Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

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Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

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Vladimir Stefanov Gerdjikov^*

Aleksander Aleksiev Stefanov

Vladimir Stefanov Gerdjikov^*

Aleksander Aleksiev Stefanov

This version is not peer-reviewed

Submitted:

25 August 2023

Posted:

30 August 2023

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Abstract

The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: 1) Lax representation $[L,M]=0$; 2)~construction of fundamental analytic solutions (FAS); 3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem (RHP) $\xi^+(x,t,\lambda) =\xi^-(x,t,\lambda) G(x,t\lambda)$ on a contour $\Gamma$ with sewing function $G(x,t,\lambda)$; 4) soliton solutions and possible applications.Step 1) involves several assumptions: the choice of the Lie algebra $\mathfrak{g}$ underlying $L$, as well as its dependence on the spectral parameter, typically linear or quadratic in $\lambda$. In the present paper we propose another approach which substantially extends the classes of integrable NLEE.Its first advantage is that one can effectively use any polynomial dependence in both $L$ and $M$. We use the following steps: A) Start with canonically normalized RHP with predefined contour $\Gamma$; B) Specify the $x$ and $t$ dependence of the sewing function defined on $\Gamma$; C) Introduce convenient parametrization for the solutions $ \xi^\pm(x,t,\lambda)$ of the RHP and formulate the Lax pair and the nonlinear evolution equations (NLEE); D) use Zakharov-Shabat dressing method to derive their soliton solutions. This needs correctly taking into account the symmetries of the RHP. E) Define the resolvent of the Lax operator and use it analyze its spectral properties.

Keywords:

Subject: Physical Sciences - Mathematical Physics

1. Introduction

In 1968 one of the great discoveries in mathematical physics took place. Its authors: P. Lax, C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura and N. J. Zabusky after several years of analysis proved that the KdV equation can be exactly solved by the inverse scattering method (ISM). This was the first and for some time, the only NLEE that could be solved exactly. Soon after that it was demonstrated that the KdV is completely integrable infinite-dimensional Hamiltonian system; its action-angle variables were found by Zakharov and Faddeev [137]. The whole story is well described by N. J. Zabusky in his review paper [128].

The second big step in this direction followed in 1971 by the seminal paper of Zakharov and Shabat who discovered the second equation integrable by the ISM: the nonlinear Schrödinger (NLS) equation [132]; in 1973 the same authors demonstrated that the NLS equation is integrable also under nonvanishing boundary conditions [133]. Both versions of NLS equations described interesting and important physical applications in nonlinear optics, plasma physics, hydrodynamics and others. This inspired many scientists, mathematicians and physicists alike to join the scientific community interested in the study of soliton equations. As a result new soliton equations started to appear one after another. Here we only mention the modified KdV (mKdV) equation [125], the N-wave equations [131], the Manakov system known also as the vector NLS equation [92] and many others. Many of them have already been included in monographs: see e.g. [1,10,18,64,104] and the numerous references therein.

The first few NLEE were related to the algebra

s l (2)

, so the corresponding inverse scattering problem could be solved using the famous Gelfand-Levitan-Marchenko (GLM) equation. For the Manakov system it was necessary to use

2 \times 2

block matrix Lax operator, so the GLM equation was naturally generalized also for that case. However, the ISM for the N-wave system came up to be substantially more difficult. Indeed, for the

2 \times 2

(block) matrix Lax operator the Jost solutions possess analyticity properties which are basic for the GLM eq. However the Lax pair for the N-wave system is the generalized

s l (n)

Zakharov-Shabat system:

\begin{matrix} L ψ & \equiv i \frac{\partial ψ}{\partial x} + ([J, Q (x, t)] - λ J) ψ (x, t, λ) = 0, M ψ \equiv i \frac{\partial ψ}{\partial t} + ([K, Q (x, t)] - λ K) ψ (x, t, λ) = 0, \\ J & = diag (a_{1}, a_{2}, \dots, a_{n}), K = diag (b_{1}, b_{2}, \dots, b_{n}), \end{matrix}

(1.1)

where

a_{k}

and

b_{k}

are real constants such that

tr J = 0

tr K = 0

. Without restrictions we can assume that

a_{1} > a_{2} > \dots > a_{n}

. In this case only the first and the last column of the corresponding Jost solutions allow analytic extension in the spectral parameter

λ

. This, however, was not enough to derive GLM equation. It was Shabat who discovered the way out of this difficulty [106,107]. He was able to modify the integral equations for the Jost solutions into integral equations that provide the fundamental analytic solutions (FAS)

χ^{+} (x, t, λ)

and

χ^{+} (x, t, λ)

of L which allowed analytic extensions for

Im λ > 0

and

Im λ < 0

respectively. As a result the interrelation between the FAS and the sewing function

G_{0} (x, t, λ)

\begin{matrix} χ^{+} (x, t, λ) & = χ^{-} (x, t, λ) G_{0} (t, λ), Im λ = 0, \\ i \frac{\partial G_{0}}{\partial t} & = λ [K, G_{0} (x, t, λ)] . \end{matrix}

(1.2)

can be reformulated as Riemann-Hilbert problem (RHP). Now we can solve the ISP for L by using the RHP with canonical normalization:

\begin{matrix} ξ^{+} (x, t, λ) & = ξ^{-} (x, t, λ) G (x, t, λ), Im λ = 0, \\ i \frac{\partial G_{0}}{\partial x} & = λ [J, G_{0} (x, t, λ)], i \frac{\partial G_{0}}{\partial t} = λ [K, G_{0} (x, t, λ)] . \end{matrix}

(1.3)

\begin{matrix} lim_{λ \to \infty} ξ^{\pm} (x, t, λ) = 1 1, \end{matrix}

(1.4)

The normalization condition (1.4) ensures that the RHP has unique regular solution [104]. It also allowed Zakharov and Shabat to develop their dressing method, which enables one to calculate the N-soliton solutions not only for the N-wave system, but also for the whole hierarchy of NLEE related to L (1.1); see [134,135] and also [32,104]. In short, the dressing Zakharov-Shabat factor came up as one of the most effective methods for a) constructing soliton solutions, and b) understand that the dressed Lax operator has some additional discrete eigenvalues, as well as the explicit form of the dressed FAS.

The GZS system has natural reductions after which the potential

Q (x, t)

and J belong either to

s o (n)

or to

s p (2 r)

algebra. Other reductions were proposed by Mikhailov [96] which substantially enlarged the classes of integrable NLEE. Some of these reductions require that J has complex-valued eigenvalues. Constructing FAS for such systems poses additional difficulties, which were overcome by Beals and Coifman [7] for the GZS systems related to

s l (n)

algebra. Later the results of [7] were extended first for the systems related to

s o (n)

or to

s p (2 r)

algebra, (see [64]) as well as to Mikhailov’s reductions [57,64].

The FAS play an important role in soliton theory. Indeed, they can be used to introduce

1.: Scattering data The minimal sets of scattering data are determined by the asymptotics of

$lim_{x \to - \infty} χ^{\pm} (x, t, λ) e^{i J λ x} = S^{\pm} (λ, t) or lim_{x \to \infty} χ^{\pm} (x, t, λ) e^{i J λ x} = T^{\pm} (λ, t) .$

Here $S^{\pm} (t, λ)$ and $T^{\pm} (λ, t)$ are the factors of the Gauss decompositions of the scattering matrix $T (λ, t)$ .
2.: Resolvent The FAS determine the kernel of the resolvent $R^{\pm} (x, y, t, λ)$ of L. Applying contour integration method on $R^{\pm} (x, y, t, λ)$ one can derive the spectral expansions for L, i.e. the completeness relation of FAS.
3.: Dressing method Zakharov-Shabat dressing method is a very effective and convenient method to construct the class of reflectionless potentials of L and to derive the soliton solutions of the NLEE. The simplest dressing factor $u (x, t, λ)$ has pole singularities at $λ_{1}^{\pm}$ , which determine the new discrete eigenvalues that are added to the spectrum of the initial Lax operator.
4.: Generalized Fourier transforms Here we start with GZSh system related to a simple Lie algebra $g$ with Cartan-Weyl basis $H_{j}$ , $E_{α}$ [67] and construct the so-called 'squared solutions'

$e_{α}^{\pm} (x, t, λ) = π_{0 J} (χ^{\pm} E_{α} {(χ^{\pm})}^{- 1} (x, t, λ) and h_{j}^{\pm} (x, t, λ) = π_{0 J} (χ^{\pm} H_{j} {(χ^{\pm})}^{- 1} (x, t, λ),$

where $π_{0 J}$ is the projector onto the image of the operator ${ad}_{J}$ . It is known that the 'squared solution' are complete set of functions in the space of allowed potentials [30]. In particular, if we expand the potential $Q (x, t)$ over the 'squared solutions' the expansion coefficients will provide the minimal set of scattering data. Similarly, the expansion coefficients of ${ad}_{J}^{- 1} δ Q$ are the variations of the minimal set of scattering data. Therefore the 'squared solutions' can be viewed as FAS in the adjoint representation of $g$ , see [2,12,30,45,59,62,63,64,78,108,121] as well as [12,24,39,65].
5.: Hierarchies of Hamiltonian structures The GFT described above allow one to prove that each of the NLEE related to L allows a hierarchy of Hamiltonian structures. More precisely, each NLEE allows a hierarchy of Hamiltonians $H^{(n)}$ and a hierarchy of symplectic forms $Ω^{(n)}$ (or a hierarchy of Poisson brackets) such that for any n they produce the relevant NLEE. [30,81,86]
6.: Complete integrability and action-angle variables. Starting from the famous paper by Zakharov and Faddeev [137] it is known that some of the NLEE allow action-angle variables. The difficulty here is that these NLEE are Hamiltonian system with infinitely many degrees of freedom. Therefore the strict derivation of the proof must be based on the completeness relation for the 'squared solutions'. In fact VG and E. Khristov [45,59] (see also [63]) proposed the so-called 'symplectic basis' of squared solutions, which maps the variation of the potential $σ_{3} δ Q (x, t)$ of the AKNS system to the variation of the action-variables. Unfortunately for many multi-component systems such bases are not yet known.

The above arguments lead us to the hypothesis that we could use more effective approach to the integrable NLEE which starts from the RHP rather than from a specific Lax operator. In the first part of this paper we will demonstrate that FAS could be constructed and used also for quadratic pencils. We also formulate explicitly the corresponding RHP. For quadratic pencils we have additional natural symmetry which maps

λ \to - λ

. This symmetry is also inherent in the contour in the complex

λ

-plane, on which the RHP is defined.

In Section 2 we first demonstrate that the well known methods for analysis of NLEE can be generalized also for Lax operators quadratic in

λ

, see eq. (2.28) below. On this level we for the first time meet with purely algebraic problem for constructing two commuting quadratic pencils. For polynomial pencils of higher orders those problems will be more and more difficult to solve. In particular we outline the construction of the FAS

χ^{+} (x, t, λ)

and

χ^{-} (x, t, λ)

which are analytic in

Ω_{0} \cup Ω_{3}

and

Ω_{2} \cup Ω_{4}

respectively, see Figure 4 below. As a result, the continuous spectrum of these Lax operators fill up the union of the real and imaginary axis of the complex

λ

-plane. As a consequence contours of the corresponding RHP will be

R \cup i R

unless an additional factor complicates the picture. Thus we see that the symmetries of the NLEE or of its Lax pair determine the contour of the RHP.

In Section 3 we remind the notion of Mikhailov’s reduction group and briefly outline characteristic contours of the relevant RHP. We also demonstrate that Zakharov–Shabat theorem is valid for a larger class of Lax operators than it was proved before. It is important to request that the RHP is canonically normalized. This ensures that the RHP has unique regular solution, which is important for the application of the Zakharov-Shabat dressing method.

Another important factor in formulating the RHP is Mikhailov’s reduction group. In Section 4 we outline some of the obvious effects which the reduction group may have on the contour of the RHP. Therefore it is not only the order of the polynomial in

λ

, but also the symmetry (the reduction group) which determine the contour of the RHP. For example, if we add an additional Mikhailov’s symmetry that maps

λ \to 1 / λ^{*}

then the corresponding Lax operator will be polynomial in

λ

and

1 / λ

which in turn will require adjustments in the techniques for deriving the dressing factors and soliton solutions.

In Section 5 we propose a parametrization of the solutions

ξ (x, t, λ)

of RHP for the class of RHP related to homogeneous spaces, see eq. (5.1). Here we require that the coefficients

Q_{s} (x, t)

provide local coordinates of the corresponding homogeneous space. Thus we are able to derive a new systems of N-wave equations, see also [36,52,53,56,74]. We also demonstrate that the dressing Zakharov-Shabat method [100,129,134,135,136] can be naturally extended to derive the soliton solutions of these new N-wave equations. At the same time the structure of the dressing factors depends substantially on the symmetries of the Lax operators. Thus even for the one-soliton solutions we need to solve linear block-matrix equations. The situation when we have two involutions: the Hermitian one

{(χ^{+} (x, t, λ))}^{†} = {(χ^{-} (x, t, λ^{*}))}^{- 1}

and the

λ \to - λ

symmetry

C_{0} χ^{\pm} (x, t, λ) C_{0}^{- 1} = χ^{\pm} (x, t, - λ)

are typical for Lax operators L related to the algebras

s l (n, C)

. But if we request in addition that L is related to symplectic or orthogonal Lie algebra then we have to deal with three involutions, and the corresponding linear equations get more involved. That is why we focus first on the one-soliton solutions. The derivation of the N soliton solutions is discussed later.

Section 6 is devoted to the MNLS equations which require the use of symmetric spaces, see Refs. [21,40,67,92,94,118,119,122,126]. We start again with the parametrization of the RHP which now must be compatible with the structure of the symmetric spaces. To us it was natural to limit ourselves to the four classes Hermitian symmetric spaces related to the non-exceptional Lie algebras, see [67]. Again we parametrize the coefficients

Q_{s} (x, t)

as local coordinates of the corresponding symmetric spaces. In fact

Q (x, t, λ)

must have the same grading as the symmetric space, but we were able to apply additional reductional requesting

Q_{2 s} = 0

, see eq. (6.11) below. Thus we formulate the typical MNLS equations related to the four classes of symmetric spaces.

In Section 7 we derive the one soliton solutions of MNLS. Again, like in Section 5, we treat separately the MNLS related to A.III type symmetric spaces, because the corresponding FAS have only two involutions. The MNLS related to C.I and D.III symmetric spaces possess three involutions; the corresponding linear equations are similar to the ones for the class of N-wave equations, but the solutions are different. The symmetric spaces of BD.I class are treated separately, because their typical representation is provided by

3 \times 3

block matrices, so many of the calculations are indeed different. At the end of this Section we derive the soliton interactions for the BD.I class of MNLS [41]. More precisely we use the asymptotic of the dressing factor for

x \to \pm \infty

applying it to the two-soliton solution in order to calculate the center-of-mass and the phase shifts of the solitons.

In Section 8 we introduce the resolvent of the Lax operators in terms of the FAS. The diagonal of the resolvent after a regularization can be expressed in terms of the solution of the RHP by

ξ^{\pm} (x, t, λ) J {\hat{ξ}}^{\pm} (x, t, λ)

; here by 'hat' we denote the inverse matrix. It can be viewed as generating functional of the integrals of motion.

