The stationary equations of a coupled nonlinear Schrödinger system

Physica D 126 (1999) 275–289

The stationary equations of a coupled nonlinear Schr¨odinger system Otis C. Wright1 Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia Received 31 March 1997; received in revised form 26 May 1998; accepted 28 September 1998 Communicated by C.K.R.T. Jones

Abstract The integrable coupled nonlinear Schr¨odinger (CNLS) equations under periodic boundary conditions are known to possess linearized instabilities in both the focussing and defocussing cases [M.G. Forest, D.W. McLaughlin, D. Muraki, O.C. Wright, Non-focussing instabilities in coupled, Integrable nonlinear Schr¨odinger PDEs, in preparation; D.J. Muraki, O.C. Wright, D.W. McLaughlin, Birefringent optical fibers: Modulational instability in a near-integrable system, Nonlinear Processes in Physics: Proceedings of III Postdam-V Kiev Workshop, 1991, pp. 242–245; O.C. Wright, Modulational stability in a defocussing coupled nonlinear Schr¨odinger system, Physica D 82 (1995) 1–10], whereas the scalar NLS equation is linearly unstable only in the focussing case [M.G. Forest, J.E. Lee, Geometry and modulation theory for the periodic Schr¨odinger equation, in: Dafermas et al. (Eds.), Oscillation Theory, Computation, and Methods of Compensated Compactness, I.M.A. Math. Appl. 2 (1986) 35–70]. These instabilities indicate the presence of crossed homoclinic orbits similar to those in the phase plane of the unforced Duffing oscillator [Y. Li, D.W. McLaughlin, Morse and Melnikov functions for NLS pde’s, Commun. Math. Phys. 162 (1994) 175–214; D.W. McLaughlin, E.A. Overman, Whiskered tori for integrable Pde’s: Chaotic behaviour in near integrable Pde’s, in: Keller et al. (Eds.), Surveys in Applied Mathematics, vol. 1, Chapter 2, Plenum Press, New York, 1995]. The homoclinic orbits and the near homoclinic tori that are connected to the unstable wave trains of the NLS and the CNLS reside in the finite-dimensional phase space of certain stationary equations [S.P. Novikov, Funct. Anal. Prilozen, 8 (3) (1974) 54–66] of the infinite hierarchy of integrable commuting flows. The correct stationary equations must be matched to the unstable torus through the analytic structure of the spectral curves [O.C. Wright, Near homoclinic orbits of the focussing nonlinear Schr¨odinger equation, preprint]. Thus, in this paper, the stationary equations of the CNLS are derived and the analytic structure of the trigonal spectral curve is examined, providing a basis for further study of the near homoclinic orbits c 1999 Elsevier Science B.V. All rights reserved. of the CNLS system. Keywords: Lax pair; Riemann surface; Integrable system; Quasiperiodic solutions; Lie algebra

1

Present address: Department of Mathematics, CB no. 3250, Phillips Hall, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA c 0167-2789/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 9 8 ) 0 0 2 7 1 - 1

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1. Introduction 1.1. The integrable system The scalar nonlinear Schr¨odinger (NLS) equation, with σ = ±1 for focussing (+) and defocussing (−), 1 σ ipt + pxx + p|p|2 = 0, 2 4

(1)

is a canonical equation for the propagation of waves in a weakly nonlinear and dispersive medium; light propagation in optical fibers being an important example [15]. A near integrable coupled version of the NLS arises in the propagation of light in a birefringent fiber [29]. The coupled system exhibits linearized instabilities in both its focussing and defocussing forms [23], even in the integrable case [33], unlike the scalar equation which is linearly unstable only in the focussing case [11]. The linearized instabilities of the scalar focussing NLS equation are known to be associated with crossed homoclinic orbits in the phase space similar to the saddle point in the phase plane of the unforced Duffing oscillator. Moreover, chaos is associated with the breaking of such homoclinic saddles when the equation is perturbed [6,16,22]. The homoclinic orbits of the NLS under periodic boundary conditions are surrounded by finite-dimensional tori in the phase space. The finite-dimensional tori near homoclinic orbits that saturate a given instability are solutions of finite-dimensional Hamiltonian systems that are given by certain stationary equations or Novikov equations [25,30] of the infinite hierarchy of commuting flows. The stationary equation that governs the saturation of a given instability must be chosen so that the branching structure of the desingularized invariant spectral curve matches that of the unstable torus [34]. Thus, to study the underlying finite-dimensional structure of the homoclinic orbits of the CNLS, the stationary equations must be derived and the analytic structure of the associated invariant spectral curve must be studied. This is non-trivial in the case of the CNLS because the spectral curve is trigonal (since the underlying Lie algebra is sl(3)), and not hyperelliptic as in the case of the scalar NLS (where the underlying Lie algebra is sl(2)). The integrable version of the coupled nonlinear Schr¨odinger equations is   σ  σ  iqt + qxx + q |p|2 + |q|2 = 0, (2) ipt + pxx + p |p|2 + |q|2 = 0, 2 2 which is itself a special integrable case of a family of coupled equations that represent the propagation of orthogonal components p and q of an electric field in a glass fiber [23,29]. The defocussing (σ = −1) and focussing (σ = +1) cases are distinguished by σ . The CNLS equations (2) are a reduction of the following generalized integrable CNLS system that is analogous to the AKNS [2,35] equations for integrating the scalar NLS equation: 1 ipt + pxx − p(pr + qs) = 0, 2 1 −irt + rxx − r(pr + qs) = 0, 2

