On the numerical evaluation of algebro-geometric solutions to integrable equations

ON THE NUMERICAL EV ALUA TION OF ALGEBR O-GEOMETRIC SOLUTIONS TO INTEGRABLE EQUA TIONS C. KALLA AND C. KLEIN Abstra ct. Ph ysically meaningful perio dic solutions to certain integrable partial dif- feren tial equations are giv en in terms of multi-dimensional theta functions asso ciated to real Riemann surfaces. T ypical analytical problems in the numerical ev aluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist ev en for almost degenerate surfaces. This allows the numerical study of solitonic limits. F or general real Riemann surfaces, the choice of a homology basis adapted to the anti- holomorphic inv olution is imp ortan t for a conv enient formulation of the solutions and smo othness conditions. Since existing algorithms for algebraic curves produce a homol- ogy basis not related to automorphisms of the curve, w e study symplectic transformations to an adapted basis and give explicit formulae for M-curves. As examples we discuss solu- tions of the Da vey-Stew artson and the m ulti-comp onen t nonlinear Sc hrö dinger equations. 1. Intr oduction The imp ortance of Riemann surfaces for the construction of almost p erio dic solutions to v arious integrable partial differen tial equations (PDEs) w as realized at the b eginning of the 1970s by No viko v, Dubro vin and Its, Matveev. The latter found the Its-Matv eev formula for the Kortew eg-de V ries (KdV) equation in terms of multi-dimensional theta functions on h yp erelliptic Riemann surfaces. Similar form ulae w ere later obtained for other in tegrable PDEs as nonlinear Schrödinger (NLS) and sine-Gordon equations. F or the history of the topic the reader is referred to the reviews [2] and [9]. Krichev er [21] show ed that theta- functional solutions to the Kadom tsev-P etviash vili equation can b e obtained on arbitrary Riemann surfaces. The problems of real-v aluedness and smo othness of these solutions were solv ed by Dubro vin and Natanzon in [11]. No vik ov criticized the practical relev ance of theta functions since no n umerical algo- rithms existed at the time to actually compute the found solutions. He suggested an effectiv e treatmen t of theta functions (see, for instance, [9]) b y a suitable parametrization of the characteristic quan tities of a Riemann surface, i.e., the perio ds of holomorphic and certain meromorphic differen tials on the given surface. This program is limited to genera smaller than 4 since so-called Schottky relations exist for higher genus b et w een the comp o- nen ts of the p erio d matrix of a Riemann surface. The task to find such relations is known as the Sc hottky problem. This led to the famous No vik ov conjecture for the Sc hottky problem that a Riemann matrix (a symmetric matrix with negativ e definite real part) is the matrix of B -perio ds of the normalized holomorphic different ials of a Riemann surface Date : Octob er 27, 2018. W e thank D. Korotkin and V. Shramchenk o for useful discussions and hints. This work has b een sup- p orted in part by the pro ject F roM-PDE funded b y the Europ ean Research Council through the Adv anced In vestigator Grant Scheme, the Conseil Régional de Bourgogne via a F ABER grant and the ANR via the program ANR-09-BLAN-0117-01. 1 2 C. KALLA AND C. KLEIN if and only if Kric hever’s form ula with this matrix yields a solution to the KP equation. The conjecture w as finally prov en by Shiota [26]. First plots of KP solutions app eared in [24] and via Sc hottky uniformizations in [4]. Since all compact Riemann surfaces can b e defined via non-singular plane algebraic curv es of the form (1.1) F ( x, y ) := N X n =1 M X m =1 a mn x m y n = 0 , x, y ∈ C , with constan t complex co efficients a nm , Deconinc k and v an Ho eij developed an approac h to the symbolic-numerical treatment of algebraic curv es. This approach is distributed as the algcurves pac k age with Maple, see [6, 7, 8]. A purely n umerical approac h to real h yp erelliptic Riemann surfaces was given in [14, 15], and for general Riemann surfaces in [16]. F or a review on computational approaches to Riemann surfaces the reader is referred to [3]. In this pap er we w an t to address typical analytical problems app earing in the numerical study of theta-functional solutions to in tegrable PDEs, and present the state of the art of the field by considering concrete examples. The case of hyperelliptic Riemann surfaces ( N = 2 in (1.1)) is the most accessible, since equation (1.1) can be solved explicitly for y , and since a basis for differentials and homology can b e giv en a priori. F amilies of h yp erelliptic curv es can b e conv eniently parametrized b y their branch p oin ts. The co des [14, 15] are able to treat effectiv ely numerically collisions of branc h points, a limit in whic h certain p erio ds of the corresp onding hyperelliptic surface div erge. If the limiting Riemann surface has genus 0 , the theta series breaks down to a finite sum which giv es for an appropriate c hoice of the c haracteristic well known solitonic solutions to the studied equation. F or solutions defined on general real algebraic curves, i.e., curv es (1.1) with all a nm real, an imp ortan t p oint in applications are realit y and smo othness conditions. These are con v eniently form ulated for a homology basis for which the A -cycles are in v ariant under the action of the an ti-holomorphic inv olution. Ho wev er, the existing algorithms for the computational treatment of algebraic curves pro duce a basis of the homology that is in general not related to p ossible automorphisms of the curve. T o implemen t the realit y and smo othness requiremen ts, a transformation to the basis for which the conditions are form ulated has to b e constructed. W e study the necessary symplectic transformations and giv e explicit relations for so-called M-curves, curves with the maximal num ber of real ov als. T o illustrate these concepts, w e study for the first time numerically theta-functional solutions to integrable equations from the family of NLS equations, namely , the m ulti- comp onen t nonlinear Schrödinger equation (1.2) i ∂ ψ j ∂ t + ∂ 2 ψ j ∂ x 2 + 2 n X k =1 s k | ψ k | 2 ! ψ j = 0 , j = 1 , . . . , n, denoted by n -NLS s , where s = ( s 1 , . . . , s n ) , s k = ± 1 , and the (2 + 1) -dimensional Dav ey- Stew artson (DS) equations, i ψ t + ψ xx − α 2 ψ y y + 2 (Φ + ρ | ψ | 2 ) ψ = 0 , Φ xx + α 2 Φ y y + 2 ρ | ψ | 2 xx = 0 , (1.3) where α = i or α = 1 and where ρ = ± 1 . Both equations (1.2) and (1.3) reduce to the NLS equation under certain conditions: the former obviously in the case n = 1 , the latter 3 if ψ is indep enden t of the v ariable y and satisfies certain b oundary conditions, for instance that Φ + ρ | ψ | 2 tends to zero when x tends to infinity . In tegrabilit y of the NLS equation was shown by Zakharov and Shabat [31] and algebro- geometric solutions were given by Its [18]. The m ulti-comp onent nonlinear Sc hrö dinger equation (1.2) in the case n = 2 , s = (1 , 1) , is called the vector NLS or Manako v system. Manak o v [23] first examined this equation as an asymptotic mo del for the propagation of the electric field in a wa v eguide. Its integrabilit y was sho wn for n = 2 by Zakharo v and Sc h ulman in [32] and for the general case in [25]. Algebro-geometric solutions to the 2-NLS s equation with s = (1 , 1) w ere presented in [12], and for the general case in [19]. The DS equation (1.3) was introduced in [5] to describ e the evolution of a three-dimensional wa ve pac k et on water of finite depth. Its integrabilit y was shown in [1], and solutions in terms of m ulti-dimensional theta functions on general Riemann surfaces were given in [22, 19]. T o ensure the correct numerical implementation of the formulae of [19], w e c hec k for eac h p oint in the spacetime whether certain identities for theta functions are satisfied. Since these identities are not used in the co de, they pro vide a strong test for the computed quan tities. Numerically the iden tities are nev er exactly satisfied, but to high precision. The co de rep orts a w arning if the residual of the test relations is larger than 10 − 6 whic h is w ell b elo w plotting accuracy . Typically the conditions are satisfied to mac hine precision 1 . In addition w e compute the solutions on a numerical grid and numerically differen tiate them. W e c hec k in this w a y for low genus that the solutions to n -NLS s and DS in terms of multi-dimensional theta functions satisfy the resp ectiv e equations to b etter than 10 − 6 . These tw o completely indep enden t tests ensure that the presented plots are showing the correct solutions to b etter than plotting accuracy . The pap er is organized as follows: in Section 2 we recall some facts from the theory of m ulti-dimensional theta functions and the theory of real Riemann surfaces, necessary to giv e theta-functional solutions to the n -NLS s and DS equations. In Section 3 w e consider the hyperelliptic case and study concrete examples of low gen us, also in almost degenerate situations. In Section 4 we consider examples of non-h yp erelliptic real Riemann surfaces and discuss symplectic transformations needed to obtain smo oth solutions. W e add some concluding remarks in Section 5. 