We end the paper by discussion and conclusions. Some technical aspects in the calculations such as the structure of the symmetric spaces, and the root systems of the simple Lie algebras as well as the Gauss decompositions of the elements of the simple Lie groups are given in the appendices.

2. From the Lax representation to the RHP

2.1. N-waves according to Manakov and Zakharov

The N-wave equations were discovered by Manakov and Zakharov in 1974 [131]. The Lax operator is the classical Zakharov-Shabat system:

\begin{matrix} L_{ZS} ψ \equiv i \frac{\partial ψ}{\partial x} + ([J, Q (x, t)] - λ J) ψ (x, t, λ) = 0, \end{matrix}

(2.1)

with real-valued diagonal

J = diag (a_{1}, a_{2}, \dots, a_{n})

. The second operator in the Lax representation is also linear in

λ

\begin{matrix} M ψ \equiv i \frac{\partial ψ_{ZS}}{\partial x} + ([K, Q (x, t)] - λ K) ψ (x, t, λ) = 0, \end{matrix}

(2.2)

with real-valued diagonal

K = diag (b_{1}, b_{2}, \dots, b_{n})

. Then the N-wave equations, which are the compatibility condition

[L, M] = 0

take the form:

\begin{matrix} [J, \frac{\partial Q}{\partial t}] - [K, \frac{\partial Q}{\partial x}] + [[J, Q (x, t)], [K, Q (x, t)]] = 0 . \end{matrix}

(2.3)

For some time solving the ISP for the Lax operator

L_{ZS}

was an open problem. However in 1974 Shabat [106,107] proved that

L_{ZS}

has fundamental solutions

χ^{+} (x, t, λ)

and

χ^{-} (x, t, λ)

which are analytic in the upper- and lower-half of

C

Let us briefly outline the construction of FAS proposed by A. B. Shabat [106,107]; for more details see also [32,104]. We will assume for simplicity that the potential

q (x)

is defined on the whole axis and satisfies the following conditions:

C.1: By $Q (x) \in M_{S}$ we mean that $Q (x)$ possesses smooth derivatives of all orders and falls off to zero for $| x | \to \infty$ faster than any power of x:

$lim_{x \to \pm \infty} {| x |}^{k} Q (x) = 0, \forall k = 0, 1, 2, \dots$
C.2: $Q (x)$ is such that the corresponding operator L has only a finite number of simple discrete eigenvalues.

We will impose also the typical reduction of the Lax operator: the GZSs with

Z_{2}

-reduction:

U^{†} (x, t, ϵ λ^{*}) = U (x, t, λ) .

(2.4)

We start by introducing the Jost solutions of L:

\begin{matrix} lim_{x \to - \infty} ϕ (x, t, λ) e^{i λ J x} = 1 1, lim_{x \to \infty} ϕ (x, t, λ) e^{i λ J x} = 1 1, \end{matrix}

(2.5)

and the scattering matrix:

\begin{matrix} T (t, λ) = ψ^{- 1} (x, t, λ) ϕ (x, t, λ) . \end{matrix}

(2.6)

The Jost solutions satisfy the following Volterra type integral equations:

\begin{matrix} {\tilde{ψ}}_{i j} (x, λ) & = δ_{i j} + i \int_{\infty}^{x} d y e^{- i λ (a_{i} - a_{j}) (x - y)} \sum_{p = 1}^{h} (a_{i} - a_{p}) Q_{i p} (y) {\tilde{ψ}}_{p j} (y, λ), \\ {\tilde{ϕ}}_{i j} (x, λ) & = δ_{i j} + i \int_{- \infty}^{x} d y e^{- i λ (a_{i} - a_{j}) (x - y)} \sum_{p = 1}^{h} (a_{i} - a_{p}) Q_{i p} (y) {\tilde{ϕ}}_{p j} (y, λ), \end{matrix}

(2.7)

where

\tilde{ψ} (x, λ) = ψ (x, λ) e^{i J λ x}

and

\tilde{ϕ} (x, λ) = ϕ (x, λ) e^{i J λ x}

. It is well known that Volterra type equations possess solutions provided the integrals of the equations are convergent. In our case this will be true for real

λ

. Indeed, in this case the exponential factors in the equations (2.7) will be bounded and the convergence of the integrals is ensured by the fact that

Q (x)

satisfies Condition 1. However, for complex

λ

the exponential factors are not growing only for the first and the last columns of the Jost solutions.

Shabat introduced the FAS

χ^{\pm} (x, λ)

of L by modifying the integral equations for them as follows:

\begin{matrix} ξ_{i j}^{+} (x, λ) & = δ_{i j} + i \int_{- \infty}^{x} d y e^{- i λ (a_{i} - a_{j}) (x - y)} \sum_{p = 1}^{h} (a_{i} - a_{p}) Q_{i p} (y) ξ_{p j}^{+} (y, λ), i \geq j; \\ ξ_{i j}^{+} (x, λ) & = i \int_{\infty}^{x} d y e^{- i λ (a_{i} - a_{j}) (x - y)} \sum_{p = 1}^{h} (a_{i} - a_{p}) Q_{i p} (y) ξ_{p j}^{+} (y, λ), i < j; \end{matrix}

(2.8)

\begin{matrix} ξ_{i j}^{-} (x, λ) & = i \int_{\infty}^{x} d y e^{- i λ (a_{i} - a_{j}) (x - y)} \sum_{p = 1}^{h} (a_{i} - a_{p}) Q_{i p} (y) ξ_{p j}^{-} (y, λ), i > j; \\ ξ_{i j}^{-} (x, λ) & = δ_{i j} + i \int_{- \infty}^{x} d y e^{- i λ (a_{i} - a_{j}) (x - y)} \sum_{p = 1}^{h} (a_{i} - a_{p}) Q_{i p} (y) ξ_{p j}^{-} (y, λ), i \leq j; \end{matrix}

(2.9)

where

ξ^{\pm} (x, λ) = χ^{\pm} (x, λ) e^{i λ J x} .

(2.10)

Now it is not difficult to check that all exponential factors in the integrands of (2.8) are falling off for

λ \in C_{+}

, while the exponential factors in the integrands of (2.9) are falling off for

λ \in C_{-}

. In other words,

ξ^{+} (x, t, λ)

allows analytic extensions for

λ

in the upper complex half-plane, while

ξ^{-} (x, t, λ)

is analytic for

λ

in the lower complex half-plane. Obviously

ξ^{+} (x, λ)

and

ξ^{-} (x, λ)

are FAS of a slightly modified Lax operator:

\begin{matrix} \tilde{L} ξ^{\pm} \equiv i \frac{\partial ξ^{\pm}}{\partial x} + [J, Q (x, t)] ξ^{\pm} (x, λ) - λ [J, ξ^{\pm} (x, λ)] = 0 . \end{matrix}

(2.11)

Theorem 1.

Let

q (x) \in M_{S}

satisfies conditions (C.1), (C.2) and let the matrix elements of J be ordered

a_{1} > a_{2} > \dots > a_{n}

. Then the solution

ξ^{+} (x, λ)

of the eqs. (2.8) (resp.

ξ^{-} (x, λ)

of the eqs. (2.9)) exists and allows analytic extension for

λ \in C_{+}

(resp. for

λ \in C_{-}

Remark 1.

Due to the fact that in eq. (2.8) we have both ∞ and

- \infty

as lower limits the equations are rather of Fredholm than of Volterra type. Therefore we have to consider also the Fredholm alternative, i.e. there may exist finite number of values of

λ = λ_{k}^{\pm} \in C_{\pm}

for which the solutions

ξ^{\pm} (x, λ)

have zeroes and pole singularities in λ. The points

λ_{k}^{\pm}

in fact are the discrete eigenvalues of

L (λ)

C_{\pm}

The reduction condition (2.4) with

ϵ = 1

means that the FAS and the scattering matrix

T (λ)

satisfy:

\begin{matrix} {(χ^{+} (x, t, λ^{*}))}^{†} = {(χ^{-} (x, t, λ))}^{- 1}, T^{†} (t, λ^{*}) = T^{- 1} (t, λ) . \end{matrix}

(2.12)

Each fundamental solution of the Lax operator is uniquely determined by its asymptotic for

x \to \infty

x \to - \infty

. Therefore in order to determine the linear relations between the FAS and the Jost solutions for

λ \in R

we need to calculate the asymptotics of FAS for

x \to \pm \infty

. Taking the limits in the right hand sides of the integral equations (2.8) and (2.9) we get:

\begin{matrix} lim_{x \to - \infty} ξ_{i j}^{+} (x, λ) = δ_{i j}, for i \geq j; lim_{x \to \infty} ξ_{i j}^{+} (x, λ) = 0, for i < j; \end{matrix}

(2.13a)

\begin{matrix} lim_{x \to - \infty} ξ_{i j}^{-} (x, λ) = δ_{i j}, for i \leq j; lim_{x \to \infty} ξ_{i j}^{-} (x, λ) = 0, for i > j; \end{matrix}

(2.13b)

This can be written in compact form using (2.10):

χ^{\pm} (x, λ) = ψ_{-} (x, λ) S^{\pm} (λ) = ψ_{+} (x, λ) T^{\mp} (λ) D^{\pm} (λ),

(2.14)

where the matrices

S^{\pm} (λ)

D^{\pm} (λ)

and

T^{\pm} (λ)

are of the form:

\begin{matrix} S^{+} (λ) = (\begin{matrix} 1 & S_{12}^{+} & \dots & S_{1 n}^{+} \\ 0 & 1 & \dots & S_{2 n}^{+} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{matrix}), T^{+} (λ) = (\begin{matrix} 1 & T_{12}^{+} & \dots & T_{1 n}^{+} \\ 0 & 1 & \dots & T_{2 n}^{+} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{matrix}), \end{matrix}

(2.15a)

\begin{matrix} D^{+} (λ) = diag (D_{1}^{+}, D_{2}^{+}, \dots, D_{n}^{+}), D^{-} (λ) = diag (D_{1}^{-}, D_{2}^{-}, \dots, D_{n}^{-}), \end{matrix}

(2.15b)

\begin{matrix} S^{-} (λ) = (\begin{matrix} 1 & 0 & \dots & 0 \\ S_{21}^{-} & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ S_{n 1}^{-} & S_{n 2}^{-} & \dots & 1 \end{matrix}), T^{-} (λ) = (\begin{matrix} 1 & 0 & \dots & 0 \\ T_{21}^{-} & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ T_{n 1}^{-} & T_{n 2}^{-} & \dots & 1 \end{matrix}), \end{matrix}

(2.15c)

Let us now relate the factors

T^{\pm} (λ)

S^{\pm} (λ)

and

D^{\pm} (λ)

to the scattering matrix

T (λ)

. Comparing (2.14) with (2.6) we find

T (λ) = T^{-} (λ) D^{+} (λ) {\hat{S}}^{+} (λ) = T^{+} (λ) D^{-} (λ) {\hat{S}}^{-} (λ),

(2.16)

i.e.

T^{\pm} (λ)

S^{\pm} (λ)

and

D^{\pm} (λ)

are the factors in the Gauss decomposition of

T (λ)

It is well known how given

T (λ)

one can construct explicitly its Gauss decomposition, see the Appendix B. Here we need only the expressions for

D^{\pm} (λ)

D_{j}^{+} (λ) = \frac{m_{j}^{+} (λ)}{m_{j - 1}^{+} (λ)}, D_{j}^{-} (λ) = \frac{m_{n - j + 1}^{-} (λ)}{m_{n - j}^{-} (λ)},

(2.17)

where

m_{j}^{\pm}

are the principal upper and lower minors of

T (λ)

of order j.

Remark 2.

Gauss decomposition of of

T (t, λ) \in S L (n)

is well known, see Appendix B and [32]. The derivation requires the knowledge of the fundamental representations of the Lie algebra

s l (n)

. If

T (t, λ)

belongs to an orthogonal or symplectic group, see [30].

Corollary 1.

The upper (resp. lower) principal minors

m_{j}^{\pm} (λ)

(resp.

m_{j}^{-} (λ)

T (λ)

are analytic functions of λ for

λ \in C_{+}

(resp. for

λ \in C_{-}

Proof.

Follows directly from theorem 1, from the limits:

lim_{x \to \infty} ξ_{j j}^{+} (x, λ) = D_{j}^{+} (λ), lim_{x \to \infty} ξ_{j j}^{-} (x, λ) = D_{j}^{-} (λ),

(2.18)

and from (2.15b) and (2.17). □

Thus the solution of the ISP for

L_{ZS}

was reduced to a RHP:

\begin{matrix} χ^{+} (x, t, λ) = χ^{-} (x, t, λ) G_{0} (t, λ), λ \in R \end{matrix}

(2.19)

on the real axis of

C

. The solutions

χ^{+} (x, t, λ)

defined by (2.10) satisfy an alternative RHP:

\begin{matrix} ξ^{+} (x, t, λ) & = ξ^{-} (x, t, λ) G (x, t, λ), λ \in R, \\ i \frac{\partial G}{\partial x} - λ [J, G (x, t, λ) = 0, i \frac{\partial G}{\partial t} - λ [K, G (x, t, λ) = 0, \end{matrix}

(2.20)

which is canonically normalized, i.e.

\begin{matrix} lim_{λ \to \infty} ξ^{\pm} (x, t, λ) = 1 1 . \end{matrix}

(2.21)

2.2. MNLS equations according to Manakov, Fordy and Kulish

The first MNLS:

\begin{matrix} i \frac{\partial \vec{q}}{\partial t} + \frac{\partial^{2} \vec{q}}{\partial x^{2}} + 2 (\vec{q}^{†}, \vec{q}) \vec{q} (x, t) = 0 \end{matrix}

(2.22)

where

\vec{q}

is a 2-component vector, was proposed by Manakov [92] in 1974. Later in their seminal paper [21] Fordy and Kulish demonstrated the deep relations between the MNLS equations and the symmetric spaces [67]. As a result the Lax representation for the MNLS takes the form:

\begin{matrix} L_{MNLS} ψ & \equiv i \frac{\partial ψ}{\partial x} + (Q (x, t) - λ J) ψ (x, t, λ) = 0, \\ M_{MNLS} ψ & \equiv i \frac{\partial ψ}{\partial t} + (V_{2} (x, t) + λ Q (x, t) - λ^{2} J) ψ (x, t, λ) = 0 \end{matrix}

(2.23)

For the Manakov model (2.22)

J = diag (1, - 1, - 1)

and

Q (x, t) = (\begin{matrix} 0 & {\vec{q}}^{T} \\ \vec{q}^{*} & 0 \end{matrix})

. Special role here is played by the Cartan element J of the corresponding simple Lie algebra. It determines the Cartan involution, which picks up the symmetric space from the corresponding Lie algebra. The Manakov model is related to the symmetric space

S U (3) / S (U (1) \times U (2))

. The symmetric spaces have been classified about a century ago by E. Cartan, see e.g. [67].