1 iqt + qxx − q(pr + qs) = 0, 2 1 −ist + sxx − s(pr + qs) = 0. 2

(3)

The generalized CNLS system (3) reduces to the CNLS equations (2) when r = −σp∗ and s = −σ q ∗ . The system (3) is an integrable system in the sense that it is equivalent to the compatibility condition of the following Lax pair [1,17,20]: Qx = [L1 , Q],

Qt = [L2 , Q],

(4)

where Q, L1 , L2 , are 3 × 3 matrices that depend on a formal parameter E. The matrices L1 and L2 are given by L1 = EA0 + A1 ,

L2 = E 2 A0 + EA1 + A2 ,

O.C. Wright / Physica D 126 (1999) 275–289

with



−2 i A0 =  0 3 0

0 1 0

 0 0, 1



0 1 A1 =  r 2 s

p 0 0

 q 0 , 0

277



pr + qs i A2 = −  2rx 4 2sx

−2px −pr −ps

 −2qx −qr  −qs

(5)

and [, ] is the usual commutator. This is the “squared eigenfunction” formulation of the Lax pair in which Q corresponds to squared eigenfunctions of a vector function Lax pair in which E is the eigenvalue. The key to studying “N-phase” or quasiperiodic solutions of this integrable system is the study of solutions Q e in the graded loop algebra sl(3) of 3 × 3 traceless matrices graded by the parameter E. A solution Q of the Lax pair (4) is equivalent to finding a solution in p, q, r, s of the CNLS system (3). Quasiperiodic solutions correspond to Q having a finite expansion in E within the graded loop algebra, provided that certain “reality constraints” are satisfied by the coefficients in the expansion and the initial conditions. Within this class of quasiperiodic solutions are the near homoclinic tori that saturate the instabilities of the CNLS equations. The equations that define the quasiperiodic solutions are the stationary equations of the infinite hierarchy of commuting flows that arises from the integrable structure. 1.2. Hierarchy of commuting time flows A hierarchy of commuting time flows exists for the CNLS system (3) when Q solves the equation QTN = [LN , Q], where LN = B0 E N + B1 E N−1 + · · · + BN for N ≥ 2 and T1 = x with L1 = A0 E + A1 and T2 = t if Bi = Ai for i = 0, 1, 2. In order for the compatibility of the x-flow and the TN -flow for Q to be equivalent to a flow of the potentials p, q, r, s, it must be that LN satisfies, for every order in E greater than 0: ∂x LN = [L1 , LN ] and in this way the Bj are determined for j = 0, 1, 2, . . . , N − 1, N via recursion relations on the entries of the Bj . The zeroth order term in the above condition produces the higher time flow equation for p, q, r, s: ∂TN L1 = {∂x LN − [L1 , LN ]}0 , where { }0 indicates that the zeroth order term in the E expansion is taken. Thus, the hierarchy of higher time flows is generated from a solution of the x-flow of the Lax pair (4) in the form of a Laurent series in E: (6) Q = Q0 + Q1 E −1 + Q2 E −2 + Q3 E −3 + · · ·  N so that LN = E Q + where the notation { }+ indicates that all the terms in the series with non-negative E exponent are taken. Notice the following: – The time flows in the hierarchy will be partial differential equations if and only if the entries in the Qi (and thus the Li ) are all differential polynomials in p, q, r, s. Otherwise integro-differential equations will arise. – Time independent solutions of higher time flow equations are equivalent to solutions Q = LN of the x-flow equation of the Lax pair which truncate at finite order in the Laurent series (6). Such solutions also satisfy an additional set of constraints, namely the stationary equations of the higher time flow. These stationary equations or multiphase equations or N-phase equations were first considered in the case of the KdV equation by Novikov

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[25] and also by Lax [18]. These equations possess multiphase or quasiperiodic solutions. The characteristic polynomial of the stationary Q is an integral of the flow and its desingularization is the desingularization of the spectral curve of spatially periodic solutions if they are considered as potentials in the vector function formulation of the Lax pair. The classification of the stationary equations and the study of the monodromy of the spectral curve are the aims of this paper. – Adams et al. [4,5] (see also Previato’s work on the NLS [26] and, with two of the previous authors on the CNLS [3]) have shown how to integrate some of the stationary equations for the CNLS system, in terms of theta functions, by mapping the equations to ellipsoidal coordinates used by Moser [24]. However, the initial value problem, which will be important in the study of the saturation of linear instabilities of the system [13,33], was not analyzed. Crucial to the solution of the initial value problem will be the monodromy of the spectral curve, since canonical cycles must be chosen to satisfy “reality” conditions that guarantee a quasiperiodic solution. Issues of monodromy have been discussed in the case of the Boussinesq equation, which also possesses a Lax operator of order 3, by McKean [21] and Previato [27,28]. However, the connection between the monodromy of the curve, the solution of the initial value problem for quasiperiodic solutions, and the classification of the stationary equations of the CNLS system remains only partially understood. – In this paper, the stationary equations of the CNLS system are derived and those which are purely differential (no integral terms) are identified. Reduction theorems which allow the construction of stationary equations with a given number of branch points in the spectral curve of invariants are presented, with a view toward future work on constructing the stationary equations that govern near homoclinic orbits of a specific linearly unstable wave train. Moreover, an unusual technical feature of the spectral curves as Riemann surfaces (“twisted tori”) and the solution space of the stationary equations is elucidated, viz. there are distinct solutions with topologically equivalent spectral curves (“tori”) but possessing different monodromy (“twist”).