2. Thet a functions and real Riemann surf aces In this section we recall basic facts on Riemann surfaces, in particular real surfaces, and m ulti-dimensional theta functions defined on them. Solutions to the n -NLS s and the DS equations in terms of theta functions will b e giv en follo wing [19]. 2.1. Theta functions. Let R g b e a compact Riemann surface of gen us g > 0 . Denote b y ( A , B ) := ( A 1 , . . . , A g , B 1 , . . . , B g ) a canonical homology basis, and b y ( ω 1 , . . . , ω g ) the basis of holomorphic differen tials normalized via (2.1) Z A k ω j = 2i π δ kj , k , j = 1 , . . . , g . The matrix B with entries B kj = R B k ω j of B -p erio ds of the normalized holomorphic dif- feren tials ω j , j = 1 , . . . , g , is symmetric and has a negativ e definite real part. The theta 1 W e w ork with double precision, i.e., a precision of 10 − 16 ; due to rounding errors this is t ypically reduced to 10 − 12 to 10 − 14 . 4 C. KALLA AND C. KLEIN function with (half in teger) characteristic δ = [ δ 1 , δ 2 ] is defined b y (2.2) Θ B [ δ ]( z ) = X m ∈ Z g exp  1 2 h B ( m + δ 1 ) , m + δ 1 i + h m + δ 1 , z + 2i π δ 2 i  , for any z ∈ C g ; here δ 1 , δ 2 ∈  0 , 1 2  g are the vectors of the characteristic δ ; h ., . i denotes the scalar pro duct h u , v i = P i u i v i for an y u , v ∈ C g . The theta function Θ[ δ ]( z ) is even if the characteristic δ is even, i.e., if 4 h δ 1 , δ 2 i is even, and o dd if the characteristic δ is o dd, i.e., if 4 h δ 1 , δ 2 i is o dd. An ev en c haracteristic is called non-singular if Θ[ δ ](0) 6 = 0 , and an o dd characteristic is called non-singular if the gradien t ∇ Θ[ δ ](0) is non-zero. The theta function with c haracteristic is related to the theta function with zero characteristic (the Riemann theta function denoted b y Θ ) as follows (2.3) Θ[ δ ]( z ) = Θ( z + 2i π δ 2 + B δ 1 ) exp  1 2 h B δ 1 , δ 1 i + h z + 2i π δ 2 , δ 1 i  . Denote by Λ the lattice Λ = { 2i π N + B M , N , M ∈ Z g } generated b y the A and B - p eriods of the normalized holomorphic differen tials ω j , j = 1 , . . . , g . The complex torus J ( R g ) = C g / Λ is called the Jacobian of the Riemann surface R g . The theta function (2.2) has the follo wing quasi-p erio dicit y prop erty with resp ect to the lattice Λ : Θ[ δ ]( z + 2i π N + B M ) (2.4) = Θ[ δ ]( z ) exp  − 1 2 h B M , M i − h z , M i + 2i π ( h δ 1 , N i − h δ 2 , M i )  . F or the formulation of solutions to physically relev ant integrable equations in terms of m ulti-dimensional theta functions, there is t ypically a preferred homology basis in whic h the solution takes a simple form. Let ( A , B ) and ( ˜ A , ˜ B ) b e arbitrary canonical homology basis defined on R g , represented here b y 2 g -dimensional v ectors. Under the change of homology basis (2.5)  A B C D   ˜ A ˜ B  =  A B  , where  A B C D  ∈ S p (2 g , Z ) is a symplectic matrix, the theta function (2.2) transforms as (2.6) Θ B [ δ ]( z ) = κ p det ˜ K exp n 1 2 ˜ z t ( ˜ K t ) − 1 B ˜ z o Θ ˜ B [ ˜ δ ]( ˜ z ) , where ˜ K = 2i π A + B ˜ B and B = 2i π (2i π C + D ˜ B ) ˜ K − 1 , (2.7) ˜ z = (2i π ) − 1 ˜ K t z , (2.8)  δ 1 δ 2  =  A − B − C D   ˜ δ 1 ˜ δ 2  + 1 2 Diag  B A t D C t  , (2.9) for an y z ∈ C g , where Diag ( . ) denotes the column v ector of the diagonal entries of the matrix. Here κ is a constant indep enden t of z and ˜ B (the exact v alue of κ is not needed for our purp oses). The Ab el map R g − → J ( R g ) is defined b y (2.10) Z p p 0 := Z p p 0 ω , for an y p ∈ R g , where p 0 ∈ R g is the base p oin t of the application, and where ω = ( ω 1 , . . . , ω g ) t is the v ector of the normalized holomorphic differentials. 5 No w let k a denote a lo cal parameter near a ∈ R g and consider the follo wing expansion of the normalized holomorphic differen tials ω j , j = 1 , . . . , g , (2.11) ω j ( p ) = ( V a,j + W a,j k a ( p ) + . . . ) d k a ( p ) , for an y p oin t p ∈ R g lying in a neigh b ourho od of a , where V a,j , W a,j ∈ C . Let us denote by D a (resp. D 0 a ) the op erator of the directional deriv ativ e along the v ector V a = ( V a, 1 , . . . , V a,g ) t (resp. W a ). A ccording to [24] and [19], the theta function sat- isfies the follo wing identities derived from F ay’s identit y [13]: (2.12) D a D b ln Θ( z ) = q 1 + q 2 Θ( z + R b a ) Θ( z − R b a ) Θ( z ) 2 , (2.13) D 0 a ln Θ( z + R b a ) Θ( z ) + D 2 a ln Θ( z + R b a ) Θ( z ) +  D a ln Θ( z + R b a ) Θ( z ) − K 1  2 + 2 D 2 a ln Θ( z ) + K 2 = 0 , for an y z ∈ C g and an y distinct p oin ts a, b ∈ R g ; here the scalars q i , K i , i = 1 , 2 dep end on the p oin ts a, b and are giv en by (2.14) q 1 ( a, b ) = D a D b ln Θ[ δ ]( R b a ) , (2.15) q 2 ( a, b ) = D a Θ[ δ ](0) D b Θ[ δ ](0) Θ[ δ ]( R b a ) 2 , (2.16) K 1 ( a, b ) = 1 2 D 0 a Θ[ δ ](0) D a Θ[ δ ](0) + D a ln Θ[ δ ]( R b a ) , (2.17) K 2 ( a, b ) = − D 0 a ln Θ( R b a ) − D 2 a ln  Θ( R b a ) Θ(0)  −  D a ln Θ( R b a ) − K 1 ( a, b )  2 , where δ is a non-singular o dd c haracteristic. 2.2. Real Riemann surfaces. A Riemann surface R g is called real if it admits an anti- holomorphic in volution τ : R g → R g , τ 2 = id. The connected comp onen ts of the set of fixed p oin ts of the anti-in v olution τ are called real ov als of τ . W e denote by R g ( R ) the set of fixed p oin ts. A ccording to Harnack’s inequalit y [17], the num b er k of real ov als of a real Riemann surface of genus g cannot exceed g + 1 : 0 ≤ k ≤ g + 1 . Curves with the maximal n um b er k = g + 1 of real ov als are called M-curv es. The complement R g \ R g ( R ) has either one or tw o connected comp onen ts. The curv e R g is called a dividing curve if R g \ R g ( R ) has tw o comp onen ts, and R g is called non-dividing if R g \ R g ( R ) is connected (notice that an M-curve is alwa ys a dividing curve). Example 2.1. Consider the hyp er el liptic curve of genus g define d by the e quation (2.18) µ 2 = 2 g +2 Y i =1 ( λ − λ i ) , wher e the br anch p oints λ i ∈ R ar e or der e d such that λ 1 < . . . < λ 2 g +2 . On such a curve, we c an define two anti-holomorphic involutions τ 1 and τ 2 , given r esp e ctively by τ 1 ( λ, µ ) = ( λ, µ ) and τ 2 ( λ, µ ) = ( λ, − µ ) . Pr oje ctions of r e al ovals of τ 1 on the λ -plane c oincide with the intervals [ λ 2 g +2 , λ 1 ] , . . . , [ λ 2 g , λ 2 g +1 ] , wher e as pr oje ctions of r e al ovals of τ 2 on the λ -plane c oincide with the intervals [ λ 1 , λ 2 ] , . . . , [ λ 2 g +1 , λ 2 g +2 ] . Henc e the curve (2.18) is an M-curve with r esp e ct to b oth anti-involutions τ 1 and τ 2 . 6 C. KALLA AND C. KLEIN Let ( A , B ) b e a basis of the homology group H 1 ( R g ) . According to Prop osition 2.2 in Vinnik o v’s pap er [30], there exists a canonical homology basis (that w e call for simplicity ‘Vinnik o v basis’ in the following) such that (2.19)  τ A τ B  =  I g 0 H − I g   A B  , where I g is the g × g unit matrix, and H is a blo c k diagonal g × g matrix, defined as follows: 1) if R g ( R ) 6 = ∅ , H =              0 1 1 0 . . . 0 1 1 0 0 . . . 0              if R g is dividing , H =          1 . . . 1 0 . . . 0          if R g is non-dividing ; rank ( H ) = g + 1 − k in b oth cases. 