In 1976 Kaup and Newell [81] derived the generalizations of NLS corresponding to Lax operator, quadratic in

λ

. This generalization of NLS now is known as the derivative NLS (DNLS), because its nonlinear terms depend on the x-derivative of q. Another form of the DNLS eq. is known since 1978 as the Gerdjikov-Ivanov (GI) equation [42,58], see also [14,19,20,90] and the references therein. It is gauge equivalent to DNLS and besides the terms with x-derivative, contains also nonlinearities of 5-th order. In Section 6 below we describe the multi-component generalizations to GI equations.

2.3. Generic Lax representation

We start with the idea of constructing more general polynomial Lax pairs:

\begin{matrix} L_{0} ψ_{0} & \equiv i \frac{\partial ψ_{0}}{\partial x} + (U (x, t, λ) - λ^{N_{1}} J) ψ_{0} (x, t, λ) = 0, \\ M_{0} ψ_{0} & \equiv i \frac{\partial ψ_{0}}{\partial t} + (V (x, t, λ) - λ^{N_{2}} K) ψ_{0} (x, t, λ) = ψ_{0} (x, t, λ) C (λ), \\ U (x, t, λ) & = \sum_{s = 1}^{N_{1}} U_{s} λ^{N_{1} - s}, V (x, t, λ) = \sum_{s = 1}^{N_{2}} V_{s} λ^{N_{2} - s} . \end{matrix}

(2.24)

The compatibility condition

[L_{0}, M_{0}] = 0

of the pair (2.24) holds for any

C (λ)

and has the form:

\begin{matrix} i \frac{\partial V}{\partial x} - i \frac{\partial U}{\partial t} + [U (x, t, λ), V [x, t, λ)] = 0 . \end{matrix}

(2.25)

Here we will assume that both

N_{1} \geq 2

and

N_{2} \geq 2

. In addition we will fix up the gauge by requiring that

\begin{matrix} U_{N_{1}} (x, t) = - J, V_{N_{2}} (x, t) = - K, \end{matrix}

(2.26)

where J and K are constant diagonal matrices.

Eq. (2.25) must hold identically with respect to the spectral parameter

λ

. Thus here there comes the first technical difficulty related with the parametrization of L and M. Indeed, let us consider the example where

N_{1} = 3

and

N_{2} = 3

. The the left hand side of (2.25) is a polynomial of order 6 with respect to

λ

, whose highest 4 coefficients are given by

\begin{matrix} λ^{6} & [K, J] = 0, & λ^{5} & - [V_{2}, J] - [K, U_{2}] = 0, \\ λ^{4} & - [V_{1}, J] + [V_{2}, U_{2}] - [K, U_{1}] = 0, \\ λ^{3} & - [V_{2}, J] + [V_{1}, U_{2}] + [V_{2}, U_{1}] - [K, U_{2}] = 0, \end{matrix}

(2.27)

The coefficient at

λ^{6}

is vanishing because J and K are diagonal. The vanishing of the coefficient at

λ^{5}

is means that

V_{2}

is expressed through

U_{2}

. Indeed, if we put

J = d i a g (a_{1}, a_{2}, \dots, a_{n})

and

K = d i a g (b_{1}, b_{2}, \dots, b_{n})

then

{(V_{2})}_{j k} = \frac{b_{j} - b_{k}}{a_{j} - a_{k}} {(U_{2})}_{j k} .

The next two relations coming from

λ^{4}

and

λ^{3}

are not so easy to satisfy.

Gel’fand and Dickey [15,25,26,27,28] provide a very effective solution to the problem. They suggest a general construction for the Lax pairs of the form (2.24) using the fractional powers of the Lax operator. Below we will outline, along with the effective parametrization of the RHP, an equivalently effective method for generic Lax representations.

2.4. Jost solutions and FAS of L

Here we start with generic Lax operator, quadratic in

λ

with vanishing boundary conditions and canonical gauge. In our case this is:

\begin{matrix} L ψ & \equiv i \frac{\partial ψ}{\partial x} + (U_{2} (x, t) + U_{1} (x, t) - λ^{2} J) ψ (x, t, λ) = 0, \\ J & = diag (a_{1}, a_{2}, \dots, a_{n}), a_{1} > a_{2} > \dots > a_{n}, tr J = 0 . \end{matrix}

(2.28)

Remark 3.

For the potentials

U_{2} (x, t)

and

U_{1} (x, t)

we assume that they

n \times n

matrices which are smooth functions of x for all values of t, tending to 0 fast enough for

x \to \pm \infty

; for simplicity we could take them to be Schwartz-type functions

The Jost solutions to (2.28) are defined as follows:

\begin{matrix} lim_{x \to \infty} ψ (x, t, λ) e^{i λ^{2} J x} = 1 1, lim_{x \to - \infty} ϕ (x, t, λ) e^{i λ^{2} J x} = 1 1 . \end{matrix}

(2.29)

Both Jost solutions are fundamental solutions: indeed they are non-degenerate

n \times n

matrix-valued functions. In what follows we will choose the potentials

U_{2}

and

U_{1}

to take values in a given simple Lie algebra

g

. Then the fundamental solutions

ψ

and

ϕ

will belong to the corresponding simple Lie group

G

It is also well known that any two fundamental solutions are linearly related. In other words:

\begin{matrix} ϕ (x, t, λ) = ψ (x, t, λ) T (t, λ) . \end{matrix}

(2.30)

The matrix

T (t, λ)

belongs to the Lie group

G

The integral equations for the Jost solutions take the form:

\begin{matrix} ψ (x, t, λ) & = 1 1 + i \int_{\infty}^{x} d y e^{- i λ^{2} J (x - y)} (U_{2} (y, t) + λ U_{1} (y, t)) ψ (y, t, λ) e^{i λ^{2} J (x - y)}, \\ ϕ (x, t, λ) & = 1 1 + i \int_{- \infty}^{x} d y e^{- i λ^{2} J (x - y)} (U_{2} (y, t) + λ U_{1} (y, t)) ϕ (y, t, λ) e^{i λ^{2} J (x - y)}, \end{matrix}

(2.31)

where

ψ (x, t, λ) = ψ (x, t, λ) e^{i λ^{2} J x}

and

ϕ (x, t, λ) = ϕ (x, t, λ) e^{i λ^{2} J x}

and

1 1

is the unit

n \times n

matrix. In components we have:

\begin{matrix} ψ_{j k} (x, t, λ) & = δ_{j k} + i \int_{\infty}^{x} d y e^{- i λ^{2} (a_{j} - a_{k}) (x - y)} {((U_{2} (y, t) + λ U_{1} (y, t)) ψ (y, t, λ))}_{j k}, \\ ϕ_{j k} (x, t, λ) & = δ_{j k} + i \int_{- \infty}^{x} d y e^{- i λ^{2} (a_{j} - a_{k}) (x - y)} {((U_{2} (y, t) + λ U_{1} (y, t)) ϕ (y, t, λ))}_{j k} . \end{matrix}

(2.32)

Note that both integral equations are Volterra type equations.

The boundary conditions on the potentials

U_{2} (x, t)

and

U_{1} (x, t)

in Remark 3 ensure that equations (2.31) always have solutions provided the exponentials

e^{- i λ^{2} (a_{j} - a_{k}) (x - y)}

do not grow for

x \to \infty

x \to - \infty

. Obviously, this holds true for

Im λ^{2} = 0

, i.e. for

λ \in R \oplus i R

. As we will see below, this is the continuous spectrum of L (2.28).

However the equations (2.31) allow also for important exceptions. These will be easier seen if we use the equations (2.32). Let us, for example consider the equations for the first (resp. for the last) columns of the Jost solutions; So we have to consider the equations (2.32) for

k = 1

(resp. for

k = n

). Let us assume also that

Im λ^{2} < 0

, i.e.

λ

is in the second or fourth quadrant of the complex

λ

-plane. Then it is easy to check that the exponential factors in all the equations for

{(ψ)}_{j 1}

j = 1, \dots, n

decay for

x, y \to \pm \infty

. The same holds true also for the equations for

{(ϕ)}_{j n}

j = 1, \dots, n

. Therefore we find that the first column of

ψ

and the last column of

ϕ

allow analytic extensions to the second and fourth quadrants of

C

. Similarly we find, that the first column of

ϕ

and the last column of

ψ

allow analytic extensions for

Im λ^{2} > 0

, or to the first and third quadrants of

C

. The other columns of

ϕ

and

ψ

are defined only on the continuous spectrum of L. Indeed, the corresponding set of equations (2.32) some of the exponential factors will decay, but others will grow up.

Nevertheless our aim will be to demonstrate that one can construct fundamental analytic solutions (FAS) for L. These will be

n \times n

matrix solutions, one of which

χ^{+} (x, t, λ)

allows analytic extension for

Im λ^{2} > 0

and the other one

χ^{-} (x, t, λ)

– for

Im λ^{2} < 0

. This can be done using Shabat’s method [106,107] based on proper modification of the integral equations (2.32). So let

χ^{\pm} (x, t, λ)

be fundamental solutions of L, i.e.

L χ^{+} (x, t, λ) = 0

and let us introduce

χ^{\pm} (x, t, λ) = χ^{\pm} (x, t, λ) e^{i λ^{2} J x}

. These solutions will be different from the Jost solutions, because, as we will see, their behavior for

x \to \pm \infty

is different.

Following Shabat’s idea we define

χ^{+} (x, t, λ)

as the solution of the following set of integral equations:

\begin{matrix} χ_{j k}^{+} (x, t, λ) & = δ_{j k} + i \int_{- \infty}^{x} d y e^{- i λ^{2} (a_{j} - a_{k}) (x - y)} {((U_{2} (y, t) + λ U_{1} (y, t)) χ^{+} (y, t, λ))}_{j k}, j \geq k \\ χ_{j k}^{+} (x, t, λ) & = i \int_{\infty}^{x} d y e^{- i λ^{2} (a_{j} - a_{k}) (x - y)} {((U_{2} (y, t) + λ U_{1} (y, t)) χ^{+} (y, t, λ))}_{j k}, j < k . \end{matrix}

(2.33)

Likewise,

χ^{-} (x, t, λ)

are defined as the solution of the following set of integral equations:

\begin{matrix} χ_{j k}^{-} (x, t, λ) & = δ_{j k} + i \int_{- \infty}^{x} d y e^{- i λ^{2} (a_{j} - a_{k}) (x - y)} {((U_{2} (y, t) + λ U_{1} (y, t)) χ^{-} (y, t, λ))}_{j k}, j \leq k \\ χ_{j k}^{-} (x, t, λ) & = i \int_{\infty}^{x} d y e^{- i λ^{2} (a_{j} - a_{k}) (x - y)} {((U_{2} (y, t) + λ U_{1} (y, t)) χ^{-} (y, t, λ))}_{j k}, j > k . \end{matrix}

(2.34)

Note that the only difference with the equations (2.32) is in index inequalities in the right hand sides; this changes the signs of the factors

a_{j} - a_{k}

There is an alternative possibility to introduce FAS with a minor change of the integral equations (2.33) and (2.34)

\begin{matrix} χ_{j k}^{+,'} (x, t, λ) & = i \int_{\infty}^{x} d y e^{- i λ^{2} (a_{j} - a_{k}) (x - y)} {((U_{2} (y, t) + λ U_{1} (y, t)) χ^{+,'} (y, t, λ))}_{j k}, j > k \\ χ_{j k}^{+,'} (x, t, λ) & = δ_{j k} + i \int_{- \infty}^{x} d y e^{- i λ^{2} (a_{j} - a_{k}) (x - y)} {((U_{2} (y, t) + λ U_{1} (y, t)) χ^{+,'} (y, t, λ))}_{j k}, j \leq k . \end{matrix}

(2.35)

Likewise,

χ^{-} (x, t, λ)

are defined as the solution of the following set of integral equations:

\begin{matrix} χ_{j k}^{-,'} (x, t, λ) & = i \int_{- \infty}^{x} d y e^{- i λ^{2} (a_{j} - a_{k}) (x - y)} {((U_{2} (y, t) + λ U_{1} (y, t)) χ^{-,'} (y, t, λ))}_{j k}, j < k \\ χ_{j k}^{-,'} (x, t, λ) & = δ_{j k} + i \int_{\infty}^{x} d y e^{- i λ^{2} (a_{j} - a_{k}) (x - y)} {((U_{2} (y, t) + λ U_{1} (y, t)) χ^{-,'} (y, t, λ))}_{j k}, j \geq k . \end{matrix}

(2.36)

Figure 1. The continuous spectrum of the Lax operator (2.28) and the contour of the related Riemann-Hilbert problem

R \cup i R

Figure 1. The continuous spectrum of the Lax operator (2.28) and the contour of the related Riemann-Hilbert problem

R \cup i R

Now we have to establish the linear relations between the FAS and the Jost solutions.

The first consequence of the equations (2.33), (2.34), (2.35) and (2.36) concerns the limits of their diagonal matrix elements for

x \to \infty

, namely:

\begin{matrix} lim_{x \to \infty} diag (χ^{\pm} (x, t, λ)) = D^{\pm} (λ) = diag (D_{1}^{\pm} (λ), D_{2}^{\pm} (λ), \dots, D_{n}^{\pm} (λ)), \\ lim_{x \to \infty} diag (χ^{\pm,'} (x, t, λ)) = D^{\pm,'} (λ) = diag (D_{1}^{\pm,'} (λ), D_{2}^{\pm,'} (λ), \dots, D_{n}^{\pm,'} (λ)), \end{matrix}

(2.37)

Obviously the matrices

D^{+} (λ)

and

D^{+,'} (λ)

are x and t independent and in addition they are analytic for

Im λ^{2} > 0

; analogously

D^{-} (λ)

and

D^{-,'} (λ)

are analytic for

Im λ^{2} < 0

Next we find that:

\begin{matrix} lim_{x \to - \infty} e^{i λ^{2} J x} χ^{\pm} (x, t, λ) & = S^{\pm} (t, λ), & lim_{x \to \infty} e^{i λ^{2} J x} χ^{\pm} (x, t, λ) & = T^{\mp} (t, λ) D^{\pm} (λ), \\ lim_{x \to \infty} e^{i λ^{2} J x} χ^{\pm,'} (x, t, λ) & = T^{\mp} (t, λ), & lim_{x \to - \infty} e^{i λ^{2} J x} χ^{\pm,'} (x, t, λ) & = S^{\pm} (t, λ) {\hat{D}}^{\pm} (λ), \end{matrix}

(2.38)

where

S^{+} (t, λ)

and

T^{+} (t, λ)

(resp.

S^{-} (t, λ)

and

T^{-} (t, λ)

) are upper-triangular matrices (resp. lower triangular) with 1 on the diagonal. Remember, since the Jost solutions and the FAS belong to the Lie group

G

, then all the limits in (2.38) must also belong to

G

From (2.38) it follows that the FAS

χ_{j k}^{\pm} (x, t, λ)

are related to the Jost solutions as follows:

\begin{matrix} χ^{\pm} (x, t, λ) & = ϕ (x, t, λ) S^{\pm} (t, λ), & χ^{\pm} (x, t, λ) & = ψ (x, t, λ) T^{\mp} (t, λ) D^{\pm} (λ), \\ χ^{\pm,'} (x, t, λ) & = ϕ (x, t, λ) S^{\pm} (t, λ) {\hat{D}}^{\pm}, & χ^{\pm,'} (x, t, λ) & = ψ (x, t, λ) T^{\mp} (t, λ), \end{matrix}

(2.39)

More detailed analysis shows that these triangular and diagonal matrices are in fact the factors in the Gauss decompositions of the scattering matrix

T (t, λ)

\begin{matrix} T (t, λ) = T^{-} (t, λ) D^{+} (λ) {\hat{S}}^{+} (t, λ) = T^{+} (t, λ) D^{-} (λ) {\hat{S}}^{-} (t, λ), \end{matrix}

(2.40)

see Appendix B.