2. The loop algebra 2.1. Definitions The basic tool for studying stationary equations of the infinite hierarchy of the CNLS system is a loop algebra of formal series of 3 × 3 matrices graded by the spectral parameter E, following the techniques of Flaschka et al. [10], in the case of the 2 × 2 AKNS system (i.e. the scalar NLS case.) P −n where Q ∈ sl(3, B), e be the Lie algebra of formal series of the type Q = ∞ Definition 1. Let G = sl(3) n n=0 Qn E for all n, and Q0 ∈ sl(3, C). Here B is the polynomial ring over the complex field C generated by the symbols p, q, r, s and equipped with the derivation ∂. The Lie algebra commutator is the usual matrix commutator extended naturally to the formal series. – Notice that if Q0 ∈ GL(3, C) then G will also have the structure of a group under formal multiplication of the infinite series. In particular, if Q ∈ G then Q−1 ∈ G also, each term in Q−1 being a finite sum of finite products of Q−1 0 and the Qi . – The above definition can be extended by equipping G with an anti-derivation ∂ −1 . So that integration constants can be well defined, we assume that ∂ −1 0 = 0. The extended ring is denoted by B+ and the extended Lie algebra by G + . – Elements of B + and G + are unique up to equivalence, namely u, v ∈ B+ are equivalent if and only if ∂u = ∂v. – If Q ∈ G + then detQ =

∞ X cn E −n , n=0

O.C. Wright / Physica D 126 (1999) 275–289

279

where each cn ∈ B + comes from finite sums and products of entries from finitely many terms in Q. Moreover, the usual product rule for determinants holds for elements of G + . The problem is to classify those Q ∈ G + which solve the equation ∂Q = [L1 , Q], and then to examine the subset of such solutions which truncate to finite order in E. Such “polynomial ansatz” Q correspond, equivalently, to stationary equations of the hierarchy or time independent solutions of different time flows in the hierarchy. – Previous experience with 2 × 2 AKNS systems [10] suggests that Q ∈ G, viz. all the entries of Q are necessarily differential polynomials of p, q, r, s. However, this is not true in the 3 × 3 case. In general Q ∈ G + . With the correct choice of Q0 , namely Q0 = A0 , it will be shown that Q ∈ G. 2.2. Generating solutions A series of lemmas is now given that will enable us to systematically generate polynomial solutions of the Lax pair. In the following sections the stationary equations will be derived and then the subset of purely differential stationary equations will be classified. Finally the spectral curves of the stationary equations will be examined in terms of genus and monodromy. Lemma 1. Q ∈ G + and ∂Q = [L1 , Q] only if ∂Q0 = 0 and [A0 , Q0 ] = 0. The proof of this lemma is a simple calculation; moreover, it shows that the entries of Q0 are constants. The restriction of zero trace Q0 is no loss of generality since the identity matrix commutes with all matrices. Lemma 2. Q ∈ G + and ∂Q = [L1 , Q] if and only if the entries of Qn+1 are given recursively in terms of the entries of Qn , denoted by qijn for n ≥ 0, by the formulae: (off-block entries) i i n n+1 n n n = i ∂q12 − p(q22 − q11 ) − qq32 , q12 2 2 i i n n+1 n n n = −i ∂q21 − r(q22 − q11 ) − sq23 q21 2 2 (block entries)

1 −1 n+1 n+1 + rq12 ) + iα, ∂ (−pq21 2 1 1 n+1 n+1 n+1 n+1 n+1 = ∂ −1 (−qq31 + sq13 ) + iδ, q23 = ∂ −1 (−qq21 + rq13 ) + iβ, 2 2 1 n+1 n+1 = ∂ −1 (−pq31 + sq12 ) + iγ . 2

n+1 n+1 n+1 = −q22 − q33 , q11 n+1 q33 n+1 q32

i i n n+1 n n n q13 = i ∂q13 − q(q33 − q11 ) − pq23 , 2 2 i i n n+1 n n n q31 = −i ∂q31 − s(q33 − q11 ) − rq32 2 2

n+1 q22 =

(7)

