay's identity and its application to integrable systems

Introduction

The well known trisecant identity discovered by Fay is a far-reaching generalization of the addition theorem for elliptic theta functions (see [START_REF] Fay | Theta functions on Rieman surfaces[END_REF]). 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 c 1 , c 2 and c 3 such that

c 1 Θ z + a c Θ z + d b + c 2 Θ z + a b Θ z + d c = c 3 Θ(z) Θ z + a c + d b , (1.1)
where Θ is the multi-dimensional theta function (2.2); here and below we use the notation b a for the Abel map (2.5) 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]. Moreover, as it was realized by Mumford, theta-functional solutions of certain integrable equations as Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP), or Sine-Gordon (SG), may be derived from Fay's trisecant identity and its degenerations (see [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF]).

In the present paper we apply Mumford's approach to the Davey-Stewartson equations and the multi-component nonlinear Schrödinger equation.

The rst main result of this paper is a new degeneration of Fay's identity (1.1). This new identity holds for two distinct points a, b on a compact Riemann surface of genus g > 0, and any z ∈ C g :

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)
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 (2.8). 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)} , (1.3)
where Z = iV a x + iW a t and A ∈ C, d ∈ C g are arbitrary constants, is a solution of the linear Schrödinger equation

i ∂ψ ∂t + ∂ 2 ψ ∂x 2 + 2 u ψ = 0, (1.4)
with the potential u(x, t) = D 2 a ln Θ(Z). When this potential is related to the function ψ by u(x, t) = ρ |ψ| 2 , with ρ = ±1, the function ψ (1.3) becomes a solution of the nonlinear Schrödinger equation (NLS)

i ∂ψ ∂t + ∂ 2 ψ ∂x 2 + 2ρ |ψ| 2 ψ = 0. (1.5)
This is the starting point of our construction of algebro-geometric solutions of the Davey-Stewartson equations and the multi-component nonlinear Schrödinger equation. The nonlinear Schrödinger equation (1.5) 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 Shabat in [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media[END_REF]. Algebro-geometric solutions of (1.5) 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].

There exist various ways to generalize the NLS equation. The rst 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, (1.6)
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 denotes the Davey-Stewartson equation when α = i, and DS2 ρ the Davey-Stewartson equation when α = 1. The Davey-Stewartson equation (1.6) 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 (1.6) do not depend on the variable y the rst equation in (1.6) reduces to the NLS equation (1.5) under appropriate boundary conditions for the function Φ + ρ |ψ| 2 in the limit when x tends to innity. Algebro-geometric solutions of the Davey-Stewartson equations (1.6) 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 paper, solutions of (1.6) are constructed from solutions of the complexied system which, after the change of coordinates ξ = 1 2 (xiαy) and η = 1 2 (x + iαy), with α = i, 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, (1.7)
ϕ ξη + 1 2 ((ψψ * ) ξξ + (ψψ * ) ηη ) = 0,
where ϕ := Φ + ψψ * . This system reduces to (1.6) under the reality condition:

ψ * = ρ ψ. (1.8)
The second main result of our paper 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). 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 + 1 4 h,
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 (1.8) 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 (1.8) is based on properties of Baker-Akhiezer functions. Our present approach based on identity (1.2) 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].

Another way to generalize the NLS equation is to increase the number of dependent variables in (1.5). 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, (1.9)
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 integrability of the two-component nonlinear Schrödinger equation (1.9) 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]; 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 third main result of this paper is the construction of smooth algebro-geometric solutions of the multi-component nonlinear Schrödinger equation (1.9) for arbitrary n ≥ 2, obtained by using (1.2). 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, (1.10)
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 (1.9) under the reality conditions

ψ * j = s j ψ j , j = 1, . . . , n. (1.11)
Algebro-geometric data associated to the solutions of (1.10) 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 } n j=1 and ψ * j n j=1

of system (1.10) 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 A j ∈ C, d ∈ C g are arbitrary constants; here the g-dimensional vector Z is a linear function of the variables x and t. Imposing the reality conditions (1.11), 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 points. Our general construction, being applied to the two-component case, gives solutions with more 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. The paper is organized as follows: in section 2 we recall some facts about the theory of Riemann surfaces, and derive a new degeneration of Fay's identity. With this degeneration, we give in Section 3 an independent derivation of smooth theta-functional solutions of the Davey Stewartson equations; this approach also provides an explicit description of the constants appearing in the solutions in terms of theta functions. In Section 4, we construct new smooth theta-functional solutions of the multicomponent NLS equation, and describe explicitely solutions of the focusing and defocusing cases. We also discuss the reduction from n-NLS to (n-1)-NLS, stationary solutions of n-NLS, and the link between solutions of n-NLS and solutions of the KP1 equation. Appendix A contains various facts from the theory of real Riemann surfaces. Appendix B contains an auxiliary computation required in the construction of algebro-geometric solutions of DS and n-NLS equations.

2

New degeneration of Fay's identity

In this section we recall some facts from the classical theory of Riemann surfaces [START_REF] Fay | Theta functions on Rieman surfaces[END_REF] and derive a new corollary of Fay's trisecant identity.

Theta functions

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

A k ω j = 2iπδ jk , j, k = 1, . . . , g. (2.1)
The matrix B = B k ω j of B-periods of the normalized holomorphic dierentials {ω j } g j=1 is symmetric and has a negative denite real part. The theta function with (half integer) characteristics 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)
here z ∈ C g is the argument and δ , δ ∈ 0, 1 2 g are the vectors of characteristics; , 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 nonsingular if Θ[δ](0) = 0, and an odd characteristic is called nonsingular if the gradient ∇Θ[δ](0) is non-zero. The theta function with characteristics is related to the theta function with zero characteristics (denoted by Θ) as follows

Θ[δ](z) = Θ(z + 2iπδ + Bδ ) exp 1 2 Bδ , δ + z + 2iπδ , δ . (2.3)
Let Λ be the lattice Λ = {2iπN+BM, N, M ∈ Z g } generated by the A and B-periods of the normalized holomorphic dierentials {ω j } g j=1 . The complex torus J = J(R g ) = C g /Λ is called the Jacobian of the Riemann surface R g . The theta function with characteristics (2.2) has the following quasi-periodicity property

Θ[δ](z + 2iπN + BM) = Θ[δ](z) exp -1 2 BM, M -z, M + 2iπ( δ , N -δ , M ) . (2.4)
Denote by µ the Abel map µ : R g -→ J dened by

µ(p) = p p 0 ω, (2.5)
for any p ∈ R g , where p 0 ∈ R g is the base point of the application, and ω = (ω 1 , . . . , ω g ) is the vector of the normalized holomorphic dierentials. In the whole paper we use the notation

b a = µ(b) -µ(a).