Remark 4.

Let us consider cubic pencils in λ assuming that the leading terms

λ^{3} J

and

λ^{3} K

of L and M respectively are such that J and K have different real eigenvalues. Their FAS can be constructed quite analogously as we did above for the quadratic pencils. The substantial difference between the cubic and the quadratic cases are in the regions of analyticity. The solutions

χ^{+} (x, t, λ)

(resp.

χ^{-} (x, t, λ)

) for the cubic pencils are analytic for

I m λ^{3} > 0

(resp.

I m λ^{3} < 0

), see sectors

Ω_{0} \cup Ω_{2} \cup Ω_{4}

(resp. sectors

Ω_{1} \cup Ω_{3} \cup Ω_{5}

) on Figure 4.

2.5. The time-dependence of $T (t, λ)$

The Lax representation of a given NLEE requires the commutativity of two ordinary differential operators:

\begin{matrix} [L, M] = 0 \end{matrix}

(2.41)

where L is given by (2.28). We assume that M has the same form as L:

\begin{matrix} M ψ \equiv (i \frac{\partial}{\partial t} + V_{2} (x, t) + λ V_{1} (x, t) - λ^{2} K) ψ (x, t, λ) = ψ (x, t, λ) C (λ) . \end{matrix}

(2.42)

Below we will treat also M-operators that are higher order polynomials of

λ

. The first remark is that the commutativity condition (2.41) must hold identically with respect to

λ

. Note also that (2.41) holds true for any

C (λ)

in (2.42). We will use this fact to determine the t-dependence of the scattering matrix

T (t, λ)

Indeed, let us consider

M ϕ (x, t, λ) = ϕ (x, t, λ) C (λ)

where

ϕ (x, t, λ)

is the Jost solution in (2.29) and let us take the limit

x \to - \infty

in (2.42). Due to the vanishing boundary conditions we get:

\begin{matrix} (i \frac{\partial}{\partial t} - λ^{2} K) e^{- i λ^{2} J x} = e^{- i λ^{2} J x} C (λ), \end{matrix}

(2.43)

i.e.

C (λ) = - λ^{2} K

. Thus we determined the function

C (λ)

for this choice of the M-operator. Let us now take the limit

x \to \infty

in (2.42). This gives:

\begin{matrix} (i \frac{\partial}{\partial t} - λ^{2} K) (e^{- i λ^{2} J x} T (t, λ)) = e^{- i λ^{2} J x} T (t, λ) C (λ), \end{matrix}

(2.44)

i.e.

\begin{matrix} i \frac{\partial T}{\partial t} - λ^{2} [K, T (t, λ)] = 0 . \end{matrix}

(2.45)

From eq. (2.45) there follow also the time-dependence of the Gauss factors:

\begin{matrix} i \frac{\partial S^{\pm}}{\partial t} - λ^{2} [K, S^{\pm} (t, λ)] = 0, i \frac{\partial T^{\pm}}{\partial t} - λ^{2} [K, T^{\pm} (t, λ)] = 0 . \end{matrix}

(2.46)

For the diagonal factors we get:

\begin{matrix} i \frac{\partial D^{\pm} (λ)}{\partial t} = 0 . \end{matrix}

(2.47)

In other words we have two sets of functions of

λ

that are t-independent. Note also, that from the canonical normalization of FAS there follows that

{lim}_{λ \to \infty} D_{k}^{\pm} = 1

. Obviously they will generate integrals of motion. Indeed, let us consider their asymptotic expansions:

\begin{matrix} ln D_{k}^{\pm} = \sum_{s = 1}^{\infty} λ^{- s} I_{k} . k = 1, \dots, n; \end{matrix}

(2.48)

Obviously

d I_{k} / d t = 0

. As we shall see below each of the integrals

I_{k}

can be expressed as an integral of the potentials

U_{2}

and

U_{1}

. The advantage of the choice (2.47) is that the integrands of

I_{k}

are local, i.e. they depend only on

U_{2}

and

U_{1}

and their x-derivatives.

Above we described the time dependence for the N-wave equations related to the Lax operators (2.28). In general, each of these operators generates a hierarchy of NLEE. Indeed the NLEE is determined by fixing up along with L also the second operator M in the Lax pair. For example, if we want to treat DNLS-type equations we need, to specify two things. First, the structure of L must be compatible with a Hermitian symmetric space [21], and second, the potential of M must polynomial of order 4 in

λ

with leading term

λ^{4} J

. Then

C (λ) = - λ^{4} J

and the time dependence of the scattering data will be given by:

\begin{matrix} i \frac{\partial T}{\partial t} - λ^{4} [J, T (t, λ)] = 0 . \end{matrix}

(2.49)

Then from eq. (2.49) there follow also the time-dependence of the Gauss factors:

\begin{matrix} i \frac{\partial S^{\pm}}{\partial t} - λ^{4} [J, S^{\pm} (t, λ)] = 0, i \frac{\partial T^{\pm}}{\partial t} - λ^{4} [J, T^{\pm} (t, λ)] = 0 . \end{matrix}

(2.50)

More details about this we will give in the Section devoted to the DNLS type equations.

3. RHP and integrable NLEE

The simplest nontrivial Lax operators, i.e. the ones that are linear in

λ

have been studied in detail by now, see [18,63,104] and the numerous references therein. That is why in this paper we will concentrate mostly on Lax operators that are quadratic in

λ

like (2.28). In the previous section we constructed the FAS of this operators for the case of vanishing boundary conditions and the result was as follows. The continuous spectrum of L fills up the union of the real and purely imaginary axis

R \cup i R

, see Figure 1; The analyticity regions for the FAS of (2.28) are

Q_{1} \cup Q_{3}

for

χ^{+}

and

Q_{2} \cup Q_{4}

for

χ^{-}

, where by

Q_{k}

k = 1, 2, 3, 4

we denote the quadrants of the complex

λ

-plane.

3.1. Uniqueness of the regular solution of RHP

The FAS derived above are also linearly related to each other. From equations (2.39) there easily follows that:

\begin{matrix} χ^{+} (x, t, λ) & = χ^{-} (x, t, λ) G_{0} (t, λ), & G_{0} & = {\hat{S}}^{-} S^{+} (t, λ), \\ χ^{+,'} (x, t, λ) & = χ^{-,'} (x, t, λ) G_{0}^{'} (t, λ), & G_{0}^{'} & = {\hat{S}}^{-,'} S^{+,'} (t, λ), \end{matrix} λ \in R \oplus i R .

(3.1)

These relations allow us to relate the Lax pair to a RHP. Indeed, our FAS are not very suitable because for large

x \to \pm \infty

they strongly oscillate (like

exp (- i λ^{2} J x)

). In order to avoid these singularities we introduce

ξ^{\pm} (x, t, λ) = χ^{\pm} (x, t, λ) exp (i λ^{2} J x)

. Then the RHP can be formulated as:

\begin{matrix} ξ^{+} (x, t, λ) & = ξ^{-} (x, t, λ) G (x, t, λ), & G (x, t, λ) & = e^{- i λ^{2} J x} G_{0} (t, λ) e^{i λ^{2} J x}, \\ ξ^{+,'} (x, t, λ) & = ξ^{-,'} (x, t, λ) G^{'} (x, t, λ), & G^{'} (x, t, λ) & = e^{- i λ^{2} J x} G_{0}^{'} (t, λ) e^{i λ^{2} J x}, \end{matrix} λ \in R \oplus i R .

(3.2)

which allows canonical normalization:

\begin{matrix} lim_{λ \to \infty} ξ^{+} (x, t, λ) = lim_{λ \to \infty} ξ^{-} (x, t, λ) = 1 1 \end{matrix}

(3.3)

In the previous Subsections we outlined how, starting with the Lax representation one can derive the RHP. Here, following Zakharov and Shabat we will demonstrate how starting from the RHP one can derive the relevant Lax representation. We will use slightly modified RHP with canonical normalization:

\begin{matrix} ξ^{+} (x, t, λ) & = ξ^{-} (x, t, λ) G (x, t, λ), & λ & \in R \oplus i R, \\ i \frac{\partial G}{\partial x} & - λ^{2} [J, G (x, t, λ)] = 0, & i \frac{\partial G}{\partial t} & - λ^{2} [K, G (x, t, λ)] = 0, \\ lim_{λ \to \pm \infty} & ξ^{\pm} (x, t, λ) = 1 1 \end{matrix}

(3.4)

compare with (3.2).

We will say that the solution

ξ_{0}^{\pm} (x, t, λ)

is regular if

ξ_{0}^{+} (x, t, λ)

and

ξ_{0}^{-} (x, t, λ)

have neither zeroes nor singularities in their regions of analyticity.

Corollary 2.

The RHP (3.4) with canonical normalization has unique regular solution.

Proof.

Let us assume that

ξ_{1}^{\pm} (x, t, λ)

is another regular solution to the RHP (3.4). Consider:

\begin{matrix} g_{0}^{\pm} (x, t, λ) = ξ_{0}^{\pm} (x, t, λ) {\hat{ξ}}_{1}^{\pm} (x, t, λ); \end{matrix}

(3.5)

we remind that

\hat{ξ} \equiv ξ^{- 1}

. Using (3.4) we easily find that:

\begin{matrix} g_{0}^{+} (x, t, λ) = ξ_{0}^{+} (x, t, λ) {\hat{ξ}}_{1}^{+} (x, t, λ) = ξ_{0}^{-} (x, t, λ) G (x, t, λ) \hat{G} (x, t, λ) {\hat{ξ}}_{1}^{-} (x, t, λ) = g_{0}^{-} (x, t, λ), \end{matrix}

(3.6)

i.e.

g_{0}^{+} (x, t, λ)

is analytic in the whole complex plane

λ

. In addition, it tends to

1 1

for

λ \to \infty

. Then according to the great Liouville theorem,

g_{0}^{+} (x, t, λ) = g_{0}^{-} (x, t, λ) = 1 1

. □

Remark 5.

The RHP (3.4) obviously allows the trivial regular solution when

ξ_{0}^{+} = ξ_{0}^{-} = 1 1

. In this case the corresponding FAS of L will take the form:

\begin{matrix} χ_{0}^{+} (x, t, λ) = χ_{0}^{-} (x, t, λ) = e^{- i λ^{2} J x - i λ^{2} K t} . \end{matrix}

(3.7)

3.2. Zakharov-Shabat theorem

Theorem 2

(Zakharov-Shabat [135]). Let

ξ^{\pm} (x, t, λ)

satisfy the RHP (3.4). Then

χ^{\pm} (x, t, λ) = ξ^{\pm} (x, t, λ) e^{- i λ^{2} J x}

are FAS of L (2.28) and M (2.42).

Proof.

Let us introduce the functions

\begin{matrix} g_{1}^{\pm} (x, t, λ) & = i \frac{\partial ξ^{\pm}}{\partial x} {\hat{ξ}}^{\pm} (x, t, λ) + λ^{2} ξ^{\pm} (x, t, λ) J {\hat{ξ}}^{\pm} (x, t, λ), \\ g_{2}^{\pm} (x, t, λ) & = i \frac{\partial ξ^{\pm}}{\partial t} {\hat{ξ}}^{\pm} (x, t, λ) + λ^{2} ξ^{\pm} (x, t, λ) K {\hat{ξ}}^{\pm} (x, t, λ), \end{matrix}

(3.8)

Then, from eqs. (3.4) we find:

\begin{matrix} g_{1}^{+} (x, t, λ) & = i \frac{\partial (ξ^{-} G)}{\partial x} \hat{G} {\hat{ξ}}^{-} (x, t, λ) + λ^{2} ξ^{-} (x, t, λ) G J \hat{G} {\hat{ξ}}^{-} (x, t, λ) \\ = i \frac{\partial ξ^{-}}{\partial x} {\hat{ξ}}^{-} (x, t, λ) + ξ^{-} (x, t, λ) (i \frac{\partial G}{\partial x} \hat{G} (x, t, λ) + λ^{2} G J \hat{G} (x, t, λ)) {\hat{ξ}}^{-} (x, t, λ) \\ = i \frac{\partial ξ^{-}}{\partial x} {\hat{ξ}}^{-} (x, t, λ) + λ^{2} ξ^{-} (x, t, λ) J {\hat{ξ}}^{-} (x, t, λ) = g_{1}^{-} (x, t, λ) . \end{matrix}

(3.9)

Thus

g_{1}^{\pm} (x, t, λ)

are analytic in the whole complex plane

C

. The canonical normalization of the RHP means, however, that

g_{1}^{\pm} (x, t, λ)

are singular for

λ \to \infty

. More specifically,

g_{1}^{\pm} (x, t, λ) - λ^{2} J

is linear function of

λ

. Using again the great Liouville theorem we conclude that there must exist the linear in

λ

function

- U_{2} (x, t) - λ U_{1} (x, t)

such that:

\begin{matrix} g_{1}^{\pm} (x, t, λ) - λ^{2} J = - U_{2} (x, t) - λ U_{1} (x, t), \end{matrix}

(3.10)

i.e.

\begin{matrix} g_{1}^{\pm} (x, t, λ) + U_{2} (x, t) + λ U_{1} (x, t) - λ^{2} J = 0 . \end{matrix}

(3.11)

Multiplying both sides of (3.11) by

ξ^{\pm} (x, t, λ)

we find that the solutions to the RHP (3.4) must satisfy the ODE:

\begin{matrix} i \frac{\partial ξ^{\pm}}{\partial x} + (U_{2} (x, t) + λ U_{1} (x, t)) ξ^{\pm} (x, t, λ) - λ^{2} [J, ξ^{\pm} (x, t, λ)] = 0 . \end{matrix}

(3.12)

It remains to insert

χ^{\pm} (x, t, λ) = ξ^{\pm} (x, t, λ) e^{- i λ^{2} J x}

into eq. (3.12) to obtain that

χ^{\pm} (x, t, λ)

are FAS of eq. (2.28).

Applying the same considerations on the second function

g_{2}^{\pm} (x, t, λ)

we find that it is analytic on the whole complex

λ

-plane and has the form:

\begin{matrix} g_{2}^{\pm} (x, t, λ) + V_{2} (x, t) + λ V_{1} (x, t) - λ^{2} K = 0 . \end{matrix}

(3.13)

This means that

ξ^{\pm} (x, t, λ)

must satisfy also

\begin{matrix} i \frac{\partial ξ^{\pm}}{\partial t} + (V_{2} (x, t) + λ V_{1} (x, t)) ξ^{\pm} (x, t, λ) - λ^{2} [K, ξ^{\pm} (x, t, λ)] = 0 . \end{matrix}

(3.14)

and therefore

χ^{\pm} (x, t, λ) = ξ^{\pm} (x, t, λ) e^{- i λ^{2} J x}

are FAS of the M-operator (2.42). □

Remark 6.