The proof is an immediate calculation using the definition of L1 = A0 E + A1 . The constants α, β, γ , δ are constants of integration. They are well defined since ∂ −1 0 = 0 by assumption. There is no loss of generality in assuming that Qn ∈ sl(3, B + ) since Q is arbitrary up to addition of constant multiples of the identity matrix. – The recursion relation for the “block” entries involves ∂ −1 so, in general, Q ∈ G + . Lemma 3. There exists a unique (up to equivalence class) invertible V ∈ G + of the form V =I+

∞ X Vn E −n , n=1

which solves ∂V = [A0 , V ] E + A1 V

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and satisfies the condition that V = I when p(∞) = q (∞) = r (∞) = s (∞) = 0. The matrix I is the 3 × 3 identity matrix, the forms of A0 and A1 are given in Eq. (5). p(∞) , q (∞) , r (∞) , s (∞) are the prolongations of the symbols p, q, r, s, with respect to ∂, i.e. the symbol and all its higher derivatives. The proof of the lemma is obtained by writing down recursion relations for the elements of Vn+1 in terms of Vn for n ≥ 0 with V0 = I, in a similar fashion to the previous recursion relations for Q. In the lemma, that particular V with constants of integration equal to 0 was chosen.The purpose of defining this V ∈ G + is in order to use V to characterize solutions Q according to the following theorem: Theorem 1. Q ∈ G + is a solution of ∂Q = [L1 , Q] if and only if Q = V XV −1 where X ∈ G satisfies [A0 , X] = 0 and ∂X = 0. The proof of the sufficiency of the condition is a direct calculation of ∂Q making use of the equation satisfied by V , the commutation of A0 and X, and the fact that both A0 and X are in the kernel of the derivation ∂. The necessity of the condition follows by repeated application of the linearity of the equation satisfied by Q and Proposition 1 and the sufficiency part of the theorem. In this way Q can be shown to possess the required form. This theorem leads immediately to the important corollary: Corollary 1. If Q ∈ G + is a solution of ∂Q = [L1 , Q] then ∂ det(zI − Q) = 0. Thus, the characteristic polynomial (possibly of infinite degree) of Q is an integral of the x-flow. The element X in the above theorem has the form X = X0 + X1 E −1 + X2 E −2 + X3 E −3 + · · · , where each Xj is a 3 × 3 matrix of “block” type, i.e. it commutes with A0 , and its entries are complex constants which are precisely the constants of integration of recursion relations (7). Thus, we may define the standard solution of the CNLS system, which can be used to write down all the stationary equations. Definition 2. The standard solution of the CNLS system is defined in terms of four complex numbers α, β, γ , δ, to be   −iα − iβ 0 0 ˆ 0 iα iβ  V −1 . Q(α, β, γ , δ) = V  0 iγ iδ Thus



−iα − iβ ˆ 0 Q(α, β, γ , δ) =  0

0 iα iγ

 0 ˆ 2 E −2 + · · · , ˆ 1 E −1 + Q iβ  + Q iδ

ˆ n = (qˆ n (α, β, γ , δ)) is a 3 × 3 matrix whose entries are in B+ . These entries are unique up to where each Q ij equivalence in B + and depend only on the complex numbers α, β, γ , δ since ∂ −1 0 = 0. 3. The stationary equations Theorem 2. (The stationary equations)Q ∈ G + is a solution of ∂Q = [L1 , Q] and QE N is a polynomial of degree N in the expansion parameter E if and only if there exist complex constants αi , βi , γi , δi for i = 0, 1, 2, . . . , N such that

O.C. Wright / Physica D 126 (1999) 275–289 N X i+1 qˆ12 (αi , βi , γi , δi ) = 0,

N X i+1 qˆ13 (αi , βi , γi , δi ) = 0,

i=0

i=0

N X

N X i+1 qˆ31 (αi , βi , γi , δi ) = 0,

i=0

i=0

i+1 qˆ21 (αi , βi , γi , δi ) = 0,

281

(8)

ˆ where qˆjnk are elements in the matrices of the standard solution Q. The equations in the above theorem are precisely the analog in the CNLS case of the multiphase or stationary equations of Novikov [25] for the KdV equation. Notice that the equations are well defined only up to the equivalence relation imposed by ∂ −1 . However, the αi , βi , γi , δi , are the only free parameters in the equations since ∂ −1 0 = 0 by assumption. The proof of the theorem is an immediate consequence of the construction of the standard solution and the linearity of the Lax equations. In particular, Q = V XV −1 , where X = XN + XN−1 E −1 + · · · + X0 E −N + X−1 E −N −1 + · · · and



−i(αj + δj ) 0 Xj =  0

0 iαj iγj

 0 iβj  , iδj

with complex constant entries. Then (N ) X N i ˆ i , βi , γi , δi )E , Q(α QE = i=0

(9)

(10)