2) if R g ( R ) = ∅ , (i.e. the curve do es not ha v e real ov al), 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 o dd. No w let us choose the canonical homology basis in H 1 ( R g ) satisfying (2.19), take a, b ∈ R g and assume that τ a = a and τ b = b . Denote by ` a contour connecting the p oints a and b which do es not in tersect the canonical homology basis. Then the action of τ on the generators ( A , B , ` ) of the relativ e homology group H 1 ( R g , { a, b } ) is giv en by (2.20)   τ A τ B τ `   =   I g 0 0 H − I g 0 N t M t 1     A B `   , where the v ectors N , M ∈ Z g are related b y (see [19]) (2.21) 2 N + H M = 0 . 7 2.3. Theta-functional solutions of the n -NLS s equation. Algebro-geometric data asso ciated to smooth theta-functional solutions of the n -NLS s equation (1.2) consist of {R g , τ , f , z a } , where R g is a compact Riemann surface of genus g > 0 dividing with resp ect to an an ti-holomorphic inv olution τ , and admitting a real meromorphic function f of degree n + 1 ; here z a ∈ R is a non critical v alue of f suc h that the fib er f − 1 ( z a ) = { a 1 , . . . , a n +1 } o v er z a b elongs to the set R g ( R ) . Let us choose natural lo cal parameters k a j near a j giv en b y the pro jection map f , namely , k a j ( p ) = f ( p ) − z a for an y p oin t p ∈ R g lying in a neigh b ourhoo d of a j . Denote by ( A , B , ` j ) the generators of the relative homology group H 1 ( R g , { a n +1 , a j } ) . Let d ∈ R g and θ ∈ R . Then the following functions ψ j , j = 1 , . . . , n , give smo oth solutions of the n -NLS s equation (1.2), see [19], (2.22) ψ j ( x, t ) = | A j | e i θ Θ( Z − d + r j ) Θ( Z − d ) exp {− i ( E j x − F j t ) } , where | A j | = | q 2 ( a n +1 , a j ) | 1 / 2 exp  1 2 h d , M j i  . The v ector M j ∈ Z g is defined b y the action of τ on the relative homology group H 1 ( R g , { a n +1 , a j } ) (see (2.20)). Moreo v er, r j = R ` j ω , and the v ector Z reads Z = i V a n +1 x + i W a n +1 t, where v ectors V a n +1 and W a n +1 are defined in (2.11). The scalars E j , F j are giv en by (2.23) E j = K 1 ( a n +1 , a j ) , F j = K 2 ( a n +1 , a j ) − 2 n X k =1 q 1 ( a n +1 , a k ) , and scalars q i , K i , i = 1 , 2 are defined in (2.14)-(2.17). According to [19], necessary condi- tions for the functions ψ j in (2.22) to solve the n -NLS s equation are the identities (2.12) and (2.13) with ( a, b ) := ( a n +1 , a j ) . The signs s 1 , . . . , s n in (1.2) are giv en by (2.24) s j = exp { i π (1 + α j ) } , where α j ∈ Z denote certain intersection indices on R g defined as follows: let ˜ a n +1 , ˜ a j ∈ R g ( R ) lie in a neighbourho o d of a n +1 and a j resp ectiv ely such that f ( ˜ a n +1 ) = f (˜ a j ) . Denote b y ˜ ` j an orien ted contour connecting ˜ a n +1 and ˜ a j . Then (2.25) α j = ( τ ˜ ` j − ˜ ` j ) ◦ ` j is the intersection index of the closed contour τ ˜ ` j − ˜ ` j and the contour ` j ; this index is computed in the relativ e homology group H 1 ( R g , { a n +1 , a j } ) . In particular, it was shown in [19] that solutions of the fo cusing n -NLS s equation, i.e., for s = (1 , . . . , 1) , are obtained when the branch p oints of the meromorphic function f are pairwise conjugate. 2.4. Theta-functional solutions of the DS equations. No w let us in tro duce smo oth solutions of the DS equations. In characteristic co ordinates ξ = 1 2 ( x − i α y ) , η = 1 2 ( x + i α y ) , α = i or 1 , 8 C. KALLA AND C. KLEIN the DS equations (1.3) tak e the form i ψ t + 1 2 ( ∂ ξ ξ + ∂ η η ) ψ + 2 φ ψ = 0 , ∂ ξ ∂ η φ + ρ 1 2 ( ∂ ξ ξ + ∂ η η ) | ψ | 2 = 0 , (2.26) where φ := Φ + ρ | ψ | 2 , ρ = ± 1 . Recall that DS1 ρ denotes the Da v ey-Stew artson equation when α = i (in this case ξ and η are both real), and DS2 ρ when α = 1 (in this case ξ and η are pairwise conjugate). In b oth cases, for the DS1 ρ and DS2 ρ equations, the solutions ha ve the form [22, 19]: (2.27) ψ ( ξ , η , t ) = | A | e i θ Θ( Z − d + r ) Θ( Z − d ) exp  − i  G 1 ξ + G 2 η − G 3 t 2  , (2.28) φ ( ξ , η , t ) = 1 2 (ln Θ( Z − d )) ξ ξ + 1 2 (ln Θ( Z − d )) η η + h 4 . Here r = R b a ω for some distinct p oin ts a, b ∈ R g , and the v ector Z is defined as (2.29) Z = i  κ 1 V a ξ − κ 2 V b η + ( κ 2 1 W a − κ 2 2 W b ) t 2  . Moreo v er, the scalars G 1 , G 2 and G 3 read (2.30) G 1 = κ 1 K 1 ( a, b ) , G 2 = κ 2 K 1 ( b, a ) , (2.31) G 3 = κ 2 1 K 2 ( a, b ) + κ 2 2 K 2 ( b, a ) + h, where the scalars K 1 , K 2 are defined in (2.16) and (2.17). As shown in [19], necessary conditions for the functions ψ (2.27) and φ (2.28) to solve the DS equations are the identities (2.12) and (2.13). Algebro-geometric data asso ciated to smo oth solutions (2.27), (2.28) of the DS1 ρ equa- tion consist of {R g , τ , a, b, k a , k b } , where R g is a compact Riemann surface of gen us g > 0 , dividing with resp ect to an an ti-holomorphic in v olution τ , a, b are t wo distinct points in R g ( R ) , and k a , k b denote lo cal parameters near a and b respectively whic h satisfy k a ( τ p ) = k a ( p ) for an y p lying in a neigh b ourho od of a , and k b ( τ p ) = k b ( p ) for an y p lying in a neigh b ourho od of b . The remaining quantities satisfy the conditions: d ∈ R g , θ , h ∈ R , κ 2 ∈ R \ { 0 } , and (2.32) κ 1 = − ρ ˜ κ 2 1 κ 2 q 2 ( a, b ) exp  1 2 h B M , M i + h r + d , M i  , for some ˜ κ 1 ∈ R , where M ∈ Z g is defined in (2.20). The scalar | A | is given by | A | = | ˜ κ 1 κ 2 q 2 ( a, b ) | exp {h d , M i} , where the quan tity q 2 is defined in (2.15). Algebro-geometric data asso ciated to smo oth solutions (2.27), (2.28) of the DS2 ρ equa- tion consist of {R g , τ , a, b, k a , k b } , where R g is a compact Riemann surface of genus g > 0 with an an ti-holomorphic in v olution τ , a, b are t w o distinct p oin ts suc h that τ a = b , and k a , k b denote lo cal parameters near a and b resp ectiv ely which satisfy k b ( τ p ) = k a ( p ) for an y point p lying in a neighbourho o d of a . Moreov er, d ∈ i R g , θ , h ∈ R , κ 1 , κ 2 ∈ C \ { 0 } satisfy κ 1 = κ 2 , and the scalar | A | is given by | A | = | κ 1 | | q 2 ( a, b ) | 1 / 2 . Smo oth solutions of the DS2 + equation are obtained when the curv e R g is an M-curve with resp ect to τ , whereas solutions to DS2 − are smo oth if the asso ciated Riemann surface 9 do es not ha v e real ov al with resp ect to τ , and if there is no pseudo-real function of degree g − 1 on it (i.e., function which satisfies f ( τ p ) = − f ( p ) − 1 ), see [22]. Remark 2.1. The sy mmetric structur e of the DS e quations (2.26) with r esp e ct to ξ and η implies that a solution ψ = Ψ( ξ , η , t ) to DS1 + le ads to a solution Ψ( − ξ , η , t ) of DS1 − . 3. Hyperelliptic case Here w e consider concrete examples for the solutions, in terms of multi-dimensional theta functions, to DS and n -NLS s on h yp erelliptic Riemann surfaces. W e first review the n umerical metho ds to visualize the solutions and discuss how the accuracy is tested. 3.1. Computation on real hyperelliptic curv es. The simplest example of algebraic curv es are hyperelliptic curves, (3.1) µ 2 = ( Q 2 g +2 i =1 ( λ − λ i ) , without branc hing at infinity Q 2 g +1 i =1 ( λ − λ i ) , with branc hing at infinity , where g is the genus of the Riemann surface, and where we hav e for the branch p oints λ i ∈ C the relations λ i 6 = λ j for i 6 = j . If the num b er of finite branch p oin ts is o dd, the curv e is branc hed at infinit y . Recall that all Riemann surfaces of gen us 2 are hyperelliptic, and that the in volution σ which interc hanges the sheets, σ ( λ, µ ) = ( λ, − µ ) , is an automorphism on an y h yp erelliptic curve in the form (3.1). A vector of holomorphic differentials for these surfaces is given by (1 , λ, . . . , λ g − 1 ) t d λ/µ . F or a real hyperelliptic curve, the branch p oints are either real or pairwise conjugate. As we saw in Example 2.1, if all branch p oin ts λ i are real and ordered such that λ 1 < . . . < λ 2 g +2 , the h yp erelliptic curve is an M-curve with resp ect to b oth anti-holomorphic inv olutions τ 1 and τ 2 defined in the example. The other case of in terest in the con text of smo oth solutions to n -NLS s and DS are real curves without real branch p oin t. F or the inv olution τ 1 , a curve given by µ 2 = Q g +1 i =1 ( λ − λ i )( λ − λ i ) , with λ i ∈ C \ R , i = 1 , . . . , g + 1 , in this case is dividing (t w o p oints whose pro jections on to C hav e resp ectiv ely