Zakharov-Shabat theorem initially was proven for Lax operators with linear dependence on the spectral parameter λ. The above proof is elementary generalization of the original idea. Obviously it can easily be extended for any polynomial Lax pair, see e.g. (2.24). The details of these generalizations are left to the readers.

4. Mikhailov’s reduction groups and the contours of RHP

4.1. General theory

The reduction groups introduced by [96] are a powerful tool for deriving new integrable equations, admitting Lax representation. It has been substantially developed since its discovery, see [9,22,31,37,88,89,120,127]. Mikhailov’s reductions were used in the seminal paper of Drinfeld and Sokolov [16] as an important tool to analyze the gradings of simple Lie algebras and their consequences for the integrable equations. It also stimulated the development of the infinite dimensional Lie algebras and Kac-Moody algebras [8,16,67].

A reduction group

G_{R}

is a finite group acting on the solution set of (2.24) which preserves the Lax representation [96], i.e. it ensures that the reduction constraints are automatically compatible with the evolution.

G_{R}

must have two realizations: i)

G_{R} \subset Aut g

and ii)

G_{R} \subset Conf C

, i.e. as conformal mappings of the complex

λ

-plane. To each

g_{k} \in G_{R}

we relate a reduction condition for the Lax pair as follows [96]:

C_{k} (L (Γ_{k} (λ))) = η_{k} L (λ), C_{k} (M (Γ_{k} (λ))) = η_{k} M (λ),

(4.1)

where

C_{k} \in Aut g

and

Γ_{k} (λ) \in Conf C

are the images of

g_{k}

and

η_{k} = 1

- 1

depending on the choice of

C_{k}

. Since

G_{R}

is a finite group then for each

g_{k}

there exist an integer

N_{k}

such that

g_{k}^{N_{k}} = 1 1

The finite subgroups of

Conf (C)

were classified by Klein, see [13]. They consist of two infinite series: i)

Z_{h}

- cyclic group of order h; ii)

D_{h}

- dihedral group of order

2 h

; and the groups related to the Platonic solids: tetrahedron, cube, octahedron, dodecahedron and icosahedron. In what follows we will most attention to

Z_{h}

and

D_{h}

; although examples of systems with Platonic solids as group of reductions are also known, [82,88,89,95]

It is important to note that the form of the equations depends not only on the chosen reduction group, but also on its realization.

It is well known, that every finite group can be imbedded as a subgroup of some finite symmetric group

S_{m}

; this is the group of permutations of the numbers

{1, 2, \dots, m}

. The group

Z_{h}

consists of all cyclic permutations of the numbers

1, 2, \dots, h

The symmetric group

A_{h}

is the commutator subgroup of the symmetric group

S_{h}

with index 2 and has therefore

h! / 2

elements. It is the kernel of the signature group homomorphism

sgn : S_{h} \to {1, - 1}

explained under symmetric group. It is isomorphic to

D_{h}

Generically each of these groups is rigorously defined by their genetic codes. In other words one introduces one or more generating elements of the group and defines the relationes they must satisfy. For the two types of groups we have:

\begin{matrix} S_{2} & : s_{1}^{2} & = 1 1, \\ S_{h} & : s_{1}^{h} & = 1 1, \\ D_{h} & : s_{1}^{2} & = 1 1, s_{2}^{2} = 1 1, {(s_{1} s_{2})}^{h} = 1 1, \end{matrix}

(4.2)

These are the formal definitions of these groups. Below we will outline their realizations as subgroups of the group of automorphisms

Aut g

of the algebra

g

, as well as subgroups of the conformal group. These realizations are specific both for the explicit form of the corresponding NLEE as well as for the spectral properties of their Lax operators.

Other important facts are the orbits and the fundamental domains of the groups. Each element

Γ_{k}

G_{R}

is of finite order, i.e. there exist an integer

n_{k}

such that

Γ_{k}^{n_{k}} = 1 1

. Acting on a point, say

λ_{0}

it produces an orbit in the complex

λ

-plane consisting of the points

{Γ_{k}^{s} λ_{0}, s = 1, \dots, n_{k}

. Then by fundamental domain of

G_{R}

we mean the manifold

A_{G_{R}}

which contains only one point of each orbit. The orbits depend not only on the element

Γ_{k}

, but also on the specific realization of

G_{R}

. Below we will specify the fundamental domains for each of the realizations of the reduction groups.

We start with

Z_{2}

-reductions, (or involutions) for two of the best known classes of NLEE:

4.2. Involutive reductions

We will start with typical

Z_{2}

reductions (involutions) on the Lax representations (2.24) [96]:

\begin{matrix} 1) & C_{1} (U^{†} (κ_{1} (λ))) & = U (λ), & C_{1} (V^{†} (κ_{1} (λ))) & = V (λ), \\ 2) & C_{2} (U^{T} (κ_{2} (λ))) & = - U (λ), & C_{2} (V^{T} (κ_{2} (λ))) & = - V (λ), \\ 3) & C_{3} (U^{*} (κ_{1} (λ))) & = - U (λ), & C_{3} (V^{*} (κ_{1} (λ))) & = - V (λ), \\ 4) & C_{4} (U (κ_{2} (λ))) & = U (λ), & C_{4} (V (κ_{2} (λ))) & = V (λ), \end{matrix}

(4.3)

C_{j}

above we have denoted involutive automorphisms

C_{j}^{2} = 1 1

of the simple Lie algebra

g

in which

U (x, t, λ)

and

v (x, t, λ)

take values.

Working with Lax operators which are quadratic pencils it is natural to impose two basic symmetries: i) the Hermitian symmetry 1) in (4.3) with

C_{1} = 1 1

and

κ_{1} (λ) = λ^{*}

; and the symmetry 2) in (4.3) mapping

λ \to - λ

. A third involution iii) may appear if we request in addition that U and V belong to a simple Lie algebra of the series

B_{r}

C_{r}

D_{r}

. This means:

\begin{matrix} U (x, t, λ) = - S_{a} U^{T} (x, t, λ) S_{a}^{- 1}, V (x, t, λ) = - S_{a} V^{T} (x, t, λ) S_{a}^{- 1}, \end{matrix}

where the matrices

S_{a}

are introduced in the Appendix A.

Note that equations (4.3) in fact take into account the all typical external automorphisms of the simple Lie algebras. Therefore we need to consider only those realizations of

C_{j}

which are elements of the Weyl group

W_{g}

g

, or belong to the Cartan subgroup of

g

. This means that

C_{j}

may have the form:

\begin{matrix} C_{j} \to S_{j} \equiv S_{β_{1}} S_{β_{2}} \dots S_{β_{k}}, or C_{j} \to h_{j} = exp (π i H_{{\vec{b}}_{j}}), \end{matrix}

(4.4)

where the roots

β_{1}, β_{2}, \dots, β_{k}

are all orthogonal to each other. This will ensure that

S_{j}

is an involution:

S_{j}^{2} = 1 1

. Similarly the vector

{\vec{b}}_{j}

must be such that

{\vec{b}}_{j} = \sum_{s = 1}^{r} k_{j, s} ω_{s}

, where

k_{j, s}

are nonnegative integers and

ω_{s}

are the fundamental weights of

g

. Then the similarity transformations with

h_{j}

will also have the property

h_{j}^{2} = 1 1

Figure 2. Examples of involutions for Lax operators linear in

λ

. The continuous spectrum fills up the real axis; the FAS

χ^{\pm} (x, t, λ)

are analytic for

λ \in C_{\pm}

respectively. Left panel: for reductions of types

C_{1}

and

C_{3}

the discrete eigenvalues come in complex-conjugate pairs, shown by × and ∘; Right panel: for reductions of type

C_{2}

and

C_{3}

there are two types of eigenvalues: pairs of purely imaginary ones

\pm i λ_{0}

shown by ∘; quadruplets

\pm μ

and

\pm μ^{*}

shown by ×

Figure 2. Examples of involutions for Lax operators linear in

λ

. The continuous spectrum fills up the real axis; the FAS

χ^{\pm} (x, t, λ)

are analytic for

λ \in C_{\pm}

respectively. Left panel: for reductions of types

C_{1}

and

C_{3}

the discrete eigenvalues come in complex-conjugate pairs, shown by × and ∘; Right panel: for reductions of type

C_{2}

and

C_{3}

there are two types of eigenvalues: pairs of purely imaginary ones

\pm i λ_{0}

shown by ∘; quadruplets

\pm μ

and

\pm μ^{*}

shown by ×

Figure 3. Examples of involutions for Lax operators quadratic in

λ

; the continuous spectrum fills up

R \oplus i R

. Left panel: for reductions of types

C_{1}

and

C_{3}

the discrete eigenvalues come in complex-conjugate pairs, shown by ×; for reductions of type

C_{2}

they come in pairs shown by ∘. In both cases the contour of the RHP is the real axis. Right panel: the first involution maps

λ \to λ^{- 1}

; or to

λ \to - λ^{- 1}

; the second one maps

μ \to μ^{*, - 1}

or to

μ \to - μ^{*, - 1}

. In both cases the contour of the RHP is given by the unit circle

{| λ |}^{2} = 1

,119]

Figure 3. Examples of involutions for Lax operators quadratic in

λ

; the continuous spectrum fills up

R \oplus i R

. Left panel: for reductions of types

C_{1}

and

C_{3}

the discrete eigenvalues come in complex-conjugate pairs, shown by ×; for reductions of type

C_{2}

they come in pairs shown by ∘. In both cases the contour of the RHP is the real axis. Right panel: the first involution maps

λ \to λ^{- 1}

; or to

λ \to - λ^{- 1}

; the second one maps

μ \to μ^{*, - 1}

or to

μ \to - μ^{*, - 1}

. In both cases the contour of the RHP is given by the unit circle

{| λ |}^{2} = 1

,119]

Figure 4. Left panel: contour of a RHP with

Z_{3}

symmetry. The FAS

χ^{+} (x, λ)

is analytic in the sectors

Ω_{0} \cup Ω_{2} \cup Ω_{4}

, while

χ^{-} (x, λ)

is analytic in the sectors

Ω_{1} \cup Ω_{3} \cup Ω_{5}

; Right panel: contour of a RHP with

Z_{3}

symmetry with additional involution mapping

λ \to λ^{- 1}

Figure 4. Left panel: contour of a RHP with

Z_{3}

symmetry. The FAS

χ^{+} (x, λ)

is analytic in the sectors

Ω_{0} \cup Ω_{2} \cup Ω_{4}

, while

χ^{-} (x, λ)

is analytic in the sectors

Ω_{1} \cup Ω_{3} \cup Ω_{5}

; Right panel: contour of a RHP with

Z_{3}

symmetry with additional involution mapping

λ \to λ^{- 1}

Another important factor for the correct realization of the reduction groups is

N_{1}

– the leading power

λ

in the Lax operator (2.24). In the majority of cases people consider generalized Zakharov-Shabat systems, i.e.

N_{1} = 1

. Let us combine this choice with real-valued eigenvalues of J and vanishing boundary conditions on

Q (x, t)

. Then the analysis of the corresponding integral equations for the Jost solutions show that the FAS

χ_{0}^{+} (x, t, λ)

(resp.

χ_{0}^{-} (x, t, λ)

) of

L_{0}

have analyticity properties for

Im λ > 0

(resp.

Im λ < 0

). That means that the contour of the corresponding RHP coincides with the real axis

R

. Then the spectrum of

L_{0}

consists of continuous part filling up

R

and pairs of complex-valued discrete eigenvalues

λ_{j}^{\pm} \in C_{\pm}

, see Figure 3. As fundamental domain of this group we can choose the upper half-plane

C_{+}

of the complex

λ

-plane.

In what follows we will concentrate on Lax operators with

N_{1} > 1

; most often we will have

N_{1} = 2

. In Section 2 we proved that

χ^{+} (x, t, λ)

(resp.

χ^{-} (x, t, λ)

) of L (2.28) have analyticity properties for

Im λ^{2} > 0

(resp.

Im λ^{2} < 0

). That means that the contour of the corresponding RHP coincides with the union of the real and purely imaginary axis

R \oplus i R

. Then the spectrum of L consists of continuous part filling up

R \oplus i R

. Typically such Lax operators have additional symmetry

λ \leftrightarrow - λ

, so the discrete eigenvalues come in quadruplets

\pm λ_{j}^{+} \in Q_{1} \cup Q_{3}

and

\pm λ_{j}^{-} \in Q_{2} \cup Q_{4}

, see Figure 3. Effectively in these cases we deal with

Z_{2} \otimes Z_{2}

= Then as fundamental domain of this realizations of the group we can choose the first quadrant

Q_{1}

of the complex

λ

-plane.

4.3. $Z_{h}$ reduction groups

We will consider here the cyclic groups

Z_{h}

of order

h > 2

. These groups have only one generating elements:

\begin{matrix} Z_{h} : s^{h} = 1 1 . \end{matrix}

(4.5)

The cyclic group has h elements:

1 1, s^{k}, k = 1, \dots, h - 1

; typically its realization on the complex

λ

-plane is given by

s (λ) = λ ω

, where

ω = exp (2 π i / h)

4.4. $D_{h}$ reduction groups

We will consider here the dihedral groups

D_{h}

of order

h > 2

. These groups have two generating elements:

\begin{matrix} D_{h} : r^{2} = s^{h} = 1 1, s r s^{- 1} = s^{- 1} . \end{matrix}

(4.6)

The dihedral group

D_{h}

has

2 h

elements:

{s^{k}, r s^{k}, k = 1, \dots, h}

and allows several inequivalent realization on the complex

λ

-plane. Some of them are:

\begin{matrix} (i) & s (λ) & = λ ω, & r (λ) & = ϵ λ^{*}, & (i i) & s (λ) & = λ ω, & r (λ) & = \frac{ϵ}{λ^{*}}, \\ (i i i) & s (λ) & = λ ω, & r (λ) & = ϵ λ, & (i v) & s (λ) & = λ ω, & r (λ) & = \frac{ϵ}{λ}, \end{matrix}

where again

ω = exp (2 π i / h)

and

ϵ = \pm 1

. An important realization in the case of a

D_{2}

reduction group is given by

(v) s (λ) = λ^{*}, r (λ) = \frac{ϵ}{λ} .

(4.7)

5. Parametrizing the RHP with canonical normalization

An important tool in our investigation is the theory of the simple Lie algebras and the methods of their gradings.

The reason to limit our selves with the simple Lie algebras is due to the fact, that we need to have unique solution of the inverse spectral problem of the Lax operator. The mapping between the potential and the scattering matrix for generic, linear in

λ

operators have been studied using the Wronskian relations [10,30,63]. They require the existence of a non-degenerate metric. A metric, characteristic for the Lie algebras is the famous Killing form, which is non-degenerate for the semi-simple Lie algebras. In fact we will limit ourselves by considering only the simple Lie algebras.