Fay's identity and previously known degenerations

Let us introduce the prime-form which is given by

E(a, b) = Θ[δ]( a b ) h δ (a)h δ (b) , (2.6)
a, b ∈ R g ; h δ (a) is a spinor dened by h 2 δ (a) = g j=1 ∂Θ[δ]
∂z j (0)ω j (a), where δ = [δ δ ] is a non-singular odd characteristic (the prime form is independent of the choice of the characteristic δ). 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 , (2.7)
where a, b, c, d ∈ R g and all integration contours do not intersect cycles of the canonical homology basis. Let us now discuss degenerations of identity (2.7).

Let k a (p) denote a local parameter near a ∈ R g , where p lies in a neighbourhood of a. 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! + . . . dk a (p), (2.8)
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 ):

D a F (z) = g j=1 ∂ z j F (z)V a,j = ∇F (z), V a , (2.9)
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 ):

D a F (z) = g j=1 ∂ z j F (z)W a,j = ∇F (z), W a .
Then for any z ∈ C g and any distinct points a, b ∈ R g , the following well-known degenerated version of Fay's identity holds (see [START_REF] Mumford | Tata Lectures on Theta. I and II[END_REF])

D a D b ln Θ(z) = q 1 (a, b) + q 2 (a, b) Θ(z + b a ) Θ(z + a b ) Θ(z) 2 , (2.10)
where the scalars q 1 (a, b) and q 2 (a, b) are given by

q 1 (a, b) = D a D b ln Θ[δ]( b a
), (2.11)

q 2 (a, b) = D a Θ[δ](0) D b Θ[δ](0) Θ[δ]( b a ) 2 , (2.12)
where δ is a non-singular odd characteristic. Notice that q 1 (a, b) and q 2 (a, b) depend on the choice of local parameters k a and k b near a and b respectively.

New degeneration of Fay's identity

Algebro-geometric solutions of the Davey-Stewartson equations and the multi-component NLS equation constructed in this paper are obtained by using the following new degenerated version of Fay's identity.

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 (a, b) 2 + 2 D 2 a ln Θ(z) + K 2 (a, b) = 0, (2.13)
where the scalars K 1 (a, b) and K 2 (a, b) are given by

K 1 (a, b) = 1 2 D a Θ[δ](0) D a Θ[δ](0) + D a ln Θ[δ]( b a ) , (2.14)
and

K 2 (a, b) = -D a ln Θ( b a ) -D 2 a ln Θ( b a )Θ(0) -D a ln Θ( b a ) -K 1 (a, b) 2 .
(2.15)

Proof. We start from the following lemma Lemma 2.1. 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 ,

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, (2.16)
where the scalar K 1 (b, c) is dened in (2.14).

Proof of Lemma 2.1. Let us introduce the notations Θ ab = Θ(z + b a ω) and Θ = Θ(z). Dierentiating (2.7) 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 (2.17) = 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, 3, 4, are given by

p 1 (a, b, c) = - E(c, b) E(a, b) ∂ kx E(a, x) E(c, x) x=b , p 2 (a, b, c) = E(a, c) E(a, b) ∂ kx E(x, b) E(c, x) x=b , (2.18
)

p 3 (a, b, c) = E(a, c) E(a, b) ∂ 2 kx E(x, b) 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 . (2.19)
Dierentiating (2.17) 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, (2.20)
where the scalar K depends on the points b, c, but not on the vector z ∈ C g . 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. ( 2
-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 + 2p 1 2 - 1 4
p 3 p 2 + 2 p 1 2 -p 4 + 2 D 2 b ln Θ.
By Lemma 2.1, 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 + 2p 1 2 + 2 D 2 b ln Θ = 0. ( 2
z ∈ C g 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.
Then (2.22) 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 2.1). Due to the fact that on each Riemann surface R g , there exists

a positive divisor d 1 + ... + d g of degree g such that vectors ω(d 1 ) dk d 1 , ..., ω (dg)
dk dg are linearly independent (see [START_REF] Klein | Ernst equation, Fay identities and variational formulas on hyperelliptic curves[END_REF], 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) (2.23)
for any z ∈ C g . Interchanging a and b, and changing the variable z ↔ -z in (2.23) we get (2.13). The expression (2.15) for the scalar K 2 (a, b) follows from (2.23) putting z = 0.

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

ξ = 1 2 (x -iαy), η = 1 2 (x + iαy), α = i, 1.
In these coordinates the Davey Stewartson equations (1.6) become

i ψ t + D 1 ψ + 2 φ ψ = 0, D 2 φ + ρ D 1 |ψ| 2 = 0, (3.1)
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

To construct algebro-geometric solutions of (3.1), let us rst introduce the complexied 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, (3.2)
ϕ ξη + 1 2 ((ψψ * ) ξξ + (ψψ * ) ηη ) = 0,
where ϕ := Φ + ψψ * . This system reduces to (3.1) under the reality condition:

ψ * = ρ ψ, (3.3)
which leads to ϕ = φ. Theta functional solutions of system (3.2) are given by Theorem 3.1. 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 (3.2)

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 , (3.4) ϕ(ξ, η, t) = 1 2 (ln Θ(Z -d)) ξξ + 1 2 (ln Θ(Z -d)) ηη + 1 4 h.
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 , (3.5)
where the vectors V a , V b and W a , W b were introduced in (2.8). 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), (3.6)
G 3 = κ 2 1 K 2 (a, b) + κ 2 2 K 2 (b, a) + h, (3.7)
and scalars q 2 (a, b), K 1 (a, b), K 2 (a, b) are dened in (2.12), (2.14), (2.15) respectively.

Proof. Substitute functions (3.4) in the rst equation of system (3.2) 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 (2.13), the last equality holds for any z ∈ C g , and in particular for z = Zd. In the same way, it can be checked that functions (3.4) satisfy the second equation of system (3.2). Moreover, from (2.10) 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 (3.4) satisfy the last equation of system (3.2).