a p ositiv e and a negative imaginary part cannot b e connected by a contour whic h does not cross a real ov al), whereas a curv e giv en by µ 2 = − Q g +1 i =1 ( λ − λ i )( λ − λ i ) has no real o v al, and vice versa for the inv olution τ 2 . In the follo wing, w e will only consider real hyperelliptic curv es without branc hing at infinit y and write the defining equation in the form µ 2 = ( λ − ξ )( λ − η ) Q g i =1 ( λ − E i )( λ − F i ) . It is p ossible to in tro duce a con venien t homology basis on the related surfaces, see Fig. 1 for the case η = ξ . The simple form of the algebraic relation betw een µ and λ for hyperelliptic curves mak es the generation of very efficient n umerical co des possible, see, for instance, [14, 15] for details. These co des allo w the treatment of almost degenerate Riemann surfaces, i.e., the case where the branch p oin ts almost collide pairwise, where the distance of the branch p oin ts is of the order of mac hine precision: | E i − F i | ∼ 10 − 14 . The homology basis Fig. 1 is adapted to this kind of degeneration. The Ab el map R b a ω b et w een tw o p oints a and b is computed in the follo wing wa y: the sheet iden tified at the p oin t a = ( λ ( a ) , µ ( a )) (where w e take for µ the ro ot computed b y Matlab) is lab eled sheet 1 , and at the p oint ( λ ( a ) , − µ ( a )) , sheet 2 . Then the ramification p oin t whose pro jection to the λ -sphere has the minimal distance to λ ( a ) is determined. F or simplicit y we assume alw a ys that this is the p oin t ξ in Fig. 1 (for another branc h p oint, this leads to the addition of half-perio ds, see e.g. [2]). This means we compute R b a ω as R b a ω = R b ξ ω − R a ξ ω . The c hoice of a branc h p oin t as the base p oin t of the Ab el map has the 10 C. KALLA AND C. KLEIN N .. j j E 1 F N+1 E g F g E N+1 F 1 E N F Figure 1. Homolo gy b asis on r e al hyp er el liptic curves, c ontours on she et 1 ar e solid, c ontours on she et 2 ar e dashe d. A -cycles ar e the close d c ontours entir ely on she et 1. adv antage that a change of sheet of a p oin t a just implies a change of sign of the integral: R ( λ ( a ) ,µ ( a )) ξ ω = − R ( λ ( a ) , − µ ( a )) ξ ω . T o compute the in tegral R a ξ ω , one has to analytically con tin ue µ on the connecting line b et w een λ ( a ) and ξ onto the λ -sphere. Whereas the ro ot µ is not supp osed to hav e any branching on the considered path, the square ro ot in Matlab is branched on the negative real axis. T o analytically contin ue µ on the path [ λ ( a ) , ξ ] , w e compute the Matlab ro ot at some λ j ∈ [ λ ( a ) , ξ ] , j = 0 , . . . , N c and analytically con tinue it starting from µ ( a ) by demanding that | µ ( λ j +1 ) − µ ( λ j ) | < | µ ( λ j +1 ) + µ ( λ j ) | . The so defined sheets will b e denoted here and in the following by num b ers, i.e., a p oin t on sheet 1 with pro jection λ ( a ) into the base is denoted by ( λ ( a )) (1) . Th us the computation of the Ab el map is reduced to the computation of line integrals on the connecting line b et w een λ ( a ) and ξ in the complex λ -plane. F or the numerical computation of such integrals we use Clenshaw-Curtis integration (see, for instance, [27]): to compute an integral R 1 − 1 h ( x ) d x , thi s algorithm samples the integrand on the N c + 1 Cheb yshev collocation p oin ts x j = cos( j π / N c ) , j = 0 , . . . , N c . The in tegral is appro xi- mated as the sum: R 1 − 1 h ( x ) d x ∼ P N c j =0 w j h ( x j ) (see [27] on ho w to obtain the w eights w j ). It can b e shown that the conv ergence of the integral is exp onen tial for analytic func- tions h as the ones considered here. T o compute the Ab el map, one uses the transformation λ → λ ( a )(1 + x ) / 2 + ξ (1 − x ) / 2 , to the Clenshaw-Curtis in tegration v ariable. The same pro cedure is then carried out for the integral from ξ to b . The theta functions are appro ximated as in [14] as a sum, (3.2) Θ B [ δ ]( z ) ∼ N θ X m 1 = − N θ . . . N θ X m g = − N θ exp  1 2 h B ( m + δ 1 ) , m + δ 1 i + h m + δ 1 , z + 2i π δ 2 i  . The p erio dicit y prop erties of the theta function (2.4) mak e it p ossible to write z = z 0 + 2i π N + B M for some N , M ∈ Z g , where z 0 = 2i π α + B β with α i , β i ∈ ] − 1 2 , 1 2 ] for i = 1 , . . . , g . The v alue of N θ is determined b y the condition that all terms in (2.2) with | m i | > N θ are smaller than mac hine precision, which is controlled by the largest eigenv alue of the real part of the Riemann matrix (the one with minimal absolute v alue since the real part is negativ e definite), see [14, 16]. 11 T o con trol the accuracy of the numerical solutions, we use essentially t w o approaches. First w e c heck the theta iden tit y (2.13), whic h is the underlying reason for the studied functions being solutions to n -NLS s and DS, at eac h point in the spacetime. This test requires the computation of theta deriv atives not needed in the solution (whic h sligh tly reduces the efficiency of the co de since additional quan tities are computed), but pro vides an immediate chec k whether the solution satisfies (2.13) with the required accuracy . Since this iden tit y is not implemented in the co de, it pro vides a strong test. This ensures that all quan tities en tering the solution are computed with the necessary precision. In addition, the solutions are computed on Chebyshev collocation p oin ts (see, for instance, [27]) for each of the ph ysical v ariables. This can b e used for an expansion of the computed solution in terms of Chebyshev p olynomials, a so-called sp ectral metho d having in practice exp onential con v ergence for analytic functions as the ones considered here. Since the deriv atives of the Cheb yshev polynomials can b e expressed linearly in terms of Chebyshev p olynomials, a deriv ative acts on the space of p olynomials via a so called differentiation matrix. With these standard Chebyshev differen tiation matrices (see [27]), the solution can b e n umerically differen tiated. The computed deriv atives allo w to chec k with whic h n umerical precision the PDE is satisfied by a n umerical solution. With these tw o indep enden t tests, we ensure that the shown solutions are correct to muc h better than plotting accuracy (the co de rep orts a warning if the ab ov e tests are not satisfied to better than 10 − 6 ). 3.2. Solutions to the DS equations. The elliptic solutions are the well known trav elling w a ve solutions and will not b e discussed here. The simplest examples we will consider for the DS solutions are giv en on h yp erelliptic curves of gen us 2 . As we saw in Section 2.4, for DS1 ρ realit y and smo othness conditions imply that the branch p oin ts of the curv e are either all real (M-curve) or all pairwise conjugate (dividing curv e). The p oin ts a and b m ust pro ject to real p oin ts on the λ -sphere and must b e stable under the anti-holomorphic in v olution τ (we use here τ = τ 1 , as defined in Example 2.1, except for DS2 − ). F or DS2 ρ , we hav e τ a = b where the pro jection of a onto the λ -sphere is the conjugate of the pro jection of b . F or DS2 + the curve m ust hav e only real branch p oints (M-curve), whereas for DS2 − it m ust hav e no real ov al. F or DS we will mainly give plots for fixed time since for low genus, the solution is essen tially tra v elling in one direction. F or higher genus, w e sho w a more interesting time dep endence in Fig. 9. W e first consider the defo cusing v ariants, DS1 + and DS2 + on M-curv es. In gen us 2 w e study the family of curves with the branc h p oin ts − 2 , − 1 , 0 , , 2 , 2 +  for  = 1 and  = 10 − 10 . In the former case the solutions will b e p eriodic in the ( x, y ) -plane, in the latter almost solitonic since the Riemann surface is almost degenerate (in the limit  → 0 the surface degenerates to a surface of genus 0 ; the resulting solutions are discussed in more detail in [20]). T o obtain non-trivial solutions in the solitonic limit, we use d = 1 2 [ 1 1 0 0 ] t in all examples. In Fig. 2 it can b e seen that these are in fact dark solitons , i.e., the solutions tend asymptotically to a non-zero constant and the solitons thus represent ‘shadows’ on a bac kground of light. The well kno wn features from soliton collisions for (1+1)-dimensional in tegrable equations, namely , the propagation without change of shap e, and the unchanged shap e and phase shift after the collision, can be seen here in the ( x, y ) -plane. The corresp onding solutions to DS2 + can b e seen in Fig. 3. W e only show the square mo dulus of the solution here for simplicity . F or the real and the imaginary part of such a solution for the DS1 − -case, see Fig. 6. Because of remark 2.1 all solutions shown for DS1 + 12 C. KALLA AND C. KLEIN on M-curves are after the change of co ordinate ξ → − ξ solutions to DS1 − . F or this reason DS1 − solutions on M-curv es will not b e presen ted here. Figure 2. Solution (2.27) to the DS1 + e quation at t = 0 on a hyp er el liptic curve of genus 2 with br anch p oints − 2 , − 1 , 0 , , 2 , 2 +  and a = ( − 1 . 