We will limit ourselves also by considering only two families of NLEE. The first family is known as the N-wave equations, discovered by Zakharov and Manakov [131], see Section 2.1 above. Typically they contain first order derivatives in both x and t and quadratic nonlinearities. In this Section we will describe a new class of N-wave equations whose Lax operators are both quadratic in

λ

[36,52]. We will see, that they have higher order nonlinearities.

The second family of NLEE we will focus on are the multi-component NLS (MNLS) equations. It is well known that they are related to the symmetric spaces [21]. Their Lax operators will also be quadratic in

λ

, so they will be multicomponent generalizations of the derivative MLS eq. [81] and GI equations [19,20,42,58].

5.1. Generic parametrization of the RHP with canonical normalization

We can introduce a parametrization for

ξ^{\pm} (x, t, λ)

using its asymptotic expansion:

\begin{matrix} ξ^{\pm} (x, t, λ) = exp (Q (x, t, λ)), Q (x, t, λ) = \sum_{s = 1}^{\infty} λ^{- s} Q_{s} (x, t) . \end{matrix}

(5.1)

Obviously, if we want that

ξ^{\pm} (x, t, λ)

be elements of a simple Lie group

G

, then the coefficients

Q_{s} (x, t)

must be elements of the corresponding simple Lie algebra

g

. In addition we request that

Q_{s}

provides local coordinates of the corresponding homogeneous space. Besides, the solution

ξ^{\pm} (x, t, λ)

is canonically normalized, because

\begin{matrix} lim_{λ \to \infty} ξ^{\pm} (x, t, λ) = 1 1 . \end{matrix}

(5.2)

The most general parametrization of

ξ^{\pm} (x, t, λ)

requires that

Q_{s} (x, t)

are generic elements of the algebra

g

. However such approach has a disadvantage: the corresponding NLEE involve too many independent functions. There are two ways to avoid it: first, we can fix up the gauge of the Lax operators; second we can and will impose reductions of Mikhailov type. Typically we will fix the gauge by requesting that the leading terms in the Lax operators are chosen as diagonal constant matrices, i.e. constant elements of the Cartan subalgebra

h

. Another important issue is to explain how, using

Q_{s} (x, t)

from eq. (5.1) we can parameterize any generic Lax pair related to that RHP.

Let us choose, following the ideas of Gel’fand and Dickey

\begin{matrix} L ψ & \equiv \frac{\partial ψ}{\partial x} - U (x, t, λ) ψ (x, t, λ) = 0, U (x, t, λ) = {(λ^{N_{1}} ξ J ξ^{- 1} (x, t, λ))}_{+}, \\ M ψ & \equiv \frac{\partial ψ}{\partial t} - V (x, t, λ) ψ (x, t, λ) = 0, V (x, t, λ) = {(λ^{N_{2}} ξ K ξ^{- 1} (x, t, λ))}_{+}, \end{matrix}

(5.3)

where the subscript + means that we retain only the non-negative in

λ

terms in the right hand sides of (5.3) and explain how one can calculate

U (x, t, λ)

and

V (x, t, λ)

. First, we remind that since

J, K \in h

and

ξ^{\pm} (x, t, λ) \in G

, then both

U (x, t, λ), V (x, t, λ) \in g

. From the general theory of Lie algebras we know that

\begin{matrix} ξ^{\pm} J {\hat{ξ}}^{\pm} (x, t, λ) & = J + \sum_{k = 1}^{\infty} \frac{λ^{- k}}{k!} {ad}_{Q}^{k} J = J - \sum_{k = 1}^{\infty} λ^{- k} V_{k} . \end{matrix}

(5.4)

where

{ad}_{Q} Y \equiv [Q, Y]

{ad}_{Q}^{2} \equiv [[Q, [Q, Y]]

etc. The first few coefficients in these expansions take the form:

\begin{matrix} V_{1} & = {ad}_{J} Q_{1}, V_{2} = {ad}_{J} Q_{2} - \frac{1}{2} {ad}_{Q_{1}}^{2} J, \\ V_{3} & = {ad}_{J} Q_{3} - \frac{1}{2} ({ad}_{Q_{1}} {a d}_{Q_{2}} + {ad}_{Q_{2}} {ad}_{Q_{1}}) J - \frac{1}{6} {ad}_{Q_{1}}^{3} J, \\ V_{4} & = {ad}_{J} Q_{4} - \frac{1}{2} ({ad}_{Q_{1}} {a d}_{Q_{3}} + {ad}_{Q_{2}}^{2} + {ad}_{Q_{3}} {ad}_{Q_{1}}) J \\ - \frac{1}{6} ({ad}_{Q_{1}}^{2} {ad}_{Q_{2}} + {ad}_{Q_{1}} {ad}_{Q_{2}} {ad}_{Q_{1}} + {ad}_{Q_{2}} {ad}_{Q_{1}}^{2}) J - \frac{1}{24} {ad}_{Q_{1}}^{4} J . \end{matrix}

(5.5)

Thus we see that

U (x, t, λ)

and

V (x, t, λ)

are parameterized by the first few coefficients

Q_{s}

Another formula from the general theory of Lie algebras which we will need below is:

\begin{matrix} i \frac{\partial ξ^{\pm}}{\partial x} {\hat{ξ}}^{\pm} (x, t, λ) & = i \frac{\partial Q}{\partial x} + i \sum_{k = 1}^{\infty} \frac{1}{(k + 1)!} {ad}_{Q}^{k} \frac{\partial Q}{\partial x} = i \sum_{k = 1}^{\infty} X_{k} λ^{- k}, \end{matrix}

(5.6)

In general:

\begin{matrix} i X_{k} & = i \frac{\partial Q_{k}}{\partial x} + i \sum_{s = 1}^{k} \frac{1}{(s + 1)!} \sum_{j_{1} + j_{2} + \dots + j_{s} = k}^{} {ad}_{Q_{j_{1}}} {ad}_{Q_{j_{2}}} \dots {ad}_{Q_{j_{s - 1}}} \frac{\partial Q_{j_{s}}}{\partial x} . \end{matrix}

(5.7)

The first few coefficients are:

\begin{matrix} X_{1} & = \frac{\partial Q_{1}}{\partial x}, X_{2} = \frac{\partial Q_{2}}{\partial x} + \frac{1}{2} {ad}_{Q_{1}} \frac{\partial Q_{1}}{\partial x}, \\ X_{3} & = \frac{\partial Q_{3}}{\partial x} + \frac{1}{2} ({ad}_{Q_{1}} \frac{\partial Q_{2}}{\partial x} + {ad}_{Q_{2}} \frac{\partial Q_{1}}{\partial x}) + \frac{1}{6} {ad}_{Q_{1}}^{2} \frac{\partial Q_{1}}{\partial x}, \\ X_{4} & = \frac{\partial Q_{4}}{\partial x} + \frac{1}{2} ({ad}_{Q_{1}} \frac{\partial Q_{3}}{\partial x} + {ad}_{Q_{2}} \frac{\partial Q_{2}}{\partial x} + {ad}_{Q_{3}} \frac{\partial Q_{1}}{\partial x}) \\ + \frac{1}{6} ({ad}_{Q_{1}}^{2} \frac{\partial Q_{2}}{\partial x} + {ad}_{Q_{1}} {ad}_{Q_{2}} \frac{\partial Q_{1}}{\partial x} + {ad}_{Q_{2}} {ad}_{Q_{1}} \frac{\partial Q_{1}}{\partial x}) + \frac{1}{24} {ad}_{Q_{1}}^{3} \frac{\partial Q_{1}}{\partial x}, \end{matrix}

(5.8)

The effectiveness of the general form of the Lax pair (5.3) follows from the relation which is easy to check:

[λ^{N_{1}} ξ J ξ^{- 1} (x, t, λ), λ^{N_{2}} ξ K ξ^{- 1} (x, t, λ)] = λ^{N_{1} + N_{2}} ξ [J, K] ξ^{- 1} (x, t, λ) = 0,

because the matrices K and J are diagonal. Therefore, the commutator

[U (x, t, λ), V (x, t, λ)]

must contain only negative powers of

λ

In addition we may impose on

Q

Mikhailov type reductions. Each of them uses a finite order automorphism, which introduces a grading in the algebra

g

. Below we will use several types of

Z_{2}

-reductions based on automorphisms of order 2 of the Lie algebra:

\begin{matrix} 1) & Q^{†} (x, t, κ_{1} (λ)) & = - Q (x, t, λ), & κ_{1} (λ) & = λ^{*}, \\ 2) & Q^{T} (x, t, λ) & = - S_{a} Q (x, t, λ) S_{a}^{- 1}, & κ_{2} (λ) & = \pm λ, \\ 3) & Q^{*} (x, t, κ_{3} (λ)) & = - Q (x, t, λ), & κ_{3} (λ) & = λ^{*}, \\ 4) & C_{0} Q (x, t, λ) C_{0}^{- 1} & = Q (x, t, - λ), & C_{0}^{2} & = 1 1, C_{0} \in h . \end{matrix}

(5.9)

compare with (4.3). The last reduction

C_{0}

is typical for Lax operators which are quadratic in

λ

Another important

Z_{2}

reduction is provided by the Cartan involutions

J

, which determines the hermitian symmetric spaces [67] and acts on

Q (x, t, λ)

as follows:

\begin{matrix} Q (x, t, λ) & = J Q (x, t, - λ) J . \end{matrix}

(5.10)

5.2. The family of N-wave equations with cubic nonlinearities

If we generalize to Lax pairs quadratic in

λ

we find:

\begin{matrix} L_{Nw} ψ & \equiv i \frac{\partial ψ}{\partial x} + (U_{2} (x, t) + λ [J, Q (x, t)] - λ^{2} J) ψ (x, t, λ) = 0, \\ M_{Nw} ψ & \equiv i \frac{\partial ψ}{\partial x} + (V_{2} (x, t) + λ [K, Q (x, t)] - λ^{2} K) ψ (x, t, λ) = 0, \end{matrix}

(5.11)

where

U_{2}

V_{2}

, Q again belong to a simple Lie algebra

g

, J and K are constant elements of

h

. Examples of N-wave type equations will be given below; here we just note that they contain first order derivatives with respect to x and t and cubic (not quadratic) nonlinearities with respect to

Q_{j k}

Below we will impose two types of Mikhailov reductions:

\begin{matrix} C_{0} U (x, t, λ) C_{0}^{- 1} = U (x, t, - λ), & C_{0} V (x, t, λ) C_{0}^{- 1} = V (x, t, - λ), \\ U (x, t, λ) = - U^{†} (x, t, λ^{*}), & V (x, t, λ) = - V^{†} (x, t, λ^{*}), \end{matrix}

(5.12)

where

\begin{matrix} U (x, t, λ) = U_{2} (x, t) + λ [J, Q] - λ^{2} J, V (x, t, λ) = V_{2} (x, t) + λ [K, Q] - λ^{2} K . \end{matrix}

(5.13)

and

C_{0} \in Aut g

C_{0}^{2} = 1 1

. In particular for the n-wave equations (see eqs. (2.3) and (2.1), (2.2)), we get

Q = Q^{†}

and J and K must be real. For the FAS and the scattering matrix these reductions give:

\begin{matrix} χ^{\pm} (x, t, λ) & = {(χ^{\mp})}^{†} (x, t, λ^{*}), T_{Nw} (λ, t) = T_{Nw}^{†} (λ^{*}, t), \\ C_{0} χ^{\pm} (x, t, λ) C_{0}^{- 1} & = χ^{\pm} (x, t, - λ), C_{0} T_{Nw} (λ, t) C_{0}^{- 1} = T_{Nw} (- λ, t), \end{matrix}

(5.14)

see [36,52].

First we will derive the N-wave equations in general form; then we will illustrate them by a couple of examples. Using the generic parametrization (5.1) we obtain:

\begin{matrix} U (x, t, λ) & = U_{2} + λ U_{1} - λ^{2} J, & U_{2} & = [J, Q_{2}] + \frac{1}{2} [Q_{1}, U_{1}], & U_{1} & = [J, Q_{1}], \\ V (x, t, λ) & = V_{2} + λ V_{1} - λ^{2} K, & V_{2} & = [K, Q_{2}] + \frac{1}{2} [Q_{1}, V_{1}], & V_{1} & = [K, Q_{1}], \end{matrix}

(5.15)

The compatibility condition in this case is:

\begin{matrix} i \frac{\partial}{\partial x} (V_{2} + λ V_{1}) - i \frac{\partial}{\partial t} (U_{2} + λ U_{1}) + [U_{2} + λ U_{1} - λ^{2} J, V_{2} + λ V_{1} - λ^{2} K] = 0 . \end{matrix}

(5.16)

It must hold identically with respect to

λ

. It is easy to check that the coefficients at

λ^{4}

and

λ^{3}

vanish. Some more efforts are needed to check that the coefficient at

λ^{2}

\begin{matrix} - [J, V_{2}] + [U_{1}, V_{1}] - [U_{2}, K] = 0 \end{matrix}

(5.17)

also vanishes identically due to the proper parametrization of U and V. The compatibility conditions must hold identically with respect to

λ

. The first three of these relations:

\begin{matrix} λ^{4} : [J, K] = 0, λ^{3} : [K, U_{1}] - [J, V_{1}] = 0, λ^{2} : [K, U_{2}] - [J, V_{2}] + [U_{1}, V_{1}] = 0, \end{matrix}

(5.18)

are satisfied identically due to the correct parametrization of

ξ^{\pm} (x, t, λ)

. In more details

\begin{matrix} U_{1} & = [J, Q_{1}], U_{2} = - \frac{1}{2} [U_{1}, Q_{1}] + [J, Q_{2}], \\ V_{1} & = [K, Q_{1}], V_{2} = - \frac{1}{2} [V_{1}, Q_{1}] + [K, Q_{2}] . \end{matrix}

(5.19)

The last two coefficients at

λ^{1}

and

λ^{0}

vanish provided

Q_{1} (x, t)

and

Q_{2} (x, t)

satisfy the following N-wave type equations:

\begin{matrix} λ^{1} & : & - i \frac{\partial U_{1}}{\partial t} + i \frac{\partial V_{1}}{\partial x} + [U_{2}, V_{1}] + [U_{1}, V_{2}] = 0, \\ λ^{0} & : & - i \frac{\partial U_{2}}{\partial t} + i \frac{\partial V_{2}}{\partial x} + [U_{2}, V_{2}] = 0 . \end{matrix}

(5.20)

Note, that while

U_{1}

and

V_{1}

are linear in

Q_{1}

U_{2}

and

V_{2}

are quadratic in

Q_{1}

. Therefore the nonlinearities in this N-wave equations are cubic in

Q_{1}

We assume that the root system of

g

is split into

Δ = Δ_{0} \cup Δ_{1}

, such that

\begin{matrix} C_{0} E_{α} C_{0}^{- 1} = E_{α}, α \in Δ_{0}, C_{0} E_{α} C_{0}^{- 1} = - E_{α}, α \in Δ_{1} . \end{matrix}

(5.21)

We also denote positive and negative roots by a plus or minus superscript. Then considering (5.12) we must have:

\begin{matrix} C_{0} Q_{1} (x, t) C_{0}^{- 1} = - Q_{1} (x, t), C_{0} Q_{2} (x, t) C_{0}^{- 1} = Q_{2} (x, t), \\ Q_{2 s - 1} (x, t) = \sum_{α \in Δ_{1}^{+}}^{} (q_{2 s - 1, α} E_{α} - q_{2 s - 1, α}^{*} E_{- α}), Q_{2 s} (x, t) = \sum_{α \in Δ_{0}^{+}}^{} (q_{2 s, α} E_{α} - q_{2 s, α}^{*} E_{- α}), \end{matrix}

(5.22)

It is easy to check that this choice of

Q_{s}

is compatible with the following two involutions of the RHP

\begin{matrix} a) & Q (x, t, λ) & = Q^{†} (x, t, - λ^{*}) = \sum_{s = 1}^{\infty} Q_{s} (x, t) λ^{- s}, Q_{s}^{†} (x, t) = - Q_{s} (x, t), \\ b) & Q (x, t, λ) & = C_{0} Q (x, t, - λ) C_{0}^{- 1}, C_{0} Q_{2 s - 1} C_{0}^{- 1} = - Q_{2 s - 1}, C_{0} Q_{2 s} C_{0}^{- 1} = Q_{2 s}, \end{matrix}

(5.23)

which means that

\begin{matrix} a) {(ξ^{\pm})}^{†} (x, t, λ) = {(ξ^{\mp})}^{- 1} (x, t, λ^{*}), b) C_{0} ξ^{\pm} (x, t, - λ) C_{0}^{- 1} = ξ^{\pm} (x, t, - λ) . \end{matrix}

(5.24)

Example 1

(6-wave type equations:

s l (4) - c a s e

). The involution is given by

C_{0} = exp (π i H_{e_{2}}) = d i a g (1, - 1, - 1, 1) .