+

where { }+ indicates that only the non-negative powers of E are extracted from the infinite series. The off-block entries of the order E −1 term are the stationary equations. The x-flow Lax equation is satisfied to every order by construction of the standard solution, except at the lowest order, E −1 , at which point an additional constraint is imposed, namely the stationary equations of the theorem. Notice in the above expansion of Q that the complex constants αi , βi , γi , δi for i = 0, 1, . . . , N are free parameters which determine the stationary equations. However, for i = −1, −2, −3, . . . these constants are actually interdependent integrals of motion, with values determined by the initial conditions of the stationary equations. Corollary 2 (Focussing and defocussing reality constraints on the coefficients). The stationary equations are consistent when r = −σp ∗ and s = −σ q ∗ if and only if αi and δi are real and βi = γi∗ for i = 0, 1, 2, . . . , N. Corollary 3 (Focussing and defocussing CNLS subalgebras). If r = −σp∗ and s = −σ q ∗ and α and δ are real ˆ ˆ and β = γ ∗ then Q(α, β, γ , δ) ∈ su(3) e when σ = 1 (focussing) and Q(α, β, γ , δ) ∈ su(1, e 2) when σ = −1 (defocussing). Remember that su(3) e and su(1, e 2) are special unitary loop algebras of 3 × 3 matrices in which each coefficient matrix is in sl(3) and also a fixed point of the automorphism −J Qt J −1 = Q∗ ,

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where



1 J = 0 0 and



−1 J = 0 0

 0 0 1

0 1 0

0 1 0

 0 0 1

for su(3) e

for su(1, e 2).

Corollary 4 (Genus of the stationary equation spectral curve). Suppose that Q ∈ G + is such that QE N is a polynomial solution of degree N of the Lax equation: ∂Q = [L1 , Q] and the complex numbers αi , βi , γi , δi for i = 0, 1, 2, . . . , N are the parameters which appear in the stationary equations. Then, the characteristic polynomial of QE N , if it is irreducible, is a three-sheeted (trigonal) compact connected covering of the Riemann E sphere which can possess a maximum of 6N square root type branch points, 2 + 5α δ + 2δ 2 − β γ 6= 0. viz. a maximum genus of 3N − 2, if and only if (αN − δN )2 + 4βN γN 6= 0 and 2αN N N N N N The proof of the corollary follows from that of the theorem. Up to equivalence, Q can be expanded as Q = V XV −1 , where   0 0 −i(αN + δN ) 0 iαN iβN  + O(E −1 ). X= 0 iγN iδN Then det(zI − QE N ) = det(zI − XE N ) = z3 + A(E)z + B(E) = 0 and 2 2 + δN + αN δN + βN γN )E 2N + · · · + a1 E + a0 , A(E) = (αN

B(E) = i(αN + δN )(βN γN − αN δN )E 3N + · · · + b1 E + b0 . The discriminant of the cubic polynomial is 2 2 + 5αN δN + 2δN − βN γN )2 E 6N + O(E 6N −1 ). 4A3 + 27B 2 = ((αN − δN )2 + 4βN γN )(2αN

Thus, in general, the Riemann surface of the cubic polynomial has 6N distinct square root branch points over finite locations and is regular over the point at infinity. The Riemann–Hurwitz relation gives the genus as 3N − 2 for the trigonal covering of the Riemann sphere. 4. Reductions of the stationary equations Of primary interest are the stationary equations of the focussing and defocussing CNLS equations. In this case r = −σp ∗ and s = −σ q ∗ and so the coefficients in the stationary equations must satisfy the reality constraints: αi and δi are real and βi = γi∗ for all i = 0, ..., N, in the expansion of X in Eq. (9). In addition to these reality conditions, it is assumed in this section that the characteristic polynomial of QE N in Eq. (10) is irreducible. It is also assumed that the leading coefficients αN , βN , γN , δN are not all zero.

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283

Table 1 N

0≤K ≤N −1

2N − 2 ≤ G(N, K) ≤ 3N − 3

1 2 3 4

0 0,1 0,1,2 0,1,2,3

0 2,3 4,5,6 6,7,8,9

Max. G = 3N − 2 1 4 7 10

The maximum genus of stationary equations of order N in the X expansion was given in the previous section as 3N − 2. However, various reductions may be imposed by assuming certain symmetries in the coefficients, so as to reduce the genus of the spectral curve. The possible reductions fall into two categories, characterized by the following lemma and two reduction theorems: 2 +5α δ +2δ 2 −β γ = 0 Lemma 4. If αN and δN are real and βN = γN∗ then (αN −δN )2 +4βN γN = 0 and 2αN N N N N N if and only if αN = βN = γN = δN = 0.

The point of this lemma is that there are exactly two mutually exclusive ways to reduce the genus of the spectral curve of the focussing and defocussing CNLS stationary equations from the generic maximum value of 3N − 2. Theorem 3 (First reduction theorem). If (αN − δN )2 + 4|βN |2 = 0 for N ≥ 1 then assume that there is a largest integer K such that 0 ≤ K ≤ N − 1 and (αK − δK )2 + 4|βK |2 6= 0. Under this reduction, the maximum genus of the spectral curve of the stationary equations is G(N, K) = 2N + K − 2. Under the assumptions of the theorem, Xi = 3αi A0 for i = K + 1, K + 2, . . . , N and either αK 6= δK or βK 6= 0 so that the discriminant of the characteristic polynomial is 4A3 + 27B 2 = ((αK − δK )2 + 4|βK |2 )E 4N+2K + O(E 4N+2K−1 ). In this case, 2N − 2K branch points have migrated to the point over infinity and “pinched” together and so do not appear in the desingularization of the Riemann surface. In general, 4N + 2K square root branch points remain over finite points, giving the genus, by the Riemann–Hurwitz relation, as 2N + K − 2. 2 + 5α δ + 2δ 2 − |β |2 = 0 for N ≥ 1 then assume there exists Theorem 4 (Second reduction theorem). If 2αN N N N N a largest integer K such that 0 ≤ K ≤ N − 1 and     αN βN αK βK 6= cK γK δK γN δN