9) (1) , b = ( − 1 . 1) (2) for  = 1 on the left and  = 10 − 10 , the almost solitonic limit, on the right. Figure 3. Solution (2.27) to the DS 2 + e quation at t = 0 on a hyp er el liptic curve of genus 2 with br anch p oints − 2 , − 1 , 0 , , 2 , 2 +  and a = ( − 1 . 5 + 2i) (1) , b = ( − 1 . 5 − 2i) (2) for  = 1 on the left and  = 10 − 10 , the almost solitonic limit, on the right. In the same w a y one can study , on a gen us 4 h yp erelliptic curv e, the formation of the dark 4-soliton for these t wo equations. W e consider the curve with branch p oin ts − 4 , − 3 , − 2 , − 2 + , 0 , , 2 , 2 + , 4 , 4 +  for  = 1 and  = 10 − 10 , and use d = 1 2 [ 1 1 1 1 0 0 0 0 ] t . The DS1 + solutions for this curv e can b e seen in Fig. 4. The corresp onding solutions to DS2 + is sho wn in Fig. 5. Solutions to the fo cusing v arian ts of these equations can b e obtained on hyperelliptic curv es with pairwise conjugate branch p oin ts. F or such solutions the solitonic limit cannot b e obtained as ab o ve since the quotient of theta functions in (2.27) tends to a constan t in this case. T o obtain the w ell-known bright solitons (solutions tend to zero at spatial infinit y) in this wa y , the hyperelliptic curve has to b e completely degenerated (all branch 13 Figure 4. Solution (2.27) to the DS1 + e quation at t = 0 on a hyp er el liptic curve of genus 4 with br anch p oints − 4 , − 3 , − 2 , − 2 + , 0 , , 2 , 2 + , 4 , 4 +  and a = ( − 3 . 9) (1) , b = ( − 3 . 1) (2) for  = 1 on the left and  = 10 − 10 , the almost solitonic limit, on the right. Figure 5. Solution (2.27) to the DS2 + e quation at t = 0 on a hyp er el liptic curve of genus 4 with br anch p oints − 4 , − 3 , − 2 , − 2 + , 0 , , 2 , 2 + , 4 , 4 +  and a = ( − 1 . 5 + 2i) (1) , b = ( − 1 . 5 − 2i) (1) for  = 1 on the left and  = 10 − 10 , the almost solitonic limit, on the right. p oin ts must collide pairwise to double p oints) which leads to limits of the form ’ 0 / 0 ’ in the expression for the solution (2.27) which are not con venien t for a numerical treatmen t; see [20] for an analytic discussion. Therefore we only consider non-degenerate h yp erelliptic curv es here. T o obtain smo oth solutions, we use d = 0 . A solution in genus 2 of the DS1 − equation is studied on the curve with the branch p oin ts − 2 ± i , − 1 ± i , 1 ± i in Fig. 6. A typical example of a DS1 − solution on a hyperelliptic curve of genus 4 with branch p oin ts − 2 ± i , − 1 ± i , ± i , 1 ± i , 2 ± i is shown in Fig. 7. Smo oth solutions to DS2 − can b e obtained on Riemann surfaces without real ov al for p oin ts a and b satisfying τ a = b . As men tioned ab o ve, h yperelliptic curv es of the form µ 2 = − Q 2 g +2 i =1 ( λ − λ i ) with pairwise conjugate branc h p oin ts ha v e no real o v al for the standard in v olution τ 1 as defined in Example 2.1. On the other hand, surfaces defined b y the algebraic equation µ 2 = Q 2 g +2 i =1 ( λ − λ i ) hav e no real ov al for the in volution τ 2 (see Example 2.1). W e will consider here τ 2 for the same curves as for DS1 − . An example for 14 C. KALLA AND C. KLEIN Figure 6. Solution (2.27) to the DS1 − e quation at t = 0 on a hyp er el liptic curve of genus 2 with br anch p oints − 2 ± i , − 1 ± i , 1 ± i and a = ( − 4) (1) , b = ( − 3) (2) . The squar e mo dulus of the solution is shown on the left, r e al and imaginary p arts on the right. Figure 7. Solution (2.27) to the DS1 − e quation at t = 0 on a hyp er el liptic curve of genus 4 with br anch p oints − 2 ± i , − 1 ± i , ± i , 1 ± i , 2 ± i and a = ( − 4) (1) , b = ( − 3) (2) . gen us 2 can be seen in Fig. 8. An example for a DS2 − solution of gen us 4 can b e seen in Fig. 9. 3.3. Solutions to the n -NLS s equation. A straightforw ard w ay to obtain solutions (2.22) to the n -NLS s equation is giv en on an ( n + 1) -sheeted branched co vering of the complex plane, an approach that will b e studied in more detail in the next section. As can b e seen from the pro of of Theorem 4.1 in [19], the crucial p oint in the construction of these solutions is the fact that P n +1 k =1 V a k = 0 . This implies that it is also p ossible to construct theta-functional n -NLS s solutions on hyperelliptic surfaces b y introducing constants γ k via P n +1 k =1 γ k V a k = 0 in the follo wing corollary of Theorem 4.1 in [19]: 15 Figure 8. Solution to the DS 2 − e quation at t = 0 on a hyp er el liptic curve of genus 2 with br anch p oints − 2 ± i , − 1 ± i , 1 ± i and a = ( − 1 . 5 + 2i) (1) , b = ( − 1 . 5 − 2i) (2) . Figure 9. Solution to the DS 2 − e quation for sever al values of t on a hy- p er el liptic curve of genus 4 with br anch p oints − 2 ± i , − 1 ± i , ± i , 1 ± i , 2 ± i and a = ( − 1 . 5 + 2i) (1) , b = ( − 1 . 5 − 2i) (2) . 16 C. KALLA AND C. KLEIN Corollary 3.1. L et R g b e a r e al hyp er el liptic curve of genus g > 0 and denote by τ an anti-holomorphic involution. Cho ose the c anonic al ho molo gy b asis which satisfies (2.19). T ake n ≥ g and let a 1 , . . . , a n +1 ∈ R g ( R ) not r amific ation p oints having distinct pr oje ction λ ( a j ) , j = 1 , . . . , n + 1 , onto the λ -spher e. Denote by ` j an oriente d c ontour b etwe en a n +1 and a j which do es not interse ct cycles of the c anonic al homolo gy b asis. L et d R ∈ R g , T ∈ Z g , and define d = d R + i π 2 ( diag ( H ) − 2 T ) . T ake θ ∈ R and let γ g +1 , . . . , γ n ∈ R b e arbitr ary c onstants with γ n +1 = 1 . Put ˆ s = ( sign ( γ 1 ) s 1 , . . . , sign ( γ n ) s n ) wher e s j is given in (2.24), and the sc alars γ j , j = 1 , . . . , g , ar e define d by P n +1 k =1 γ k V a k = 0 . Then the fol lowing functions ψ j , j = 1 , . . . , n , give solutions of the n -NLS ˆ s e quation (1.2) (3.3) ψ j ( x, t ) = | γ j | 1 / 2 | A j | e i θ Θ( Z − d + r j ) Θ( Z − d ) exp {− i ( E j x − F j t ) } , wher e | A j | = | q 2 ( a n +1 , a j ) | 1 / 2 exp  1 2 h d , M j i  . Her e Z = i V a n +1 x + i W a n +1 t , wher e the ve ctors V a n +1 and W a n +1 wer e intr o duc e d in (2.11), and r j = R ` j ω . The sc alars E j , F j ar e given by E j = K 1 ( a n +1 , a j ) , F j = K 2 ( a n +1 , a j ) − 2 n X k =1 γ k q 1 ( a n +1 , a k ) , wher e q i , K i for i = 1 , 2 ar e define d in (2.14)-(2.17). If R g is dividing and if d ∈ R g , functions (3.3) give smo oth solutions of n -NLS ˆ s . As an example we consider, as for DS in gen us 2 , the family of curves with the branch p oin ts − 2 , − 1 , 0 , , 2 , 2 +  for  = 1 and  = 10 − 10 . In the former case the solutions will b e p eriodic in the ( x, t )-plane, in the latter almost solitonic. T o obtain non-trivial solutions in the solitonic limit, w e use d = 1 2 [ 1 1 0 0 ] t in all examples. In Fig. 10 w e show the case a 1 = ( − 1 . 9) (1) , a 2 = ( − 1 . 1) (1) and a 3 = ( − 1 . 8) (1) , whic h leads to a solution of 2-NLS ˆ s with ˆ s = ( − 1 , − 1) . In terc hanging a 2 and a 3 in the abov e example, w e obtain a solution to 2-NLS ˆ s with ˆ s = (1 , − 1) in Fig. 11. ï 10 ï 5 0 5 10 ï 10 ï 5 0 5 10 72 72.5 73 x t | s 1 | 2 ï 10 ï 5 0 5 10 ï 10 ï 5 0 5 10 0 0.2 0.4 x t | s 2 | 2 ï 10 ï 5 0 5 10 ï 10 ï 5 0 5 10 72 74 76 x t | s 1 | 2 ï 10 ï 5 0 5 10 ï 10 ï 5 0 5 10 0 0.1 0.2 x t | s 2 | 2 Figure 10. Solution (3.3) to the 2-NLS ˆ s e quation with ˆ s = ( − 1 , − 1) on a hyp er el liptic curve of genus 2 with br anch p oints − 2 , − 1 , 0 , , 2 , 2 +  and a 1 = ( − 1 . 9) (1) , a 2 = ( − 1 . 1) (1) and a 3 = ( − 1 . 