(5.25)

The potentials are:

\begin{matrix} Q_{1} & = (\begin{matrix} 0 & q_{1} & q_{2} & 0 \\ - q_{1}^{*} & 0 & 0 & q_{3} \\ - q_{2}^{*} & 0 & 0 & q_{4} \\ 0 & - q_{3}^{*} & - q_{4}^{*} & 0 \end{matrix}), Q_{2} = (\begin{matrix} 0 & 0 & 0 & q_{6} \\ 0 & 0 & q_{5} & 0 \\ 0 & - q_{5}^{*} & 0 & 0 \\ - q_{6}^{*} & 0 & 0 & 0 \end{matrix}), \\ J & = d i a g (a_{1}, a_{2}, - a_{2}, - a_{1}), K = d i a g (b_{1}, b_{2}, - b_{2}, - b_{1}) . \end{matrix}

(5.26)

The corresponding NLEE ensure that the coefficients at

λ^{1}

and

λ^{0}

also vanish. These give:

\begin{matrix} - i (a_{1} - a_{2}) \frac{\partial q_{1}}{\partial t} + i (b_{1} - b_{2}) \frac{\partial q_{1}}{\partial x} + κ (q_{1} (| q_{3} |^{2} - | q_{2} |^{2}) + 2 (q_{2} q_{5}^{*} + q_{6} q_{3}^{*})) = 0, \\ - i (a_{2} + a_{1}) \frac{\partial q_{2}}{\partial t} + i (b_{2} + b_{1}) \frac{\partial q_{2}}{\partial x} + κ (q_{2} (| q_{1} |^{2} - | q_{4} |^{2}) + 2 (q_{1}^{*} q_{5} - q_{4}^{*} q_{6})) = 0, \\ - i (a_{2} + a_{1}) \frac{\partial q_{3}}{\partial t} + i (b_{2} + b_{1}) \frac{\partial q_{3}}{\partial x} - κ (q_{3} (| q_{1} |^{2} - | q_{4} |^{2}) - 2 (q_{1}^{*} q_{6} - q_{4} q_{5})) = 0, \\ - i (a_{1} - a_{2}) \frac{\partial q_{4}}{\partial t} + i (b_{1} - b_{2}) \frac{\partial q_{4}}{\partial x} - κ (q_{4} (| q_{3} |^{2} - | q_{2} |^{2}) + 2 (q_{2}^{*} q_{6} + q_{3} q_{5}^{*})) = 0 \end{matrix}

(5.27)

where

κ = a_{1} b_{2} - a_{2} b_{1}

and

\begin{matrix} - 2 i a_{2} \frac{\partial q_{5}}{\partial t} + 2 i b_{2} \frac{\partial q_{5}}{\partial x} - i a_{1} \frac{\partial (q_{3} q_{4}^{*} - q_{1}^{*} q_{2})}{\partial t} + i b_{1} \frac{\partial (q_{3} q_{4}^{*} - q_{1}^{*} q_{2})}{\partial x} \\ + κ (q_{1}^{*} q_{2} - q_{3} q_{4}^{*}) (| q_{1} |^{2} + | q_{2} |^{2} + | q_{3} |^{2} + | q_{4} |^{2}) - 2 κ q_{5} (| q_{1} |^{2} - | q_{2} |^{2} - | q_{3} |^{2} + | q_{4} |^{2}) = 0, \\ - 2 i a_{1} \frac{\partial q_{6}}{\partial t} + 2 i b_{1} \frac{\partial q_{6}}{\partial x} - i a_{2} \frac{\partial (q_{1} q_{3} - q_{2} q_{4})}{\partial t} + i b_{2} \frac{\partial (q_{1} q_{3} - q_{2} q_{4})}{\partial x} \\ + κ (q_{1} q_{3} - q_{2} q_{4}) (| q_{1} |^{2} + | q_{2} |^{2} + | q_{3} |^{2} + | q_{4} |^{2}) + 2 κ q_{6} (| q_{1} |^{2} - | q_{2} |^{2} - | q_{3} |^{2} + | q_{4} |^{2}) = 0 . \end{matrix}

(5.28)

Example 2

(4-wave type equations:

s o (5) - c a s e

). The involution is given by

C_{0} = exp (π i H_{e_{1}}) = d i a g (- 1, 1, 1, 1, - 1) .

(5.29)

We first choose the potentials

Q_{1}

Q_{2}

, J, K and the involution

C_{0}

as follows:

\begin{matrix} Q_{1} (x, t) = (\begin{matrix} 0 & q_{1} & q_{2} & q_{3} & 0 \\ - q_{1}^{*} & 0 & 0 & 0 & q_{3} \\ - q_{2}^{*} & 0 & 0 & 0 & - q_{2} \\ - q_{3}^{*} & 0 & 0 & 0 & q_{1} \\ 0 & - q_{3}^{*} & q_{2}^{*} & - q_{1}^{*} & 0 \end{matrix}), Q_{2} (x, t) = (\begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & q_{4} & 0 & 0 \\ 0 & - q_{4}^{*} & 0 & q_{4} & 0 \\ 0 & 0 & - q_{4}^{*} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}), \\ J = d i a g (a_{1}, a_{2}, 0, - a_{2}, - a_{1}), K = d i a g (b_{1}, b_{2}, 0, - b_{2}, - b_{1}) . \end{matrix}

(5.30)

It is easy to check that

C_{0} Q_{1} C_{0}^{- 1} = - Q_{1}

and

C_{0} Q_{2} C_{0}^{- 1} = Q_{2}

, and consequently the FAS of

L_{Nw}

(5.11) satisfies

C_{0} χ^{\pm} (x, t, λ) C_{0}^{- 1} = χ^{\pm} (x, t, - λ)

. The corresponding equations (5.20) become:

\begin{matrix} - 2 i (a_{1} - a_{2}) \frac{\partial q_{1}}{\partial t} + 2 i (b_{1} - b_{2}) \frac{\partial q_{1}}{\partial x} + κ (q_{1} | q_{2} |^{2} + q_{2}^{2} q_{3}^{*} - 2 q_{2} q_{4}^{*}) = 0, \\ - 2 i a_{1} \frac{\partial q_{2}}{\partial t} + 2 i b_{1} \frac{\partial q_{2}}{\partial x} - κ (q_{2} (| q_{2} |^{2} - | q_{3} |^{2}) + 2 q_{3} q_{4}^{*} - 2 q_{1} q_{4}) = 0, \\ - 2 i (a_{1} + a_{2}) \frac{\partial q_{3}}{\partial t} + 2 i (b_{1} + b_{2}) \frac{\partial q_{3}}{\partial x} - κ (q_{3} | q_{2} |^{2} + q_{2}^{2} q_{1}^{*} - 2 q_{2} q_{4}) = 0, \end{matrix}

(5.31)

and

\begin{matrix} - 2 i a_{2} \frac{\partial q_{4}}{\partial t} + 2 i b_{2} \frac{\partial q_{4}}{\partial x} - i (2 a_{1} - a_{2}) \frac{\partial (q_{2} q_{1}^{*})}{\partial t} + i (2 b_{1} - b_{2}) \frac{\partial (q_{2} q_{1}^{*})}{\partial x} - i (2 a_{1} + a_{2}) \frac{\partial (q_{3} q_{2}^{*})}{\partial t} \\ + i (2 b_{1} + b_{2}) \frac{\partial (q_{3} q_{2}^{*})}{\partial x} + κ (| q_{1} |^{2} (3 q_{3} q_{2}^{*} + q_{2} q_{1}^{*}) + | q_{3} |^{2} (3 q_{2} q_{1}^{*} + q_{3} q_{2}^{*}) + 2 q_{4} (| q_{1} |^{2} - | q_{3} |^{2})) = 0, \end{matrix}

(5.32)

5.3. The main idea of the dressing method

In this section we will generalize Zakharov-Shabat dressing method [134,135] for the quadratic pencils. We will start with the simplest possible form of the dressing factor which generates the one-soliton solutions. We do this for two reasons. The first one is that due to the additional involutions inherent in the quadratic pencils the dressing factors for the one-soliton solutions require solving block-matrix linear equations. The other reason is that we will be able to calculate the asymptotics of the one-soliton dressing factors which will allow us to study the soliton interactions for the corresponding NLEE. The N-soliton solutions can be derived either by repeating N-times the one-soliton dressing or by considering dressing factors whose pole singularities determined by

λ_{k}^{\pm}

k = 1, 2, \dots, N

. In this case one has to solve much more complicated block-matrix linear equations.

In order to avoid unnecessary repetitions of formulae we will introduce the notations for the 'naked' and one-soliton solutions FAS of the Lax pairs.

\begin{matrix} L_{0} χ_{0}^{\pm} (x, t, λ) = 0, M_{0} χ_{0}^{\pm} (x, t, λ) = 0, L_{1 s} χ_{1}^{\pm} (x, t, λ) = 0, M_{1 s} χ_{1}^{\pm} (x, t, λ) = 0, \end{matrix}

(5.33)

where

L_{0}

and

M_{0}

are the Lax pair whose potentials

U_{k}

and

V_{k}

are vanishing. By

L_{1 s}

and

M_{1 s}

we denote the Lax pair whose potentials are provided by the one-soliton solutions of the corresponding NLEE. Each time from the context it will be clear which specific La pair we are considering.

For the N-wave systems the 'naked' FAS are given by:

\begin{matrix} χ_{0}^{\pm} (x, t, λ) = exp (- i λ^{2} (J x + K t)), χ_{1}^{\pm} (x, t, λ) = u (x, t, λ) χ_{0}^{\pm} (x, t, λ), \end{matrix}

(5.34)

while for the NLS-type equations

\begin{matrix} χ_{0}^{\pm} (x, t, λ) = exp (- i J (λ^{2} x + λ^{4} t)), χ_{1}^{\pm} (x, t, λ) = u (x, t, λ) χ_{0}^{\pm} (x, t, λ), \end{matrix}

(5.35)

where the dressing factor

u (x, t, λ)

will be calculated below for each of the relevant cases. The specific form of J and K in (5.34) depends on the specific choice of the corresponding homogeneous space. Likewise the specific form of J in (5.35) is determined by the choice of the relevant symmetric space.

Each dressing factor is a fractional linear function of the spectral parameter

λ

. As such we will use:

\begin{matrix} c_{1} (λ) = \frac{λ - λ_{1}^{+}}{λ - λ_{1}^{-}}, c_{1} (- λ) = \frac{λ + λ_{1}^{+}}{λ + λ_{1}^{-}}, λ_{1}^{\pm} = μ_{1} \pm ν_{1}, c_{1}^{*} (λ^{*}) = \frac{1}{c_{1} (λ)} . \end{matrix}

(5.36)

Indeed,

c_{1} (- λ)

comes up naturally due to the symmetry

λ \to - λ

. By

λ_{1}^{\pm}

we denote constants such that

Im {(λ_{1}^{\pm})}^{2} ≷ 0

; i.e

μ_{1} ν_{1} ≷ 0

. As we shall see below

λ_{1}^{\pm}

- λ_{1}^{\pm}

and their hermitian conjugate determine the discrete eigenvalues of

L_{1 s}

The generic form of the dressing factors is the same for both types of NLEE considered above. If we impose only types of symmetries on L and M, such as:

\begin{matrix} U^{†} (x, t, λ^{*}) = - U (x, t, λ), C_{0} U (x, t, - λ) C_{0}^{- 1} = U (x, t, λ), \end{matrix}

(5.37)

and similar relations for

V (x, t, λ)

. Here

C_{0}

is constant diagonal matrix such that

C_{0}^{2} = 1 1

. Then

u (x, t, λ)

must satisfy:

\begin{matrix} u^{†} (x, t, λ^{*}) = u^{- 1} (x, t, λ), C_{0} u (x, t, - λ) C_{0}^{- 1} = u (x, t, λ), \end{matrix}

(5.38)

then

u (x, t, λ)

and its inverse have the form

\begin{matrix} u (x, t, λ) & = 1 1 + (c_{1} (λ) - 1) | N_{1} 〉 〈 m_{1} | + (c_{1} (- λ) - 1) C_{0} | N_{1} 〉 〈 m_{1} | C_{0}^{- 1}, \\ u^{- 1} (x, t, λ) & = 1 1 + (\frac{1}{c_{1} (λ)} - 1) | n_{1} 〉 〈 M_{1} | + (\frac{1}{c_{1} (- λ)} - 1) C_{0} | n_{1} 〉 〈 M_{1} | C_{0}^{- 1} . \end{matrix}

(5.39)

Here the 'polarization' vectors

| N_{1} 〉

〈 m_{1} |

| n_{1} 〉

and

〈 M_{1} |

determine the residues of u and

u^{- 1}

. These residues for the one-soliton case will be evaluated explicitly below for each of the NLEE we consider.