The solutions (3.4) 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,
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 , (3.8)
where β, µ 1 , µ 2 are arbitrary complex numbers (β = 0), leads to a dierent family of solutions of the complexied system (3.2). 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 , (3.9)
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 , (3.10)
where

λ i = κ i µ i β -1 and α = h(1 -β 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, (3.11)
where ρ = ±1. Here ξ, η, t are real variables. Algebro-geometric solutions of (3.11) are constructed from solutions ψ, ψ * (3.4) 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 (see Appendix A.1). Let us choose the homology basis satisfying (A.2). Then the solutions of (3.11) are given by Theorem 3.2. Let a, b ∈ R g (R) be distinct points with local parameters satisfying k a (τ p) = k a (p) for any p lying in a neighbourhood of a, and k b (τ p) = k b (p) for any p lying in a neighbourhood of b. Denote by {A, B, } the standard generators of the relative homology group

H 1 (R g , {a, b}) (see Appendix A.2). Let d R ∈ R g , T ∈ Z g , and dene d = d R + iπ 2 (diag(H)-2 T). Morover, 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 , (3.12)
where M ∈ Z g is dened in (A.13). 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 , (3.13) φ(ξ, η, t) = 1 2 (ln Θ(Z -d)) ξξ + 1 2 (ln Θ(Z -d)) ηη + 1 4 h, (3.14)
where

|A| = |κ 1 κ 2 q 2 (a, b)| exp { d R , M } .
Here r = ω, and the vector Z is dened in (3.5). Scalars q 2 (a, b), G 1 , G 2 and G 3 are dened in (2.12), (3.6) and (3.7) respectively.

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. Let us check that under the conditions of the theorem, the functions ψ and ψ * (3.4) satisfy the reality conditions (3.3). First of all, invariance with respect to the anti-involution τ of the points a and b implies the reality of vector (3.5):

Z = Z. (3.15)
In fact, using the expansion (2.8) of the normalized holomorphic dierentials ω j near a we get

τ * ω j (a)(p) = (V a,j + W a,j k a (p) + . . .) dk a (p),
for any point p lying in a neighbourhood of a. Then by (A.3), the vectors V a and W a appearing in expression (3.5) satisfy

V a = -V a , W a = -W a . (3.16)
The same holds for the vectors V b and W b , which leads to (3.15). Moreover, from (A.3) and (A.13) we get r = -r -2iπN -BM, (3.17)

where N, M ∈ Z g are dened in (A.13) and satisfy

2 N + HM = 0. ( 3
K 1 (a, b) = K 1 (a, b) -V a , M , K 2 (a, b) = K 2 (a, b) + W a , M , (3.19)
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 (3.3) together with (3.4) 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 -iπdiag(H), M , (3.20)
taking into account the action (A.5) of the complex conjugation on the theta function, and the quasiperiodicity (2.4) of the theta function. Let us choose a vector d ∈ C g such that

d ≡ d -iπ diag(H) mod (2iπZ g + B Z g ), which is, since d -d is purely imaginary, equivalent to d = d -iπ diag(H) + 2iπT, for some T ∈ Z g .
Here we used the action (A.4) 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), (3.21)
for some d R ∈ R g and T ∈ Z g . Therefore, all theta functions in (3.20) cancel out and (3.20) becomes

|A| 2 = -ρ κ 1 κ 2 q 2 (a, b) exp 1 2 BM, M + r + d, M . (3.22)
The reality of the right hand side of equality (3.22) can be deduced from formula (B.11) for the argument of q 2 (a, b). Moreover, it is straightforward to see from (3.21) and (3.18) that exp{ d, M } is also real. Since κ 1 , κ 2 are arbitrary real constants, we can choose κ 1 as in (3.12), 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 } .
d R ∈ R g , T ∈ Z g , θ, h, ∈ R, κ1 , κ 2 ∈ R \ {0}.
Note that by 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 the next proposition.

Since the theta function is entire, singularities of the functions ψ and φ can appear only at the zeros of their denominator. Following Vinnikov's result [START_REF] Vinnikov | Self-adjoint determinantal representions of real plane curves[END_REF] we obtain The following assertions were proved in [START_REF] Vinnikov | Self-adjoint determinantal representions 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 lying in a component T v (A.25) of the Jacobian, then the curve is dividing and v = 0. Hence d ∈ T 0 where T 0 = R g .

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, (3.23)
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 (see Section 3.2), algebro-geometric solutions of (3.23) are constructed from solutions ψ, ψ * (3.4) 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 (A.2). Then the solutions of (3.23) are given by Theorem 3.3. 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 lying in a neighbourhood of a. Denote by {A, B, } the standard generators of the relative homology group (3.24) 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:

H 1 (R g , {a, b}) (see Appendix A.2). Let T, L ∈ Z g satisfy 2 T + H L = diag(H),
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 , (3.25) φ(ξ, η, t) = 1 2 (ln Θ(Z -d)) ξξ + 1 2 (ln Θ(Z -d)) ηη + 1 4 h, (3.26)
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 whith ρ = e iπ N,L , 2. if does not cross any real oval, functions ψ and φ are solutions of DS2 ρ whith ρ = -e iπ N,L . Here r = ω, the vector Z is dened in (3.5) and vector N ∈ Z g is dened in (A.6). Scalars q 2 (a, b), G 1 , G 2 and G 3 are dened in (2.12), (3.6) and (3.7) respectively. Proof. Analogously to the proof of Theorem 3.2, let us check that under the conditions of the theorem, the functions ψ * and ψ (3.2) satisfy the reality condition (3.3). First of all, due to the fact that points a and b are interchanged by τ , the vector Z (3.5) satises Z = -Z.

(3.27)

In fact, using the expansion (2.8) 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) + . . .) dk a (p),
for any point p lying in a neighbourhood of a. Then by (A.3) 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 , (3.28)
which leads to (3.27). From (A.3) and (A.6) we get

r = r -2iπ N, (3.29)
where N ∈ Z g is dened in (A.6). By Proposition B.3, the scalar q 2 (a, b) is real. From (2.13), it is straightforward to see that the scalars K 1 (a, b) and K 2 (a, b), dened in (2.14) and (2.15), 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 (3.3) together with (3.4) leads to The sign of q 2 (a, b) in the case where τ a = b is given in Proposition B2, which completes the proof. To construct solutions associated to non-dividing Riemann surfaces, we rst observe from (3.24) that all components of the vector L cannot be even, since for non dividing Riemann surfaces the vector diag(H) contains odd coecients (see Appendix A.1). In this case, the vector N has to be computed to determine the sign ρ = -e iπ N,L in the reality condition. This vector N is dened by the action of τ on the relative homology group H 1 (R g , {a, b}) (see (A.6)). It follows that we do not have a general expression for this vector.

|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) , (3.30
To ensure the smoothness of solutions (3.25) and (3.26) 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 (KP1) equation in the case where R g admits real ovals we get Remark 3.2. Smoothness of solutions of the DS2 -equation was investigated in [START_REF] Malanyuk | Finite-gap solutions of the Davey-Stewartson equations[END_REF]. It is proved that solutions are smooth if and only if the associated Riemann surface does not have real ovals, and if there are no pseudo-real functions of degree g -1 on it (i.e. functions which satisfy f (τ p) = -f (p) -1 ).