8) (1) for  = 1 on the left and  = 10 − 10 , the almost solitonic limit, on the right. Solutions of 4-NLS ˆ s can b e studied in the same wa y on the h yp erelliptic curve of gen us 4 with branch p oin ts − 4 , − 3 , − 2 , − 2 + , 0 , , 2 , 2 + , 4 , 4 +  . W e use d = 1 2 [ 1 1 1 1 0 0 0 0 ] t and the 17 p oin ts a 1 = ( − 3 . 9) (1) , a 2 = ( − 3 . 7) (1) , a 3 = ( − 3 . 5) (1) , a 4 = ( − 3 . 3) (1) and a 5 = ( − 3 . 1) (1) . Since the vectors V a j and W a j are v ery similar in this case, the same is true for the functions ψ j . Therefore, we will only show the square mo dulus of the first comp onen t ψ 1 in Fig. 12 for ˆ s = (1 , − 1 , 1 , − 1) on the left. In terchanging a 4 and a 5 in this case, one gets a solution to 4-NLS ˆ s with ˆ s = ( − 1 , 1 , − 1 , − 1) which can b e seen on the righ t of Fig. 12. ï 5 ï 4 ï 3 ï 2 ï 1 0 1 2 3 4 5 ï 1 ï 0.5 0 0.5 1 20 40 60 x t | s 1 | 2 ï 5 ï 4 ï 3 ï 2 ï 1 0 1 2 3 4 5 ï 1 ï 0.5 0 0.5 1 0 50 100 x t | s 2 | 2 ï 5 ï 4 ï 3 ï 2 ï 1 0 1 2 3 4 5 ï 1 ï 0.5 0 0.5 1 0 50 100 x t | s 1 | 2 ï 5 ï 4 ï 3 ï 2 ï 1 0 1 2 3 4 5 ï 1 ï 0.5 0 0.5 1 0 50 100 x t | s 2 | 2 Figure 11. Solution (3.3) to the 2-NLS ˆ s e quation with ˆ s = (1 , − 1) on a hyp er el liptic curve of genus 2 with br anch p oints − 2 , − 1 , 0 , , 2 , 2 +  and a 1 = ( − 1 . 9) (1) , a 2 = ( − 1 . 8) (1) and a 3 = ( − 1 . 1) (1) for  = 1 on the left and  = 10 − 10 , the almost solitonic limit, on the right. ï 5 ï 4 ï 3 ï 2 ï 1 0 1 2 3 4 5 ï 1 ï 0.5 0 0.5 1 2 4 6 8 10 x t | s 1 | 2 ï 10 ï 8 ï 6 ï 4 ï 2 0 2 4 6 8 10 ï 5 0 5 0.9 1 1.1 1.2 1.3 1.4 x t | s 1 | 2 Figure 12. Solution to the 4-NLS ˆ s e quation on a hyp er el liptic curve of genus 4 with br anch p oints − 4 , − 3 , − 2 , − 2 + , 0 , , 2 , 2 + , 4 , 4 +  and  = 1 for a 1 = ( − 3 . 9) (1) , a 2 = ( − 3 . 7) (1) , a 3 = ( − 3 . 5) (1) , a 4 = ( − 3 . 3) (1) and a 5 = ( − 3 . 1) (1) , which le ads to ˆ s = (1 , − 1 , 1 , − 1) , on the left, and for a 1 = ( − 3 . 9) (1) , a 2 = ( − 3 . 7) (1) , a 3 = ( − 3 . 5) (1) , a 4 = ( − 3 . 1) (1) and a 5 = ( − 3 . 3) (1) , which le ads to ˆ s = ( − 1 , 1 , − 1 , − 1) on the right. The almost solitonic limit  = 10 − 10 pro duces well kno wn solitonic patterns as sho wn for instance for the DS equation in the previous subsection. Hyp erelliptic solutions to the n -NLS ˆ s equation with all signs ˆ s j satisfying ˆ s j = 1 , can b e constructed on a curve without real branch p oin ts. T o obtain smo oth solutions, we use 18 C. KALLA AND C. KLEIN d = 0 . A solution of the 2-NLS ˆ s equation is studied on the curv e of genus 2 with the branc h p oin ts − 2 ± i , − 1 ± i , 1 ± i in Fig. 13. ï 10 ï 5 0 5 10 ï 10 ï 5 0 5 10 0 10 20 x t | s 1 | 2 ï 10 ï 5 0 5 10 ï 10 ï 5 0 5 10 0 5 10 x t | s 2 | 2 Figure 13. Solution to the 2-NLS ˆ s e quation with ˆ s = (1 , 1) on a hyp er el lip- tic curve of genus 2 with br anch p oints − 2 ± i , − 1 ± i , 1 ± i and a 1 = ( − 1 . 9) (1) , a 2 = ( − 1 . 8) (2) and a 3 = ( − 1 . 1) (1) . A typical example for a hyperelliptic 4-NLS ˆ s solution with ˆ s = (1 , 1 , 1 , 1) can b e obtained on a curve of genus 4 with branch p oin ts − 2 ± i , − 1 ± i , ± i , 1 ± i , 2 ± i , as shown in Fig. 14. ï 40 ï 20 0 20 40 ï 40 ï 20 0 20 40 4.8 4.9 5 x t | s 1 | 2 ï 40 ï 20 0 20 40 ï 40 ï 20 0 20 40 0.2 0.4 0.6 0.8 1 x t | s 2 | 2 ï 40 ï 20 0 20 40 ï 40 ï 20 0 20 40 68 68.5 69 x t | s 3 | 2 ï 40 ï 20 0 20 40 ï 40 ï 20 0 20 40 0.2 0.4 0.6 x t | s 4 | 2 Figure 14. Solution to the 4-NLS ˆ s e quation with ˆ s = (1 , 1 , 1 , 1) on a hy- p er el liptic curve of genus 4 with br anch p oints − 2 ± i , − 1 ± i , ± i , 1 ± i , 2 ± i and a 1 = ( − 3 . 9) (1) , a 2 = ( − 3 . 7) (2) , a 3 = ( − 3 . 5) (1) , a 4 = ( − 3 . 3) (2) and a 5 = ( − 3 . 1) (1) . 19 4. General real algebraic cur ves The quantities entering theta-functional solutions of the DS and n -NLS s equations are related to compact Riemann surfaces. Since all compact Riemann surfaces can b e defined via compactified non-singular algebraic curv es, conv enien t computational approac hes as [6, 7] and [16] are based on algebraic curves: differen tials, homology basis and p erio ds of the Riemann surface can b e obtained in an algorithmic wa y . W e refer the reader to the cited literature for details. The identification of the sheets of the co vering defined b y the algebraic curve (1.1) via the pro jection map ( x, y ) 7→ x , is done, as in the h yp erelliptic case, b y analytic con tin uation of the ro ots y i , i = 1 , . . . , N for some non-critical p oin t x b on the x -sphere, along a set of contours sp ecified in [16]. In the con text of real algebraic curv es for which solutions of n -NLS s and DS are discussed here, an additional problem is to establish the action of the anti-holomorphic inv olution τ on p oin ts on differen t sheets. A typical problem is to find points a ∈ R g and b ∈ R g with the same pro jection on to the x -sphere such that τ a = b ; here τ is defined via τ a = ( x ( a ) , y ( a )) . T o this end, the ro ots y i , i = 1 , . . . , N , identified at x = x b , are analytically con tinued to the p oints pro jecting to x ( a ) on the x -sphere. It is then established which pairs of p oints in the different sheets satisfy τ a = b . In contrast to the h yp erelliptic curves of the previous section, it is not possible for general curv es to in troduce a priori a basis of the homology . Thus the cited co des use an algorithm by T retkoff and T retk off [28] whic h produces a homology basis for a giv en branc hing structure of the cov ering which is in general not adapted to p ossible automor- phisms of the curve. In the con text of theta-functional solutions to integrable PDEs one is often interested in real curves. As discussed in [19], the Vinniko v basis (i.e., the canon- ical homology basis which satisfies (2.19)) is conv enien t in this context. Since solutions and smo othness conditions for n -NLS s and DS equations are form ulated in this basis, a symplectic transformation relating the computed basis to the Vinnik ov basis needs to b e w ork ed out. This transformation is discussed in the presen t section and will b e applied to examples of real algebraic curv es. 4.1. Symplectic transformation. Let R g b e a real compact Riemann surface of gen us g and τ an anti-holomorphic inv olution defined on it. Let ( ν 1 , . . . , ν g ) b e a basis of holo- morphic differen tials such that (4.1) τ ∗ ν j = ν j , j = 1 , . . . , g , where τ ∗ is the action of τ lifted to the space of holomorphic differentials: τ ∗ ω ( p ) = ω ( τ p ) for an y p ∈ R g . F or an arbitrary canonical homology basis ( A , B ) , let us denote by P A and P B the matrices of A and B -p eriods of the differentials ν j : (4.2) ( P A ) ij = Z A i ν j , ( P B ) ij = Z B i ν j , i, j = 1 , . . . , g . In what follows ( A , B ) denotes the Vinniko v basis. F rom (4.1) and (2.19) w e deduce the action of the complex conjugation on the matrices P A and P B : (4.3) ( P A ) ij ∈ R , (4.4) P B = − P B + H P A . Denote by ( ˜ A , ˜ B ) the homology basis on R g pro duced b y the T retk off-T retkoff algo- rithm. F rom the symplectic transformation (2.5) w e obtain the follo wing transformation 20 C. KALLA AND C. KLEIN la w b et w een the matrices P ˜ A , P ˜ B and P A , P B defined in (4.2): (4.5)  A B C D   P ˜ A P ˜ B  =  P A P B  . Therefore, b y (4.3) one gets A Re  P ˜ A  + B Re  P ˜ B  = P A (4.6) A Im  P ˜ A  + B Im  P ˜ B  = 0 , (4.7) and b y (4.4) C Re  P ˜ A  + D Re  P ˜ B  = 1 2 H P A (4.8) C Im  P ˜ A  + D Im  P ˜ B  = Im ( P B ) . (4.9) A ccording to (4.6), the matrix A Re  P ˜ A  + B Re  P ˜ B  is in vertible, since the matrix P A of A -p eriods of a basis of holomorphic differen tials is alwa ys inv ertible (see, for instance, [3]). Moreo v er, it is well kno wn that the Riemann matrix B = 2i π P B ( P A ) − 1 has a (negativ e) definite real part, which is equal to − 2 π Im ( P B ) Im (( P A ) − 1 ) for the real matrix P A here. Then, b y (4.9) the matrix C Im  P ˜ A  + D Im  P ˜ B  is also in vertible. Lemma 4.1. The matric es A, B , C , D ∈ M g ( Z ) solving (4.6)-(4.9) satisfy: A t = Im  P ˜ B   C Im  P ˜ A  + D Im  P ˜ B  − 1 (4.10) B t = − Im  P ˜ A   C Im  P ˜ A  + D Im  P ˜ B  − 1 (4.11) C t = 1 2 A t H − R e  P ˜ B   A R e  P ˜ A  + B R e  P ˜ B  − 1 (4.12) D t = 1 2 B t H + R e  P ˜ A   A R e  P ˜ A  + B R e  P ˜ B  − 1 . (4.13) Pr o of. Recall that symplectic matrices M =  A B C D  ∈ S p (2 g , Z ) are characterized by A t D − C t B = I g , (4.14) A t C = C t A, (4.15) D t B = B t D . (4.16) Multiplying equality (4.7) from the left by the matrix C t , w e deduce from (4.14) and (4.15) that: C t A Im  P ˜ A  + C t B Im  P ˜ B  = 0 C t A Im  P ˜ A  + ( A t D − I g ) Im  P ˜ B  = 0 A t C Im  P ˜ A  + A t D Im  P ˜ B  = Im  P ˜ B  , whic h leads to (4.10). Equalit y (4.11) can b e c hec ked analogously with (4.14) and (4.16). T o pro ve (4.12), m ultiply equality (4.8) from the left b y the matrix A t . Using (4.14) and 21 (4.15) one gets: A t C Re  P ˜ A  + A t D Re  P ˜ B  = 1 2 A t H P A C t A Re  P ˜ A  + ( I g + C t B ) Re  P ˜ B  = 1 2 A t H P A C t  A Re  P ˜ A  + B Re  P ˜ B  = 1 2 A t H P A − Re  P ˜ B  , whic h by (4.6) leads to (4.12). Identit y (4.13) can b e prov ed analogously .  