for any real constant cK . Moreover, assume that (4αN + 5δN )αK + (5αN + 4δN )δK − βN γK − βK γN 6= 0. Then, the maximum genus of the spectral curve of the stationary equations under this reduction is G(N, K) = 2N + K − 2. Under the assumptions of the second reduction theorem the discriminant of the characteristic polynomial is 4A3 + 27B 2 = ((αN − δN )2 + 4|βN |2 )((4αN + 5δN )αK + (5αN + 4δN )δK − βN γK − βK γN )2 E 4N+2K +O(E 4N+2K−1 ). A table of possible spectral curves classified by the order of the stationary equations and the genus of the spectral curve may now be constructed using the two reduction theorems. The first and second reduction theorems produce the same range of genuses, even though the constraints on the equations are quite distinct. The reduction table for N ≤ 4 is Table 1.

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By using such reductions, stationary equations with a spectral curve of a required genus can be constructed. Such constructions are important in finding saturation states of linear instabilities in the CNLS system [13], as carried out for the scalar NLS equation [31,32]. 5. The differential stationary equations 5.1. The differential standard solution of the Lax pair We are now able to characterize precisely which choices of the constants of integration α, β, γ , δ in the recursion relations (7) will produce a solution Q which has only differential polynomial entries. This is equivalent to choosing the complex constants αi , βi , γi , δi in the stationary equations of the previous theorem so that the equations are purely differential equations without any integrals. By the linearity of the Lax pair equations for Q, we need only to give the necessary and sufficient conditions on Q0 = X0 that will produce Q ∈ G. The necessary condition is a simple calculation using the recursion relations of Eq. (7). Proposition 1. If Q ∈ G is a solution of ∂Q = [L1 , Q] with p and q independent and r and s independent then Q0 must be proportional to the matrix A0 . It is far from obvious that the choice Q0 = A0 (and all other constants of integration set to 0) is sufficient to ensure that the recursion relations will always produce pure differential polynomials of p, q, r, s, but the following theorem shows that it is in fact true. Theorem 5. Let QD ∈ G + be the unique solution of ∂Q = [L1 , Q] defined by Q0 = A0 and the remaining Qn are given by the recursion relations (7) in which the constants of integration are all set to 0. Then QD = V A0 V −1 ∈ G. The fact that the QD is equal to V A0 V −1 follows from the uniqueness of the recursion relations once the constants of integrations are all set to 0, and the fact that V was defined so that all the constants of integration were 0. The important part of the theorem is that the solution so defined is an element of G, viz. all the entries of QD = V A0 V −1 are differential polynomials of p, q, r, s. The proof is a generalization of an idea in [10] for the 2 × 2 AKNS case. However, unlike the 2 × 2 case, the relation det(λI − V A0 V −1 ) = det(λI − A0 ) does not lead to explicit expressions for the entries of V A0 needed to show that

(11) V −1

to all orders. Instead, explicit calculation is first

V A0 V −1 = A0 + A1 E −1 + A2 E −2 + · · · ,

(12)

where A0 , A1 , A2 ∈ sl (3, B) are as previously defined by Eqs. (5). With this information, an induction argument is now needed to show that V A0 V −1 ∈ G to all orders in E. In particular, the i–j th entry of V A0 V −1 can be written as qij = qij0 + qij1 E −1 + qij2 E −2 + · · · and we know from Eq. (11) that 2i 1 ≡ λ3 − σ1 λ2 + σ2 λ − σ3 , λ3 + λ − 3 27 where σ1 = q11 + q22 + q33 , σ2 = q11 q22 + q11 q33 + q22 q33 − q21 q12 − q31 q13 − q23 q32 , σ3 = q11 q22 q33 + q12 q13 q23 + q13 q21 q32 − q11 q32 q23 − q12 q21 q33 − q13 q31 q22 .