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

i ψt + ψξξ + 2 ρ | ψ| 2 ψ = 0,
by the substitution

ψ(ξ, t) = ψ(ξ, t) exp -2i q 1 (a, b) + 1 4 h 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 the local parameters satisfy k a (p) = k b (σp), one has

V a + V b = 0, W a + W b = 0. (3.33)
To verify (3.33), we use the action σA k = -A k of the involution σ on the A-cycles of the homology basis. Hence by (2.1) we have

2iπδ jk = σA k σ * ω j = - A k σ * ω j .
It follows that the holomorphic dierential -σ * ω j satises the normalization condition (2.1), which implies, by virtue of uniqueness of the normalized holomorphic dierentials,

σ * ω j = -ω j .
Using (2.8) we obtain

σ * ω j (a)(p) = (V b,j + W b,j k b (σp) + . . .) dk b (σp) = (V b,j + W b,j k a (p) + . . .) dk a (p),
which implies V a + V b = 0 and W a + W b = 0. Therefore, when the Riemann surface associated to solutions of DS1 ρ is hyperelliptic, assuming that a and b satisfy σa = b, and κ 1 = κ 2 = 1, by (3.33) and (2.10), under the reality condition ψ * = ρ ψ, the function φ in (3.14) satises

φ(ξ, η, t) = ρ |ψ| 2 + q 1 (a, b) + 1 4 h.
Hence the function ψ (3.13) becomes a solution of the equation i ψ t + ψ ξξ + 2 ρ |ψ| 2 + q 1 (a, b) + 1 4 h ψ = 0, with ρ = ±1, depending on the reality of the branch points as explained in Section 4.

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]. [START_REF] Bobenko | Introduction to Compact Riemann Surfaces[END_REF] Algebro-geometric solutions of the multi-component NLS equation

In this section, we present another application of the degenerated Fay identity (2.13), which leads to 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, (4.1)
where s = (s 1 , . . . , s n ), s i = ±1. Here ψ j (x, t) are complex valued functions of the real variables x and t.

Solutions of the complexied n-NLS equation

Consider rst the complexied version of the n-NLS s equation, which is 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, (4.2)
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 (4.1) under the reality conditions

ψ * j = s j ψ j , j = 1, . . . , n. (4.3)
Theta functional solutions of the system (4.2) 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 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 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 * j n j=1 are solutions of the system (4.2)

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)} . (4.4)
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. (4.5)
The vectors V a n+1 and W a n+1 are dened in (2.8), 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 ). (4.6)
The scalars q 2 (a n+1 , a j ), K 1 (a n+1 , a j ), K 2 (a n+1 , a j ) and q 1 (a n+1 , a k ) are dened in (2.12), (2.14), (2.15) and (2.11) respectively.

Proof. We start with the following technical lemma.

Lemma 4.1. 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 4.1. Assume that there exist α 1 , . . . , α n+1 ∈ C * such that n+1 k=1 α k V a k = 0. The left hand side of this equality equals the vector of B-periods (see e.g. [START_REF] Bobenko | Introduction to Compact Riemann Surfaces[END_REF]) of the normalized dierential of the second kind Ω = n+1 k=1 α k Ω

(2) 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 [START_REF] Bobenko | Introduction to Compact Riemann Surfaces[END_REF] we get

∂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 lying in a neighbourhood of a j , we deduce that n+1 k=1 V a k = 0.

To prove Theorem 4.1, substitute the functions (4.4) into the rst equation of (4.2) 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. (4.7)
It can be shown that equation (4.7) holds as follows: in (2.13), 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 + K 2 + 2 D 2 a n+1 ln Θ(z) = 0, (4.8
) 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 4.1 the sum n+1 k=1 V a k equals zero, which implies

n+1 k=1 D a k = 0.
Substituting D a n+1 instead of -n k=1 D a k in (4.8) and using (2.10) we obtain (4.7), 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 (4.13) satisfy the 2n -1 other equations of the system (4.2).

The solutions (4.4) of the complexied sytem (4.2) 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. The transformation of the local parameters given by

k a j -→ β k a j + µ k 2 a j + O k 3 a j , (4.9)
where β, µ are arbitrary complex numbers (β = 0), leads to a dierent family of solutions of the complexied system (4.2). 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 , (4.10)
where λ = µ β -1 .

Reality conditions

Algebro-geometric solutions of the n-NLS s equation (4.1) are constructed from solutions (4.4) of the complexied system by imposing the reality conditions ψ * j = s j ψ j (4.3). Let R g be a real compact Riemann surface with an anti-holomorphic involution τ . Let us choose the homology basis satisfying (A.

2). A meromorphic function

f on R g is called real if f (τ p) = f (p) for any p ∈ R g .
In the next proposition we derive theta-functional solutions of (4.1). The signs s j appearing in the reality conditions (4.3) 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 Appendix A.2.2)

τ ˜ j = ˜ j + AN j + BM j + α j S a j , (4.11)
for some α j ∈ Z, where vectors N j , M j ∈ Z g are the same as in (A.13). Then

α j = (τ ˜ j -˜ j ) • j , (4.12)
between the closed contour τ ˜ j -˜ j and the contour j ; this intersection is computed in the relative homology group H 1 (R g , {a n+1 , a j }).

Theta functional solutions of (4.1) are given by Proposition 4.1. 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 lying 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 }) (see Appendix A.2.2). Let d R ∈ R g , T ∈ Z g , and dene d = d R + iπ 2 (diag(H)-2 T).
Morover, take θ ∈ R. Then the following functions {ψ j } n j=1 are solutions of n-NLS s (4.1)

ψ j (x, t) = |A j | e iθ Θ(Z -d + r j ) Θ(Z -d) exp {i(-E j x + F j t)} , (4.13)
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 . (4.14)
Here r j = j ω, the vectors V a n+1 , W a n+1 are dened in (2.8), and the vector M j ∈ Z g is dened by the action of τ on the relative homology group H 1 R (n+1) g , {a n+1 , a j } (see (A.13)). The scalars q 2 (a n+1 , a j ) and E j , F j are introduced in (2.12) and (4.6) respectively. The signs s 1 , . . . , s n are given by

s j = exp {iπ(1 + α j ) + iπ T, M j } , (4.15)
where the intersection indices α j ∈ Z are dened in (4.12).