Remark 4.1. Lemma 4.1 implies that it is sufficien t to kno w the matrices A and B (or C and D ) to determine the symplectic matrix in (4.5). In practice, this means that a con v enient ansatz for one of the matrices has to b e found. The others then follow from the relations in Lemma 4.1. Th us to construct these matrices one first c hec ks which of the matrices Re  P ˜ A  , Re  P ˜ B  , Im  P ˜ A  , Im  P ˜ B  are inv ertible. This wa y a matrix can b e iden tified (e.g. A ) in terms of whic h the others can b e expressed. The task is thus reduced to provide an ansatz for this matrix such that the others will ha v e en tire comp onen ts. W e illustrate this approach at the example of the T rott curve b elow. Prop osition 4.1. L et ( ˜ A , ˜ B ) b e the c anonic al homolo gy b asis obtaine d with the T r etkoff- T r etkoff algorithm; we denote with a tilde the quantities expr esse d in this b asis. Under the change of homolo gy b asis (2.5), solutions of n -NLS s and DS e quations given in (2.22) and (2.27), r esp e ctively, which ar e expr esse d in the b asis satisfying (2.19), tr ansform as fol lows: the ve ctor d app e aring in the solutions b e c omes (2i π ) − 1 ˜ K t d wher e ˜ K = 2i π A + B ˜ B , and the theta function Θ = Θ B with zer o char acteristic, tr ansforms to the theta function Θ ˜ B [ ˜ δ ] with char acteristic ˜ δ = [ ˜ δ 1 , ˜ δ 2 ] given by ˜ δ 1 = 1 4 diag  B t H B − 2 R e  P ˜ A  ˜ M − 1 Im  P t ˜ A  , (4.17) ˜ δ 2 = 1 4 diag  A t H A − 2 R e  P ˜ B  ˜ M − 1 Im  P t ˜ B  , (4.18) wher e (4.19) ˜ M = Im  P t ˜ B  R e  P ˜ A  − Im  P t ˜ A  R e  P ˜ B  . Mor e over, the r e al c onstant h app e aring in (2.28) and (2.31) b e c omes h + ˜ h wher e Im ( ˜ h ) is given by (4.20) Im ( ˜ h ) = 1 2 ln (      Θ ˜ B [ ˜ δ ]( ˜ Z + ˜ r ) Θ ˜ B [ ˜ δ ]( ˜ Z − ˜ r )      ) − Im ( ˜ G 3 ) , with ˜ Z = i( ˜ W a − ˜ W b ) , and the ve ctors N , M define d in (2.20) b e c ome A t N + C t M and B t N + D t M r esp e ctively. Pr o of. Under the change of the canonical homology basis (2.5), the vector ω = ( ω 1 , . . . , ω g ) t of normalized holomorphic differen tials transforms as (4.21) ω = 2i π ( ˜ K t ) − 1 ˜ ω, where ˜ K = 2i π A + B ˜ B . A ccording to the transformation la w (2.6) of theta functions, it can b e chec k ed after straightforw ard calculations, that under this change of homology 22 C. KALLA AND C. KLEIN basis, quan tities (2.15)-(2.17) transform as: q 2 ( a, b ) = ˜ q 2 ( a, b ) exp n − ˜ r t ( ˜ K t ) − 1 B ˜ r o , (4.22) q 1 ( a, b ) = ˜ q 1 ( a, b ) + 1 2 ˜ V t a ( ˜ K t ) − 1 B ˜ V b , (4.23) (4.24) K 1 ( a, b ) = ˜ K 1 ( a, b ) + 1 2  ˜ V t a ( ˜ K t ) − 1 B ˜ r + ˜ r t ( ˜ K t ) − 1 B ˜ V a  , (4.25) K 2 ( a, b ) = ˜ K 2 ( a, b ) − 1 2  ˜ W t a ( ˜ K t ) − 1 B ˜ r + ˜ r t ( ˜ K t ) − 1 B ˜ W a  − ˜ V t a ( ˜ K t ) − 1 B ˜ V a . W e deduce that solutions of the n -NLS s and DS equations giv en in (2.22) and (2.27), re- sp ectiv ely , transform as follows: the vector d b ecomes (2i π ) − 1 ˜ K t d , and the theta function Θ = Θ B with zero characteristic, transforms to the theta function Θ ˜ B [ ˜ δ ] with character- istic ˜ δ . T o compute the vectors of the c haracteristic ˜ δ w e consider the inv ersion of the symplectic matrix in (2.5) whic h leads to  ˜ A ˜ B  =  D t − B t − C t A t   A B  . Since the characteristic used in [19] to construct solutions (2.22), (2.27) of n -NLS s and DS is zero, w e get with (2.9)  ˜ δ 1 ˜ δ 2  = 1 2 Diag  D t B C t A  (note that D t B and C t A are symmetric matrices). Substituting (4.10) and (4.11) in (4.12) (resp. (4.13)), it can b e c heck ed that C t A = 1 2  A t H A − 2 Re  P ˜ B  ˜ M − 1 Im  P t ˜ B  , D t B = 1 2  B t H B − 2 Re  P ˜ A  ˜ M − 1 Im  P t ˜ A  , with ˜ M = Im  P t ˜ B  Re  P ˜ A  − Im  P t ˜ A  Re  P ˜ B  . Moreo v er, the real constant h app earing in the solutions (2.27)-(2.28) of the Da v ey- Stew artson equations b ecomes h + ˜ h , where ˜ h is giv en by (4.26) ˜ h = − ˜ V t a ( ˜ K t ) − 1 B ˜ V a − ˜ V t b ( ˜ K t ) − 1 B ˜ V b . Notice that the construction of the solutions (2.27) giv en in [19] allo ws to express the imaginary part of the constant ˜ h (4.26) in terms of the characteristic ˜ δ . Namely , since the realit y condition (4.27) ψ ∗ = ρ ψ is satisfied for the Vinnik ov basis, where the function ψ ∗ ( ξ , η , t ) reads (4.28) 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  , 23 it also holds for the computed basis. Therefore, putting ξ = η = 0 , t = 2 , d = 0 , and taking the mo dulus of each term in (4.27) expressed in the computed basis, one gets:      Θ ˜ B [ ˜ δ ]( ˜ Z − ˜ r ) Θ ˜ B [ ˜ δ ]( ˜ Z ) exp {− i ( ˜ G 3 + ˜ h ) }      =      Θ ˜ B [ ˜ δ ]( ˜ Z + ˜ r ) Θ ˜ B [ ˜ δ ]( ˜ Z ) exp { i ( ˜ G 3 + ˜ h ) }      where ˜ Z = i( ˜ W a − ˜ W b ) . W e deduce that (4.29) Im ( ˜ h ) = 1 2 ln (      Θ ˜ B [ ˜ δ ]( ˜ Z + ˜ r ) Θ ˜ B [ ˜ δ ]( ˜ Z − ˜ r )      ) − Im ( ˜ G 3 ) .  Remark 4.2. In the c ase wher e the sp e ctr al curve is an M-curve, i.e. H = 0 , the ve ctors of char acteristic (4.17) and (4.18) do not dep end explicitly on the symple ctic matrix app e aring in the change of homolo gy b asis and ar e uniquely define d by: ˜ δ 1 = 1 2 diag  R e  P ˜ A  h Im  P t ˜ B  R e  P ˜ A  − Im  P t ˜ A  R e  P ˜ B  i − 1 Im  P t ˜ A   , (4.30) ˜ δ 2 = 1 2 diag  R e  P ˜ B  h Im  P t ˜ B  R e  P ˜ A  − Im  P t ˜ A  R e  P ˜ B  i − 1 Im  P t ˜ B   . (4.31) It w ould b e possible to compute the theta-functional solutions in the Vinnik o v basis once the symplectic transformation betw een this basis and the basis determined b y the co de is known. How ev er, since this symplectic transformation is not unique, the found Vinnik o v basis leads in general to a Riemann matrix for whic h the theta series con v erges only slo wly , i.e., the v alue N θ in (3.2) has to b e chosen very large. T o a void this problem, w e compute the theta function alw ays in the typically more con venien t T retkoff-T retkoff basis with the c haracteristic of the theta functions given by (4.17)-(4.19). 4.2. T rott curve. The T rott curv e [29] given by the algebraic equation (4.32) 144 ( x 4 + y 4 ) − 225 ( x 2 + y 2 ) + 350 x 2 y 2 + 81 = 0 is an M-curv e with resp ect to the anti-holomorphic in v olution τ defined b y τ ( x, y ) = ( x, y ) , and is of genus 3. Moreo ver, this curve has real branc h p oin ts only (and 28 real bitangen ts, namely , tangents to the curve in tw o places). Our computed matrices of ˜ A and ˜ B -perio ds read 2 P ˜ A =   0 . 0235i 0 . 0138i 0 . 0138i 0 0 . 0277i 0 − 0 . 0315 0 0 . 0250   , P ˜ B =   − 0 . 0315 + 0 . 0235i 0 . 0138i − 0 . 0250 + 0 . 0138i 0 − 0 . 025 + 0 . 0277i 0 . 0250 − 0 . 0235i 0 . 0138i 0 . 0138i   . The T rott curv e b eing an M-curv e, the v ectors of the characteristic ˜ δ satisfy (4.30) and (4.31), whic h leads to ˜ δ = 1 2 [ 0 0 0 1 1 0 ] t . 2 F or the ease of representation we only give 4 digits here, though at least 12 digits are known for these quan tities. 