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Now assume that for 0 ≤ k ≤ n and n ≥ 2 we know that qijk ∈ B. This is certainly true for n = 2. By the recursion

n+1 n+1 n+1 n+1 , q21 , q13 , q31 ∈ B also. If we consider the terms at order relations (7) we know that the off-block terms q12 −n−3 we are led to the following identity: E             3 3 i 3 3 3 n+1 n+1 n+1 − 3i qs − + 3i pr q22 + − 3i pr − + 3i qs q33 + − psq23 0≡ 2 4 2 4 4 4   i 3 n+1 − qrq32 + + b, (13) 4 4

where b ∈ B is known in terms of expressions already assumed to be in B. By the inductive hypothesis the terms n+1 n+1 n+1 n+1 , q33 , q23 , q32 can only be of the form u + ∂ −1 v where u ∈ B and ∂ −1 v is a genuine integral term q22 which cannot be simplified any further. However, integral terms of this type could not cancel each other out of n+1 n+1 n+1 n+1 , q33 , q23 , q32 ∈ B. Moreover, the above identity (13), and therefore it must be that the block terms q22 n+1 n+1 n+1 q11 = −q22 − q33 ∈ B , also, so only pure differential polynomials can be generated. The inductive step is now complete. Note that the explicit calculation of Eq. (12) was necessary since the entries of qij2 were used explicitly in the identity (13). 5.2. The differential stationary equations The finite polynomial solutions of the x-flow Lax equation can now be completely characterized in the class of differential polynomials of the potentials in terms of the standard differential solution QD = V A0 V −1 . In particular, we know that QD = A0 + A1 E −1 + A2 E −2 + · · ·   ˆ k = qˆ k and For notational purposes, let Q ij ˆ0 + Q ˆ 1 E −1 + Q ˆ 2 E −2 + · · · QD = Q Theorem 6. Q ∈ G is a solution of Qx = [L1 , Q] and a finite polynomial in E if and only if there exist constants (possibly complex) ci , α, β, γ , δ, such that N X 2α + δ 1 γ 1 i+1 qˆ12 + qˆ13 ci qˆ12 + = 0, 3 3

N X α + 2δ 1 β 1 i+1 qˆ13 + qˆ12 ci qˆ13 + = 0, 3 3

N X 2α + δ 1 β 1 i+1 ci qˆ21 + = 0, qˆ21 + qˆ31 3 3

N X α + 2δ 1 γ 1 i+1 ci qˆ31 + = 0. qˆ31 + qˆ21 3 3

i=1

i=1

i=1

i=1

The proof of the theorem follows from the properties of QD as the standard solution of the x-flow equation, viz. any other solution of the x-flow equation which possesses only differential polynomial expressions must be expressible as a linear combination of QD and its products with powers of E. Thus, when such a solution is truncated as a finite expansion in E, the resulting equations in the recursive procedure for the solution of the x-flow equation are precisely the stationary equations of the theorem. The analog of these equations was found for the KdV by Novikov [25], see also [8,14,18] for similar results. Corollary 5 (Reality constraints on the coefficients). The differential stationary equations are consistent when r = −σp ∗ and s = −σ q ∗ if and only if all the ci for i = 1, 2, . . . , N and α and δ are real and also β = γ ∗ .

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Corollary 6 (Genus of the differential stationary equations). Suppose that Q ∈ G is a solution of Qx = [L1 , Q] and such that QE N is a finite polynomial in E of degree N and that ci for i = 1, 2, . . . , N and α, β, γ , δ are the constants in the differential stationary equations of the previous theorem. Then the characteristic polynomial of Q is a three-sheeted covering of the Riemann E sphere that can obtain its maximum genus of 2N − 2 if and only if α 6= δ or βγ 6= 0. The second corollary is an immediate consequence of the first reduction theorem.

6. Twisted tori There exist distinct stationary equations that possess trigonal spectral curves of the same genus (and so the curves are topologically equivalent and the solutions have the same number of phases) but different monodromy (the topology of pairs of sheets in the realization of the curve as a three-sheeted covering of the Riemann sphere is different and so the interaction of the phases of the solutions is different). The following lemma is essentially a restatement of some of the constructions in the proof of the corollary which established the genus of the spectral curve of the stationary equations. Lemma 5. Let Q ∈ G + be a finite genus solution of ∂Q = [L1 , Q] such that QE N is a polynomial of degree N in E. Then, the spectral curve of Q is z3 + A(E)z + B (E) = det(zI − E N Q) = det(zI − E N X) = 0,

(14)

where X = XN + XN−1 E −1 + XN−2 E −2 + · · · and



−i(αj + δj ) 0 Xj =  0

0 iαj iγj

 0 iβj  , iδj

with constant elements αj , βj , γj , δj ∈ C. Moreover, A(E) and B (E) are polynomials in E with coefficients in aj , bj ∈ B + : A (E) = a2N E 2N + a2N−1 E 2N−1 + · · · + a0 , B (E) = b3N E 3K + b3N−1 E 3N−1 + · · · + b0 . The coefficients aj and bj of A and B in the characteristic polynomial of QE N are integrals of the stationary equations, viz. ∂aj = ∂bj = 0, and depend on the parameters of the system given by Xj for j = 0, 1, 2, . . . , N and the integrals of the motion Xj for j = N + 1, N + 2, N + 3, . . . Notice that only a finite number of these integrals of motion are independent, the relations determining their interdependence come from the expansion of the characteristic polynomial in Eq. (14). The entries of Xj for j = N +1, N +2, . . . are integro-differential expressions Fj (p, q, r, s) such that ∂x Fj (p, q, r, s) = 0, and so are actually interdependent integrals of the stationary equations determined by the initial conditions. Since the spectral curve (14) is trigonal, its realization as a Riemann surface has three z sheets covering the Riemann E sphere which are connected, at most, by 6N square root type branch points located at the 6N roots of the discriminant 4A3 + 27B 2 = 0, assuming that each root is distinct. Each branch point connects only two of the three sheets. The way in which the branch points are distributed between the three sheets of the spectral curve determines the monodromy. If each pair of sheets is connected by exactly 2N branch points (so that the number of