Proof. The proof follows the lines of Section 3.2, where similar statements were proven for the DS1 ρ equation. First of all, invariance with respect to the anti-involution τ of the point a n+1 implies the reality of the vector Z = i V a n+1 x + i W a n+1 t. Moreover, from (A.3) and (A.13) we get

r j = -r j -2iπN j -BM j . (4.16)
where N j , M j ∈ Z g are dened in (A.13) and satisfy 2 N j + HM j = 0. (4.17)

For j = 1, . . . , n, the action of the complex conjugation on the scalars K 1 (a n+1 , a j ) and K 2 (a n+1 , a j ) is given by (3.19), and one can directy see from (2.10) 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 . (4.18)
Under the assumptions of the theorem and by (B.11), 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 + 1 2 HM j , M j - 1 2i ( B M j , M j + 2 r j , M j ) . (4.19)
Therefore, the reality conditions (4.3) together with (4.4) 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 + 1 2 HM j , M j + d -iπdiag(H), M j , (4.20)
if one takes into account (A.5) and (2.4). Let us choose a vector d ∈ C g such that

d ≡ d -iπ diag(H) (mod (2iπZ g + B Z g )) .
Since dd is purely imaginary we have

d = d -iπ diag(H) + 2iπT, (4.21)
for some T ∈ Z g , where we have used (A.4) 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), (4.22)
for some d R ∈ R g and T ∈ Z g . Therefore, (4.20) 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 , (4.23)
which by (4.22) leads to (4.14). Moreover we deduce from (4.22) and (4.23) that s j = exp iπ(1 + α j ) + iπ 2 HM j + diag(H), M jiπ T, M j . From (4.17) and the denition of the matrix H (see Appendix A.1), it can be deduced that the quantity 1 2 HM j + diag(H), M j is even in each case, which yields (4.15). Functions ψ j given in (4.13) describe a family of algebro-geometric solutions of (4.1) 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 Proposition 3.1. In this case, the sign s j (4.15) is given by s j = exp{iπ(1 + α j )}.

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.

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 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+2+2k+1 = σ 2g+2+2k+2 =(n -k, n -k -1), k = 0, . . . , n -2.
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. 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. Here we construct solutions of the 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
+ system i ∂ψ j ∂t + ∂ 2 ψ j ∂x 2 + 2 n k=1 |ψ k | 2 ψ j = 0, j = 1, . . . , n. (4.24)
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 , refering to the focusing system (4.24). 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 Appendix A.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 (A.2). By the previous topological description of R + g,n+1 , the matrix H involved in (A.2) looks as:

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 explicitely in Figure 2 for odd genus, and in Figure 3 for even genus.

< < > > 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 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 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.

As proved in the following theorem, among all coverings having a monodromy group described in Figure 1, only the covering R + g,n+1 leads to algebro-geometric solutions of the focusing system (4.24).

Theorem 4.2. 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 g and θ ∈ R. Then the following functions {ψ j } n j=1 are smooth solutions of n-NLS + :

ψ j (x, t) = |A j | e iθ Θ(Z -d + r j ) Θ(Z -d) exp {i(-E j x + F j t)} , (4.25)
where

Z = i V a n+1 x + i W a n+1 t.
Here r j = a j a n+1 ω, the vectors V a n+1 , W a n+1 are dened in (2.8), and the vector M j ∈ Z g is dened in (A.13), according to the action of τ on the relative homology group H 1 R + g,n+1 , {a n+1 , a j } . The scalars |A j | and E j , F j are given by (4.14) and (4.6) respectively. Proof. Let us check that the conditions of the theorem imply that functions ψ j in (4.13) are 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 g (i.e. T = 0), the quantities {s j } n j=1 (4.15) become

s j = exp {iπ(1 + α j )} . (4.26)
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 4) Now let us construct solutions of the system n-NLS -

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 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. α n = (τ ˜ n -˜ n ) • n ≡ 1 (mod 2),
i ∂ψ j ∂t + ∂ 2 ψ j ∂x 2 -2 n k=1 |ψ k | 2 ψ j = 0, j = 1, . . . , n. (4.28)
As for the focusing case, let us rst describe the covering and the homology basis used in our construction of the solutions of (4.28). 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 k , z k+1 are pairwise conjugate for k = 2g + 3, . . . , 2g + 2n. Denote by R - g,n+1 this covering, refering to the defocusing system (4.28). It is straightforward to see that such a covering is an M-curve (see Appendix A.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 (A.2). Since the covering R - g,n+1 is an M-curve, the matrix H involved in (A.2) satises H = 0. Such a canonical homology basis is shown in Figure 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 (4.28) associated to the covering R - g,n+1 .

Theorem 4.3. 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 g and θ ∈ R. Then the functions {ψ j } n j=1 in (4.25) are smooth solutions of n-NLS -.

Proof. Analogously to the focusing case, one has to check that all s j = -1. Since all branch points z k are real for k = 1, . . . , 2g + 2, the intersection index α n between the closed contour τ ˜ n -˜ n and the contour n satises (see Figure 6)

α n = (τ ˜ n -˜ n ) • n ≡ 0 (mod 2), (4.29)
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 Zd and the fact that the curve is dividing.

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: 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.
Solutions ψ j construced here describe a family of smooth algebro-geometric solutions of the defocusing multi-component NLS equation depending on n -1 complex parameters: z 2g+2+2k-1 ∈ R 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 g . Remark 4.1. Smooth solutions of n-NLS s for a vector s with mixed signs can be constructed in the same way.

Stationary solutions of n-NLS

It is well-known that the algebro-geometric solutions (4.13) 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 (4.10)), 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 (4.13) under the vanishing condition

W a n+1 = 0. (4.30)
This condition is equivalent to the existence of a meromorphic function h of order two on R g , such that the point a n+1 is a critical point of h (this can be proved analogously to Lemma 4.1). Therefore, stationary solutions of the multi-component NLS can be constructed from the algebrogeometric 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 order n + 1 on R g , • z a ∈ CP 1 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 order 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 lying in a neighbourhood of a j , and k a n+1 (p) = (h(p)h(a n+1 )) 1/2 for any point p lying 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 (4.13) 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 (4.30). In this case, the modulus of solutions (4.13) do not depend on the variable t.