24 C. KALLA AND C. KLEIN A p ossible c hoice of a symplectic transformation bringing the computed basis to the Vinnik o v basis is: A =   1 0 0 0 1 0 0 0 1   , B =   − 1 0 0 0 − 1 0 0 0 0   , C =   1 0 0 0 1 0 0 0 0   , D =   0 0 0 0 0 0 0 0 1   . Note that the matrices A, B , C, D are not unique since the action (2.19) of the an ti- holomorphic inv olution on the basic cycles allows for p erm utations of A j -cycles for in- stance. These matrices can b e computed as follo ws. Since the T rott curve is an M-curv e, one has H = 0 . Moreov er, the matrix Im ( P ˜ B ) b eing inv ertible here, by (4.7) one gets: (4.33) B = − A Im  P ˜ A   Im  P ˜ B  − 1 . With (4.14) and (4.15) it follows that (4.34) A t  D + C Im  P ˜ A   Im  P ˜ B  − 1  = I 3 . The computed matrix Im  P ˜ A   Im  P ˜ B  − 1 b eing (within numerical precision) equal to Im  P ˜ A   Im  P ˜ B  − 1 =   − 1 0 0 0 − 1 0 0 0 0   , and with C, D ∈ M 3 ( Z ) , w e get from (4.34) that det A = 1 . Since A ∈ M 3 ( Z ) , the condition det A = 1 implies A ∈ Gl 3 ( Z ) . F or any A ∈ Gl 3 ( Z ) , one can see from (4.33), (4.12) and (4.13) that B , C , D ∈ M 3 ( Z ) , and therefore that the matrices A, B , C, D give a solution of (4.6)-(4.9). The choice A = I 3 leads to the ab o v e matrices. The T rott curve has real fib ers and can thus b e used to construct solutions to the 3-NLS equation via the pro jection map f : ( x, y ) 7→ x , which is a real meromorphic function of degree 4 on the curve. W e consider the p oin ts on the curve stable with resp ect to τ and pro jecting to the p oint with x = 0 . 1 in the x -sphere, and choose d = 0 . The corresp onding solution to the 3-NLS equation can b e seen in Fig. 15. A solution to the DS1 + equation on this curve can be constructed for points a and b stable with resp ect to the inv olution τ . The solution for a = ( − 0 . 2) (1) , b = (0 . 2) (2) and the choice d = 0 can b e seen in Fig. 16. Note that in accordance with Remark 2.1, one w ould obtain a solution of DS1 − for the c hoice a = ( − 0 . 2) (1) and b = (0 . 2) (1) . Similarly , a solution to the DS2 + equation can b e obtained for p oints a and b sub ject to τ a = b . F or a = (0 . 1 + i) (1) and b = (0 . 1 − i) (1) w e get Fig. 17. 4.3. Dividing curves without real branch p oin t. W e consider the curv e given by the equation (4.35) 30 x 4 − 61 x 3 y + 41 y 2 x 2 − 43 x 2 − 11 y 3 x + 42 xy + y 4 − 11 y 2 + 9 = 0 whic h was studied in [10] and [30]. It is a gen us 3 curve, dividing with resp ect to the anti- holomorphic inv olution τ , without real branch p oin t. This curve admits tw o real ov als. In this case the matrix H has the form H =   0 1 0 1 0 0 0 0 0   . 25 ï 10 ï 5 0 5 10 ï 10 ï 5 0 5 10 0 0.2 0.4 x t | s 1 | 2 ï 10 ï 5 0 5 10 ï 10 ï 5 0 5 10 0 0.5 x t | s 2 | 2 ï 10 ï 5 0 5 10 ï 10 ï 5 0 5 10 0 0.5 1 x t | s 3 | 2 Figure 15. Solution (2.22) to the 3-NLS s e quation on the T r ott curve for the p oints with x = 0 . 1 on the x -spher e. The she ets ar e identifie d at the p oints pr oje cting to x = − 1 . 0129 , (0 . 9582i , − 0 . 9582i , 0 . 1146i , − 0 . 1146i) . The ve ctor of signs e quals s = (1 , − 1 , − 1) fr om top to b ottom. Figure 16. Solution to the DS1 + e quation on the T r ott curve for the p oints a = ( − 0 . 2) (1) and b = (0 . 2) (2) at t = 0 . The p eriod matrices computed by the co de read P ˜ A =   − 0 . 2721 − 0 . 0977i − 0 . 3193 + 0 . 1914i − 1 . 0668 + 0 . 4293i 0 . 2721 + 0 . 0977i − 0 . 3193 − 0 . 3341i − 1 . 0668 − 0 . 4316i 0 . 2721 − 0 . 0977i 0 . 4676 − 0 . 3341i 0 . 7992 − 0 . 4316i   , 26 C. KALLA AND C. KLEIN Figure 17. Solution to the DS2 + e quation on the T r ott curve for the p oints a = (0 . 1 + i) (1) and b = (0 . 1 − i) (1) at t = 0 . P ˜ B =   − 0 . 2721 − 0 . 2932i − 0 . 3193 + 0 . 3341i − 1 . 0668 + 0 . 4316i 0 . 2721 + 0 . 2932i − 0 . 3193 − 0 . 7169i − 1 . 0668 − 1 . 2903i 0 . 2721 − 0 . 0977i 0 . 4676 + 0 . 1914i 0 . 7992 + 0 . 4293i   . After some calculations, one finds that the following matrices A, B , C , D provide a solution of (4.6)-(4.9): A =   − 1 2 − 1 2 − 1 0 0 2 − 1   , B =   1 0 1 0 1 0 1 0 0   , C =   1 − 1 − 1 − 1 1 − 1 0 0 1   , D =   0 1 1 1 0 1 0 0 − 1   . F rom (4.17) and (4.18) one gets for the characteristic: ˜ δ = 1 2 [ 0 0 1 1 1 0 ] t . The curv e (4.35) has real fib ers and can thus b e used to construct solutions to the fo cusing 3-NLS equation. W e consider the p oin ts on the curve with x = 2 . 5 and stable with resp ect to τ , and we choose d = 0 . The corresp onding solution to the fo cusing 3-NLS equation can b e seen in Fig. 18. A solution to the DS1 − equation can b e constructed by c ho osing the p oints a = ( − 4) (1) and b = ( − 3) (2) see Fig. 19. 4.4. F ermat curve. The F ermat curv es (4.36) y n + x n + 1 = 0 , n > 2 , n even , are real curves without real ov al with resp ect to τ . W e consider here the curv e with n = 4 that has gen us 3. The matrix H has the form H =   0 1 0 1 0 0 0 0 0   , 27 ï 20 ï 10 0 10 20 ï 20 ï 10 0 10 20 0.1 0.15 0.2 x t | s 1 | 2 ï 20 ï 10 0 10 20 ï 20 ï 10 0 10 20 0 2 4 x 10 ï 3 x t | s 2 | 2 ï 20 ï 10 0 10 20 ï 20 ï 10 0 10 20 0 0.01 0.02 x t | s 3 | 2 Figure 18. Solution to the 3-NLS s e quation on the dividing curve (4.35) of genus 3 for the p oints with x = 2 . 5 on the x -spher e. The she ets ar e identifie d at the fib er over − 2 . 1404 + 0 . 4404i , ( − 12 . 2492 + 2 . 0113i , − 5 . 1634 + 1 . 3519i , − 4 . 5915 + 0 . 9380i , − 1 . 5405 + 0 . 5429i) . The ve ctor of signs is s = (1 , 1 , 1) . Figure 19. Solution to the DS1 − e quation on the dividing curve (4.35) of genus 3 for the p oints a = ( − 4) (1) and b = ( − 3) (2) at t = 0 . and w e find P ˜ A =   0 . 9270 − 0 . 9270i − 0 . 9270i 0 0 − 1 . 8541i 0 . 9270i − 0 . 9270 − 0 . 9270i   , 28 C. KALLA AND C. KLEIN P ˜ B =   0 . 9270 + 0 . 9270i 0 . 9270 − 0 . 9270i 0 0 − 0 . 9270 + 0 . 9270i 0 . 9270 − 0 . 9270i − 0 . 9270 − 0 . 9270i − 0 . 9270i   . The follo wing matrices A, B , C , D pro vide a solution of (4.6)-(4.9): A =   0 1 1 1 0 0 0 0 1   , B =   − 1 − 2 − 1 0 0 − 1 − 1 − 1 0   , C =   0 1 0 0 0 1 1 − 1 0   , D =   0 0 − 1 0 − 1 0 0 0 1   , whic h leads to the characteristic: ˜ δ = 1 2 [ 0 0 1 0 1 0 ] t . T o construct a solution of the DS2 − equation on the F ermat curv e, we choose the p oin ts a = ( − 1 . 5 + i) (1) and b = ( − 1 . 5 − i) (3) . The resulting solution for the choice d = 0 can b e seen in Fig. 20. Figure 20. Solution to the DS2 − e quation on the F ermat curve (4.36) of genus 3 for the p oints a = ( − 1 . 5 + i) (1) and b = ( − 1 . 5 − i) (3) at t = 0 . 5. Conclusion In this paper w e ha ve presented the state of the art of the numerical ev aluation of solutions to integrable equations in terms of multi-dimensional theta functions asso ciated to real Riemann surfaces by using an approach via real algebraic curves. It w as sho wn that real hyperelliptic curves parametrized b y the branch p oin ts can b e treated with machine precision for a wide range of the parameters. Ev en almost degenerate situations where the branc h p oints coincide pairwise can b e handled as long as at least one cut sta ys finite. This approac h to real hyperelliptic curves [14, 15] is b eing generalized to arbitrary hyperelliptic curv es. As discussed in [16], the main difficulty for general algebraic curv es is the correct numer- ical identificati on of the branc h p oin ts. The case of degenerations for given branch p oin ts 29 has not yet b een studied numerically , but is planned for the future. In what concerns the solutions (2.22) to n -NLS s and similar solutions to the DS and the Kadomtsev-P etviashvili equations, the main problem in the context of real Riemann surfaces is to find the symplec- tic transformation leading to the homology basis introduced in [30], for which the solutions of the studied equations, with regularit y conditions, can b e conv enien tly formulated. This problem has b een reduced to find a single g × g -matrix for given p eriods and real ov als, the latter enco ded b y the matrix H . F or M-curv es, where the matrix H v anishes, a general for- m ula for the c haracteristic (4.17)-(4.19) could b e given. 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Shabat, Exact the ory of two-dimensional self-fo cusing and one-dimensional self- mo dulation of waves in nonline ar me dia , Sov. Phys. JETP 34 , 62–69 (1972). [32] V. Zakharo v and E. Sch ulman, T o the inte gr ability of the system of two c ouple d nonline ar Schrö dinger e quations , Physica D 4 , 270–274 (1982). Institut de Ma théma tiques de Bour gogne, Université de Bourgogne, 9 a venue Alain Sa v ar y, 21078 Dijon Cedex, France E-mail addr ess : Caroline.Kalla@u-bourgogne.fr Institut de Ma théma tiques de Bour gogne, Université de Bourgogne, 9 a venue Alain Sa v ar y, 21078 Dijon Cedex, France E-mail addr ess : Christian.Klein@u-bourgogne.fr