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branch points between any pair of sheets is the same), then we call this a symmetric monodromy. Otherwise the monodromy is called asymmetric. Theorem 7. There exist distinct stationary equations of the CNLS system (3), which possess spectral curves of the same genus (which may be arbitrarily large) but different monodromy. The proof of the theorem comes from choosing X in Eq. (9), appropriately and making use of Lemma 5. First we construct curves with symmetric monodromy by choosing E K X = XK E K + XK−1 E K−1 + · · · + X0 + X−1 E −1 + · · · , where each Xj for j = 0, 1, 2, . . . , K has the form   −iαj − iδj 0 0 0 iαj 0  Xj =  0 0 iδj and, in particular, −αK − δK 6= αK 6= δK . Moreover, we consider the limit where all the integrals of motion are set to 0, so that Xj = 0 for j ≤ −1. In this limit the characteristic polynomial of the stationary equations is reducible and factors into three factors, each factor corresponding to a separate sheet of the Riemann surface: z − (−iαK − iδK )E K − · · · − (−iα0 − iδ0 ) = 0, z − iαK E K − · · · − iα0 = 0, z − iδK E K − · · · − iδ0 = 0. For small initial data, the integrals of motion are not exactly zero but are small in absolute value so there will exist stationary equations whose spectral curve is a perturbation of the reducible one which factors into the above three sheets. In particular, points of intersection of the above three sheets will split open into pairs of genuine square root type branch points. If, for example, K = 3, α3 > 0, δ3 > 0, α3 6= δ3 and the remaining αj and δj are chosen arbitrarily then, in general, the three sheets of the factored curve will be three cubics, each pair of sheets intersecting exactly three times. See Fig. 1. Under small perturbations of the integrals of motion the 3K intersections of the sheets will split open into 6K branch points with symmetric monodromy. The genus as given by the Riemann–Hurwitz relation will be G = 3K − 2. On the other hand, we can construct curves with asymmetric monodromy by choosing X to be of the following form     −iα0 − iδ0 0 0 −2 0 0 0 iα0 0  + X−1 E −1 + · · · , X =  0 1 0  i(αJ E J + · · · + α1 E 1 ) +  0 0 iδ0 0 0 1

Fig. 1. A curve with symmetric monodromy.

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Fig. 2. A curve with asymmetric monodromy.

where αJ > 0 and α0 6= δ0 . When the limit Xj = 0 for j ≤ −1 is considered, then the three sheets factor into: z − i(−2αJ E J − 2αJ −1 E J −1 − · · · − (α0 + δ0 )) = 0, z − i(αJ E J + αJ −1 E J −1 + · · · + α1 E + α0 ) = 0, z − i(αJ E J + αJ −1 E J −1 + · · · + α1 E + δ0 ) = 0. Notice that two of the sheets are identical except for a shift in the last constant. Thus, when J = 3, for example, there will be three cubics but one pair intersects only over the point at infinity. The intersection over infinity is a repeated root of order J. See Fig. 2. In particular, there are 3J intersections between the three sheets but only 2J are over finite locations, the remaining J are all pinched together over infinity. For small non-zero initial conditions only the 2J double points over finite locations will open into 4J square root branch points (see the theorem on the genus of the spectral curve of the differential stationary equations) and remain close to their original locations. Thus, the genus of this type of curve will be G = 2J − 2. Moreover, the monodromy is asymmetric. In particular, two sets of stationary equations with spectral curves of different monodromy but the same genus occur when G = 2J − 2 = 3K − 2, viz. when G mod 6 = 4. The asymmetric curve comes from the ansatz of degree J and the symmetric curve comes from the ansatz of degree K. Of course this by no means exhausts the range of possibilities of equations having curves of the same genus but different monodromy. Even within the same ansatz it may be possible to choose the initial conditions so as to cause branch points to move to a new sheet connection, thus changing the monodromy of the curve, but not its genus.

7. Conclusion The stationary equations of the CNLS Lax pair were derived; in general, these are integro-differential equations. The stationary equations were classified according to the genus of their spectral curves using reduction theorems which impose additional constraints on the coefficients of the equations. Moreover, the subset of purely differential stationary equations was constructed. In addition, it was demonstrated that there exist stationary equations of the integrable CNLS system which have spectral curves of the same (arbitrarily large) genus but distinct monodromy. Thus, the way in which the curve or torus is “twisted” will play a significant role in the linearization of the flow on the Jacobian of the curve and the subsequent explicit construction of the solutions. This additional feature arises because the spatial Lax operator of the integrable theory is of order 3. Future work is planned to match the analytic structure of the spectral curve of the stationary equations to that of the spectral curve of a given unstable wave train. Thus, the stationary equations that govern near homoclinic orbits arising from known linearized instabilities in the system can be constructed.

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