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

It is natural to ask 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 4.3.1; 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 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 (2.12) and (4.14), the function ψ 1 tends to zero as z 2g+2n and z 2g+2n-1 coalesce. Functions {ψ j } n j=2 obtained in this limit are solutions of (n-1)-NLS + associated to the covering R + g,n . A similar degeneration produces a solution of (n-1)-NLS -from a solution of n-NLS -.

Relationship between solutions of KP1 and solutions of n-NLS

Historically, the Korteweg-de Vries equation (KdV) and its generalization to two spatial variables, the Kadomtsev-Petviashvili equations (KP), were the most important examples of applications of methods of algebraic geometry in the 1970's (see e.g. [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]). Moreover, 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 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 u u x -u xxx ) x . ( 4

.31)

Let R g be an arbitrary 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 (2.8) and let d ∈ C g . Then, according to Krichever's theorem [START_REF] Krichever | Algebro-geometric construction of the Zaharov-Shabat equations and their periodic solutions[END_REF], the function

u(x, y, t) = 2 D 2 a log Θ(i V a x + i W a y + i U a t + d) + 2 c (4.32)
is a solution of KP1; here the constant c is dened by the expansion near a of the normalized meromorphic dierential Ω

a (p) having a pole of order two at a only: Ω

a (p) = (k a (p)) -2 + c 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 as k a (p) = f (p)f (a), then formula (4.32) 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 (see Lemma 4.1), the solution (4.32) of KP1 can be rewritten as where I g is the g × g unit matrix, and H is a g × g matrix dened as follows

u(x, y, t) = -2 n j=1 D a n+1 D a j log Θ(z) + 2 c, where z = i V a n+1 x + i W a n+1 y + i U a n+1 t + d. Using corollary (2.10) 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 c. ( 4
1) if R g (R) = ∅, H =               0 1 1 0 . . . 0 1 1 0 0 . . . 0               if R g (R) is dividing, H =           1 . . . 1 0 . . . 0           if R g (R) is non-dividing, (rank(H) = g + 1 -k in both cases).
2) if R g (R) = ∅, (i.e. the curve does not have real ovals), 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 (A.2), and study the action of τ on the normalized holomorphic dierentials, and the action of the complex conjugation on the theta function with zero characteristics.

By (A.2) the A-cycles of the homology basis are invariant under τ . Due to normalization condition (2.1) this leads to the following action of τ on the normalized holomorphic dierentials τ * ω j = -ω j .

(A.3) Using (A.2) and (A.3) we get the following reality property for the matrix B of B-periods

B = B -2iπ H. (A.4)
By Proposition 2.3 in [START_REF] Vinnikov | Self-adjoint determinantal representions of real plane curves[END_REF], for any z ∈ C g , relation (A.4) implies

Θ(z) = κ Θ(z -iπ diag(H)), (A.5)
where diag(H) denotes the vector of the diagonal elements of the matrix H, and κ is a root of unity which depends on matrix H (knowledge of the exact value of κ is not needed for our purpose).

where N ∈ Z g is dened by (A.6). The last intersection index in (A. where vectors N, M ∈ Z g are the same as in (A.13).

Proof. The action of τ on A and B-cycles in (A. [START_REF] Malanyuk | Finite-gap solutions of the Davey-Stewartson equations[END_REF]) coincides with the one (A.2) in H 1 (R g ). From 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 -= NA + MB, (A.17)

where N, M ∈ Z g are related by 2 N + HM = 0. This proves (A.13).

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

3

  Algebro-geometric solutions of the Davey-Stewartson equations Here we derive algebro-geometric solutions of the Davey-Stewartson equations (1.6) using the degeneration (2.13) of Fay's identity. Let us introduce the function φ := Φ + ρ|ψ| 2 , where ρ = ±1, and the dierential operators

  and the local parameters k a and k b near a and b. The transformation of the local parameters given by

Functions

  ψ and φ given in (3.13) and (3.14) describe a family of algebro-geometric solutions of (3.11) 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

Proposition 3 . 1 .

 31 Solutions (3.13) and (3.14) are smooth if the curve R g is dividing and d ∈ R g . Assume that solutions (3.13) and (3.14) are smooth for any vector d lying in a component T v (A.25) of the Jacobian, then the curve is dividing and d ∈ R g . Proof. By (3.15) and (3.21), the vector Zd belongs to the set S 1 introduced in (A.23). Hence by Proposition A.3, the solutions are smooth if the curve is dividing (in this case diag(H)=0), and if the argument Zd of the theta function in the denominator is real, which by (3.15) leads to the choice d ∈ R g (and then T = 0 in Theorem 3.2).

  ) taking into account (A.5). Let us choose a vector d ∈ C g such that d = -diπ diag(H) + 2iπT + B L, for some vector T, L ∈ Z g . The reality of the vector d + d together with (A.d I ∈ R g , where 2 T + H L = diag(H). With this choice of vector d, (3.30) becomes |A| 2 = -ρ |κ 1 | 2 q 2 (a, b) e -r,L . (3.32) Moreover, from (3.29) we deduce that equality (3.32) holds only if ρ = -sign(q 2 (a, b)) e -iπ N,L .

Proposition 3 . 2 .

 32 Functions (3.25) and (3.26) are smooth solutions of DS2 + if the curve is an Mcurve and d ∈ i R g . Assume that the curve admits real ovals and functions (3.25), (3.26) are smooth solutions of DS2 ρ for any vector d lying in a component Tv (A.27) of the Jacobian, then the curve is an M-curve, d ∈ i R g and ρ = +1. Proof. By (3.27) and (3.31) the vector Zd belongs to the set S 2 introduced in (A.24). Hence by Proposition A.4, the solutions are smooth if the curve is an M-curve and Zd ∈ i R g which implies d ∈ i R g by (3.27) (and therefore L = T = 0).

3. 4

 4 Reduction of the DS1 ρ equation to the NLS equation Solutions of the nonlinear Schrödinger equation (1.5) can be derived from solutions of the Davey Stewartson equations, when the associated Riemann surface is hyperelliptic. Proposition 3.3. Let R g be a hyperelliptic curve of genus g which admits an anti-holomorphic 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). Then, taking κ 1 = κ 2 = 1, the function ψ in (3.13) is a solution of the equation

Figure 1 :

 1 Figure 1: Hurwitz diagram of the covering R g,n+1 .

( 4 .

 4 27) which leads to s n = 1. Intersection indices α j for j = 1, . . . , n -1 can be computed in the same way. Thereforeα 1 ≡ α 2 ≡ . . . ≡ α n ≡ 1 (mod 2),which implies s j = 1. By Proposition 3.1, smoothness of the solutions is ensured by the reality of the vector Zd and the fact that the curve is dividing.Functions ψ j given in (4.25) 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 g .4.3.2Solutions of n-NLS -.

Figure 5 :

 5 Figure 5: Homology basis on the covering R - g,n+1 . The solid line indicates the sheet n + 1, and the dashed line sheet n.

q 1

 1 .33) Now let us consider solutions ψ j , ψ * j (4.4) of the complexied multi-component NLS equation, and make the change of variables (x, t) → (x, y) and d → -i U a n+1 t + d. Then by (4.33), the complexvalued solutions u (4.32) of KP1 and solutions ψ j , ψ * j (4.4) 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), (a n+1 , a j ) + 2 c. If we impose the reality conditions (4.3), we obtain real solutions (4.32) of KP1 from our solutions (4.25) of n-NLS s equation u(x, y, t) = γ -2 n j=1 s j |ψ j (x, y, t)| 2 . (4.35)

  10) equals n, which implies τ A = A. According to (A.2), the action of τ on B-cycles in H 1 (R g \ {a, b}) is given byτ B = -B + HA + m S b , (A.11)for some m ∈ Z g . Then0 = B • = -τ B • τ = -(-B + HA + m S b ) • τ = -(-B + HA + m S b ) • (-+ NA), (A.12)where N is dened by (A.6). The last intersection index in (A.12) 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 . A.2.2 Case τ a = a and τ b = b Proposition A.2. Let us choose the canonical homology basis in H 1 (R g ) satisfying (A.2), and assume that τ a = a and τ b = b. Then 1. the action of τ on the generators {A, B, } of the relative homology group H 1 (R g , {a, b}) is given by M ∈ Z g are related by 2 N + HM = 0, (A.14) 2. the action of τ on the generators {A, B, S b } of the homology group H 1 (R g \ {a, b}) is given by

(A. 2 )

 2 , one sees that each contour C ∈ H 1 (R g ) which satises τ C = -C, can be represented by C = ÑA + MB, (A.16)

2 )

 2 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 (B.1), recall thatb ã Ω b-a (p)Since δ is an odd non singular characteristic, the expressionΘ[δ]( p b ) Θ[δ]( p a )has a simple zero at b and a simple pole at a. Therefore, if we consider ã lying in a neighbourhood of a, and b lying in a neighbourhood of b, we get (withα 1 , β 1 = 0) Θ[δ]( b b ) Θ[δ]( b a ) = α 1 k b ( b) + o(k b ( b)), (B.3) Θ[δ]( ã a ) Θ[δ]( ã b ) = β 1 k a (ã) + o(k a (ã)).(B.4) Combining (B.2) together with (B.3) and (B.4), we obtain the following relation lim b → b ã → a k a (ã) k b ( b) the denition (2.12) of q 2 (a, b), it follows from (B.3) and (B.4) that α 1 β 1 = -q 2 (a, b), which by (B.5) completes the proof. B.2 Argument of q 2 (a, b) when τ a = b Here we compute the argument of the fundamental scalar q 2 (a, b) dened in (2.12) in the case where τ a = b. Let us choose the homology basis satisfying (A.2). Proposition B.2. Let a, b ∈ R g be distinct points such that τ a = b, with local parameters satisfying the relation k b (τ p) = k a (p) for any point p lying in a neighbourhood of a. Consider a contour connecting points a and b;

q 2 (

 2 a, b) < 0, 2. if does not cross any real oval, then q 2 (a, b) > 0. Proof. Let ã, b ∈ R g lie in a neighbourhood of a and b respectively, and 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), (B.6) B.3 Argument of 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 B.3. Let a, b ∈ R g (R) with local parameters satisfying k a (τ p) = k a (p) for any point p lying in a neighbourhood of a and k b (τ p) = k b (p) for any point p lying in a neighbourhood of b.Denote by {A, B, } the generators of the relative homology groupH 1 (R g , {a, b}) (see Section A.2). Let ã, b ∈ R g (R)lie in a neighbourhood of a and b respectively, and denote by ˜ an oriented contour connecting ã and b. Then the argument of the scalar q 2 (a, b) is given byarg{q 2 (a, b)} = arg{k a (ã) k b ( b)} + π 1 + α + 1 2 HM, M -1 2i ( B M, M + 2 r, M ) , (B.11)where α equals the intersection index (τ ˜τ ˜ ) • . Here r = ω, and M ∈ Z g is dened in (A.13).Proof. From the integral representation (B.1) of q 2 (a, b) we getarg{q 2 (a, b)} = π + arg{k a (ã) k b ( b)} + Im ˜ Ω b-a (p) . (B.12)Considering the action (A.15) of τ on the A-cycles, due to the uniqueness of the normalized dierential of the third kind Ω b-a , we obtainτ * Ω b-a = Ω b-a + k M k ω k , (B.13)where ω k are the normalized holomorphic dierentials.The closed contour τ ˜ -˜ ∈ H 1 (R g ) satises τ (τ ˜ -˜ ) = -(τ ˜ -˜ ); thus by (A.16) it has the following decomposition in H 1 (R g \ {a, b}) τ ˜ -˜ = NA + MB + α S b , (B.14)for some α ∈ Z, where N, M ∈ Z g are dened in (A.13). Hence we get Im ˜ 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 cycles of the absolute homology basis, BM Ω b-a = M, r . (B.16) Hence we get Im ˜ Ω b-a (p) ≡ πα + π 2 HM, M -1 2i ( M, r + r + BM, M ) , (B.17) where r = ˜ ω. Considering the limit when ã tends to a and b tends to b, we obtain (B.11).

  Theorem 2.1. 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 ,

  2 and p 3 are dened in (2.11), (2.18), and (2.19) respectively. The change of variable z ↔ -z +

	c b in
	(2.20) leads to

  .18) From (2.13), it is straightforward to see that the scalars K 1 (a, b) and K 2 (a, b) dened by (2.14) and (2.15) satisfy

  Corollary 3.1. From Theorem 3.3 we deduce that 1. if R g is dividing and each component of L is even, functions (3.25) and (3.26) are solutions of DS2 + , 2. if R g does not have real ovals and each component of L is even, functions (3.25) and (3.26) are solutions of DS2 -.

Remark 3.1.

Remark 4.2. Repeating this degeneration n -3 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) non-linear Schrödinger equation (1.5) associated to an hyperelliptic curve with pairwise conjugate branch points (resp. real branch points).

Due to the fact that in our construction of solutions of the multi-component NLS equation, the local parameters are dened by the meromorphic function f , complex solutions (4.34) and real solutions (4.35) of KP1 obtained in this way form only a subclass of Krichever's solutions. I thank C. Klein, who interested me in the subject, and D. Korotkin for carefully reading the manuscript and providing valuable hints. I am grateful to B. Dubrovin and V. Shramchenko for useful discussions. This work has been supported in part by the project FroM-PDE funded by the European Research Council through the Advanced Investigator Grant Scheme, the Conseil Régional de Bourgogne via a FABER grant and the ANR via the program ANR-09-BLAN-0117-01.

A Real Riemann surfaces

In this section, we recall some facts from the theory of real compact Riemann surfaces. Following [START_REF] Vinnikov | Self-adjoint determinantal representions of real plane curves[END_REF], we introduce a symplectic basis of cycles on R g and study reality properties of various objects on the Riemann surface R g associated to this basis.

A.1 Action of τ on the homology group H 1 (R g )

A Riemann surface R g is called real if it admits an anti-holomorphic involution τ : R g → R g , τ 2 = 1.

The connected components of the set of xed points of the anti-involution τ are called real ovals of τ . We denote by R g (R) the set of xed points. Assume that R g (R) consists of k real ovals, with 0 ≤ k ≤ g + 1. The curves with the maximal number of real ovals, k = g + 1, are called M-curves. The complement R g \ R g (R) has either one or two connected components. The curve R g is called

is connected (notice that an M-curve is always a dividing curve).

Example A.1. Consider the hyperelliptic Riemann surface of genus g dened by the equation

where the branch points λ k ∈ R are ordered such that λ 1 < . . . < λ 2g+1 . On such a Riemann surface, 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 intervals [λ 1 , λ 2 ], . . . , [λ 2g+1 , +∞], and projections of real ovals of τ 2 on the λ-plane coincide with the intervals

Hence the curve (A.1) is an M-curve with respect to both anti-involutions τ 1 and τ 2 .

Denote by {A, B} the set of generators of the homology group H 1 (R g ), where A = (A 1 , . . . , A g ) T and B = (B 1 , . . . , B g ) T . According to Proposition 2.2 in [START_REF] Vinnikov | Self-adjoint determinantal representions of real plane curves[END_REF], there exists a canonical homology basis such that

A.2 Action of τ on H 1 (R g \ {a, b}) and H 1 (R g , {a, b})

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 relative homology group H 1 (R g , {a, b}). We consider the case where τ a = b, and the case where τ a = a, τ b = b. Denote by {A, B, } the generators of the relative homology group H 1 (R g , {a, b}), where is a 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.

A.2.1

Case τ a = b Proposition A.1. Let us choose the canonical homology basis in H 1 (R g ) satisfying (A.2), and assume that τ a = b. Then 1. the action of τ on the generators {A, B, } of the relative homology group H 1 (R g , {a, b}) is given by

for some N ∈ Z g , 2. the action of τ on the generators {A, B, S b } of the homology group

where vector N ∈ Z g is the same as in (A.6).

Proof. The action of τ on A and B-cycles in (A.6) coincides with the one (A.2) in H 1 (R g ). From (A.2), 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 closed contour τ + ∈ H 1 (R g ) can be written as

for some N ∈ Z g . This proves (A.6).

Now let us prove (A.7)

. By (A.2), the cycles τ A admit the following decomposition in

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 (A.9)

Now let us prove (A.15). By (A.2), the cycles τ A admit the following decomposition in

for some n ∈ Z g . Therefore, we get from (A.18)

where N, M ∈ Z g is dened by (A.13). The last intersection index in (A. [START_REF] Raina | Fay's Trisecant Identity and Conformal Field Theory[END_REF]) equals -(n + M), which gives τ A = A -M S b . According to (A.2), the action of τ on B-cycles in H 1 (R g \ {a, b}) is given by

for some m ∈ Z g . Then

where N, M ∈ Z g are dened by (A. A.3 Action of τ on the Jacobian and theta divisor of real Riemann surfaces

In this part, we review known results [START_REF] Vinnikov | Self-adjoint determinantal representions 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. Let us choose the canonical homology basis satisfying (A.2) and consider the Jacobian J = J(R g ) of the real Riemann surface R g . The Abel map (2.5) µ : R g -→ J can be extended linearly to all divisors on R g , which denes a map on linear equivalence classes of divisors. The anti-holomorphic involution τ on R g gives rise to an anti-holomorphic involution on the Jacobian J: if D is a positive divisor of degree n on R g , then τ D is the class of the point ( In this section 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 notations:

The following proposition was proved in [START_REF] Vinnikov | Self-adjoint determinantal representions of real plane curves[END_REF].

Proposition A.3. The set S 1 is a disjoint union of the tori T v dened by

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

(A.26)

The following proposition was proved in [START_REF] Dubrovin | Real theta-function solutions of the Kadomtsev-Petviashvili equation[END_REF].

Proposition A.4. The set S 2 is a disjoint union of the tori Tv dened by

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.

B

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

This section is devoted to the computation of arg{q 2 (a, b)}, where q 2 (a, b) is dened by (2.12). As before, R g denotes a real compact Riemann surface of genus g with an anti-holomorphic involution τ . The argument of q 2 (a, b) is computed both in the case τ a = b, as well as in the case τ a = a, τ b = b.

B.1 Integral representation for q 2 (a, b)

Assume that a, b ∈ R g can be connected by a contour which does not intersect basic 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. dened in (2.12) admits the following integral representation

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. Let us now consider cases (1) and ( 2) separatly.

Case [START_REF] Anker | On the Soliton Solutions of the Davey-Stewartson Equation for Long Waves[END_REF]. 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,

where we used the action (A.6) of τ on the contour , and the fact that the intersection index between c ˜ 2 and A-cycles is zero by (B.9). Hence the intersection index α satises

which by (B.6) leads to q 2 (a, b) < 0.

Case [START_REF] Arbarello | Fay's trisecant formula and a characterization of Jacobian Varieties[END_REF]. 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

Thus the intersection index α = (τ ˜ + ˜ ) • is odd, which leads to q 2 (a, b) > 0.