5.4. Dressing of N-wave equations: two involutions

We start with the N-wave type on homogeneous spaces with two involutions. Using the equations (5.33) we derive the following equation for the dressing factor:

\begin{matrix} i \frac{\partial u}{\partial x} + (U_{2; 1 s} + λ U_{1; 1 s} - λ^{2} J) u (x, t, λ) - u (x, t, λ) (U_{20} + λ U_{10} - λ^{2} J) = 0, \end{matrix}

(5.40)

which also must hold identically with respect to

λ

. This can be verified by taking the residues of the left hand sides of (5.40) for

λ = λ_{1}^{-}

and equating them to 0. This gives:

\begin{matrix} (i \frac{\partial | N_{1} 〉}{\partial x} + (U_{2; 1 s} + λ_{1}^{-} U_{1; 1 s} - {(λ_{1}^{-})}^{2} J) | N_{1} 〉) 〈 m_{1} | \\ + | N_{1} 〉 (i \frac{\partial 〈 m_{1} |}{\partial x} - 〈 m_{1} | (U_{20} + λ_{1}^{-} U_{10} - {(λ_{1}^{-})}^{2} J) = 0, \end{matrix}

(5.41)

from which one easily finds, see e q. (5.34):

\begin{matrix} | N_{1} 〉 = χ_{1}^{-} (x, t, λ_{1}^{-}) | N_{10} 〉, 〈 m_{1} | = 〈 m_{10} | {\hat{χ}}_{0}^{-} (x, t, λ_{1}^{-}) . \end{matrix}

(5.42)

Similarly, we can use the equation satisfied by

\hat{u} \equiv u^{- 1} (x, t, λ)

which reads:

\begin{matrix} i \frac{\partial \hat{u}}{\partial x} + (U_{20} + λ U_{10} - λ^{2} J) \hat{u} (x, t, λ) - \hat{u} (x, t, λ) (U_{2; 1 s} + λ U_{1; 1 s} - λ^{2} J) = 0 . \end{matrix}

(5.43)

Putting the residue of (5.43) at

λ_{1}^{+}

to 0 we get:

\begin{matrix} (i \frac{\partial | n_{1} 〉}{\partial x} + (U_{20} + λ_{1}^{+} U_{10} - {(λ_{1}^{+})}^{2} J) | n_{1} 〉) 〈 M_{1} | \\ + | n_{1} 〉 (i \frac{\partial 〈 M_{1} |}{\partial x} - 〈 m_{1} | (U_{2; 1 s} + λ_{1}^{+} U_{1; 1 s} - {(λ_{1}^{+})}^{2} J) = 0, \end{matrix}

(5.44)

The result is, see eq. (5.34):

\begin{matrix} | n_{1} 〉 = χ_{0}^{+} (x, t, λ_{1}^{+}) | n_{10} 〉, 〈 M_{1} | = 〈 M_{10} | {\hat{χ}}_{1}^{+} (x, t, λ_{1}^{+}) . \end{matrix}

(5.45)

Remark 7.

We note that the vectors

| n_{10} 〉

| N_{10} 〉

〈 m_{10} |

and

〈 M_{10} |

are constants, which must satisfy the (5.38). Due to the same reductions we must also have

λ_{1}^{-} = {(λ_{1}^{+})}^{*}

. We have also chosen

C_{0}

to be constant diagonal matrix whose matrix elements equal

\pm 1

Thus, if we know the regular solutions

χ_{0}^{\pm} (x, t, λ)

then we have derived explicitly the x and t dependence of the vectors

| n_{1} 〉

and

〈 m_{1} |

. In addition we know that

u \hat{u} = 1 1

also must hold identically with respect to

λ

. That means that the residues:

\begin{matrix} \underset{λ = λ_{1}^{-}}{Res} u (x, t, λ) u^{- 1} (x, t, λ) = 0, \underset{λ = λ_{1}^{+}}{Res} u (x, t, λ) u^{- 1} (x, t, λ) = 0 . \end{matrix}

(5.46)

must vanish. Inserting u and

\hat{u}

from eq. (5.39) we obtain the equations:

\begin{matrix} 〈 M_{1} | & = 〈 m_{1} | {(〈 m_{1} | n_{1} 〉 1 1 + \frac{λ_{1}^{+} - λ_{1}^{-}}{λ_{1}^{+} + λ_{1}^{-}} 〈 m_{1} | C_{0} | n_{1} 〉 C_{0})}^{- 1}, \\ | N_{1} 〉 & = {(〈 m_{1} | n_{1} 〉 1 1 - \frac{λ_{1}^{+} - λ_{1}^{-}}{λ_{1}^{+} + λ_{1}^{-}} 〈 m_{1} | C_{0} | n_{1} 〉 C_{0})}^{- 1} | n_{1} 〉 . \end{matrix}

(5.47)

In the specific calculations below we will use more convenient notations, namely:

\begin{matrix} E_{01}^{\pm} & = exp (- i {(λ_{1}^{\pm})}^{2} (J x + K t)) = diag (e^{\pm z_{1} - i ϕ_{1}}, \dots, e^{\pm z_{n} - i ϕ_{n}}), λ_{1}^{\pm} = μ_{1} \pm i ν_{1}, \\ z_{k} & = 2 μ_{1} ν_{1} (J_{k} x + K_{k} t), ϕ_{k} = (μ_{1}^{2} - ν_{1}^{2}) (J_{k} x + K_{k} t), κ_{1} = \frac{λ_{1}^{+} - λ_{1}^{-}}{λ_{1}^{+} + λ_{1}^{-}} = i \frac{ν_{1}}{μ_{1}}, \\ | n_{1} 〉 & = E_{01}^{+} | n_{10} 〉, 〈 m_{1} | = 〈 m_{10} | {\hat{E}}_{01}^{-} . \end{matrix}

(5.48)

where

μ_{1} > 0

and

ν_{1} > 0

. The functions

z_{k} (x, t)

and

ϕ_{k} (x, t)

are linear functions of x and t; for each specific example they will be given explicitly.

The last step we need to do is to determine the corresponding singular potentials

U_{1}

and

U_{2}

. To this end we come back to the equation (5.40) for the dressing factor and study its limit for

λ \to \infty

. Its left hand side is a quadratic polynomial of

λ

. Skipping the details we obtain:

\begin{matrix} U_{1; 1 s} - U_{10} & = (λ_{1}^{+} - λ_{1}^{-}) [J, W^{+}], W^{\pm} = \mp | N_{1} 〉 〈 m_{1} | + C_{0} | N_{1} 〉 〈 m_{1} | C_{0}, \\ U_{2; 1 s} - U_{20} & = - (λ_{1}^{+} - λ_{1}^{-}) (U_{1} W^{-} - W^{-} U_{20} + λ_{1}^{-} [J, W^{-}]) \end{matrix}

(5.49)

We we put

U_{10} = U_{20} = 0

we get simplified expression for the one-soliton solution:

\begin{matrix} U_{1; 1 s} & = (λ_{1}^{+} - λ_{1}^{-}) [J, W^{-}], \\ U_{2; 1 s} & = - (λ_{1}^{+} - λ_{1}^{-}) (U_{1; 1 s} W^{+} + λ_{1}^{-} [J, W^{-}]) . \end{matrix}

(5.50)

More explicit expressions for

U_{1; 1 s}

and

U_{2; 1 s}

in terms of hyper-trigonometric functions will be given below for each of the examples.

Example 3

(One soliton solutions,

s l (4)

case). The Lax representation in that case is provided by the operators (5.11) where J, K,

Q_{1}

and

Q_{2}

are given by (5.26). The 'naked' polarization vectors

〈 m_{1} |

(5.42) and

| n_{1} 〉

(5.45) become:

\begin{matrix} 〈 m_{1} | = (m_{10, 1} e^{z_{1} + i ϕ_{1}}, m_{10, 2} e^{z_{2} + i ϕ_{2}}, m_{10, 3} e^{- z_{2} - i ϕ_{2}}, m_{10, 2} e^{- z_{1} - i ϕ_{1}}), | n_{1} 〉 = (\begin{matrix} n_{10, 1} e^{z_{1} - i ϕ_{1}} \\ n_{10, 2} e^{z_{2} - i ϕ_{2}} \\ n_{10, 3} e^{- z_{2} + i ϕ_{2}} \\ n_{10, 4} e^{- z_{1} + i ϕ_{1}} \end{matrix}) . \end{matrix}

(5.51)

Taking into account the typical hermitian reduction of L and M we find

{〈 m_{1} | = (| n_{1} 〉)}^{†}

and

{〈 m_{10} | = (| n_{10} 〉)}^{†}

. The dressed polarization vectors defined by (5.47) are equal to:

\begin{matrix} 〈 M_{1} | & = (\frac{m_{1; 1}}{R_{1}^{+}}, \frac{m_{1; 2}}{R_{1}^{-}}, \frac{m_{1; 3}}{R_{1}^{-}}, \frac{m_{1; 4}}{R_{1}^{+}}), | N_{1} 〉 = (\begin{matrix} {(R_{1}^{-})}^{- 1} & 0 & 0 & 0 \\ 0 & {(R_{1}^{+})}^{- 1} & 0 & 0 \\ 0 & 0 & {(R_{1}^{+})}^{- 1} & 0 \\ 0 & 0 & 0 & {(R_{1}^{-})}^{- 1} \end{matrix}) | n_{1} 〉, \\ R_{1}^{\pm} & = 〈 m_{1} | n_{1} 〉 \pm \frac{i ν_{1}}{μ_{1}} 〈 m_{1} | C_{0} | n_{1} 〉, \\ 〈 m_{1} | n_{1} 〉 & = 2 η_{01} cosh (2 z_{1} + ζ_{01}) + 2 η_{02} cosh (2 z_{2} + ζ_{02}), η_{01} = | n_{10, 1} n_{10, 4} |, η_{02} = | n_{10, 2} n_{10, 3} |, \\ 〈 m_{1} | C_{0} | n_{1} 〉 & = 2 η_{01} cosh (2 z_{1} + ζ_{01}) - 2 η_{02} cosh (2 z_{2} + ζ_{02}), ζ_{01} = ln \frac{| n_{10, 1} |}{| n_{10, 4} |}, ζ_{02} = ln \frac{| n_{10, 2} |}{| n_{10, 3} |}, \end{matrix}

(5.52)

They also satisfy

{〈 M_{1} | = (| N_{1} 〉)}^{†}

. Therefore

\begin{matrix} W^{+} & = - 2 (\begin{matrix} 0 & n_{1, 1} m_{1, 2} & n_{1, 1} m_{1, 3} & 0 \\ n_{2, 1} m_{1, 1} & 0 & 0 & n_{1, 2} m_{1, 4} \\ n_{1, 3} m_{1, 1} & 0 & 0 & n_{1, 3} m_{1, 4} \\ 0 & n_{1, 4} m_{1, 2} & n_{1, 4} m_{1, 3} & 0 \end{matrix}), \\ W^{-} & = 2 (\begin{matrix} n_{1, 1} m_{1, 1} & 0 & 0 & n_{1, 1} m_{1, 4} \\ 0 & n_{1, 2} m_{1, 2} & n_{1, 2} m_{1, 3} & 0 \\ 0 & n_{1, 3} m_{1, 2} & n_{1, 3} m_{1, 3} & 0 \\ n_{1, 4} m_{1, 1} & 0 & 0 & n_{1, 4} m_{1, 4} \end{matrix}) \end{matrix}

(5.53)

\begin{matrix} q_{1} (x, t) & = - \frac{4 i ν_{1} (a_{1} - a_{2}) m_{1, 2} n_{1, 1}}{R_{1}^{+}}, q_{2} (x, t) = - \frac{4 i ν_{1} (a_{1} + a_{2}) m_{1, 3} n_{1, 1}}{R_{1}^{+}} \\ q_{3} (x, t) & = - \frac{4 i ν_{1} (a_{1} + a_{2}) m_{1, 4} n_{1, 2}}{R_{1}^{-}}, q_{4} (x, t) = - \frac{4 i ν_{1} (a_{1} - a_{2}) m_{1, 4} n_{1, 3}}{R_{1}^{-}} \\ q_{5} (x, t) & = - \frac{8 ν_{1} n_{1, 1} m_{1, 4} ((a_{1} - a_{2}) n_{1, 2} m_{1, 2} + (a_{1} - a_{2}) n_{1, 3} m_{1, 3}) - i λ_{1}^{-} a_{1} R_{1}^{-}}{R_{1}^{+} R_{1}^{-}}, \\ q_{6} (x, t) & = \frac{8 ν_{1} n_{1, 2} m_{1, 3} ((a_{1} - a_{2}) n_{1, 1} m_{1, 1} - (a_{1} + a_{2}) n_{1, 4} m_{1, 4}) - i λ_{1}^{+} a_{2} R_{1}^{+}}{R_{1}^{+} R_{1}^{-}}, \end{matrix}

(5.54)

\begin{matrix} q_{1} (x, t) & = \frac{4 i ν_{1} μ_{1} (a_{1} - a_{2}) | n_{10, 1} n_{10, 2} | e^{z_{1} + z_{2} + i ({\tilde{ϕ}}_{2} - {\tilde{ϕ}}_{1})} [λ_{1}^{+} η_{01} cosh ({\tilde{z}}_{1}) + λ_{1}^{-} η_{02} cosh ({\tilde{z}}_{2})]}{(μ_{1}^{2} + ν_{1}^{2}) (η_{01}^{2} {cosh}^{2} ({\tilde{z}}_{1}) + η_{02}^{2} {cosh}^{2} ({\tilde{z}}_{2})) + 2 η_{10} η_{20} (μ_{1}^{2} - ν_{1}^{2}) cosh ({\tilde{z}}_{1}) cosh ({\tilde{z}}_{2})}, \\ q_{2} (x, t) & = \frac{4 i ν_{1} μ_{1} (a_{1} + a_{2}) | n_{10, 3} n_{10, 2} | e^{z_{1} - z_{2} - i ({\tilde{ϕ}}_{1} + {\tilde{ϕ}}_{2})} [λ_{1}^{+} η_{01} cosh ({\tilde{z}}_{1}) + λ_{1}^{-} η_{02} cosh ({\tilde{z}}_{2})]}{(μ_{1}^{2} + ν_{1}^{2}) (η_{01}^{2} {cosh}^{2} ({\tilde{z}}_{1}) + η_{02}^{2} {cosh}^{2} ({\tilde{z}}_{2})) + 2 η_{10} η_{20} (μ_{1}^{2} - ν_{1}^{2}) cosh ({\tilde{z}}_{1}) cosh ({\tilde{z}}_{2})}, \\ q_{3} (x, t) & = \frac{4 i ν_{1} μ_{1} (a_{1} + a_{2}) | n_{10, 4} n_{10, 2} | e^{- z_{1} + z_{2} - i ({\tilde{ϕ}}_{1} + {\tilde{ϕ}}_{2})} [λ_{1}^{-} η_{01} cosh ({\tilde{z}}_{1}) + λ_{1}^{+} η_{02} cosh ({\tilde{z}}_{2})]}{(μ_{1}^{2} + ν_{1}^{2}) (η_{01}^{2} {cosh}^{2} ({\tilde{z}}_{1}) + η_{02}^{2} {cosh}^{2} ({\tilde{z}}_{2})) + 2 η_{10} η_{20} (μ_{1}^{2} - ν_{1}^{2}) cosh ({\tilde{z}}_{1}) cosh ({\tilde{z}}_{2})}, \\ q_{4} (x, t) & = \frac{4 i ν_{1} μ_{1} (a_{1} - a_{2}) | n_{10, 1} n_{10, 2} | e^{- z_{1} - z_{2} + i ({\tilde{ϕ}}_{2} - {\tilde{ϕ}}_{1})} [λ_{1}^{-} η_{01} cosh ({\tilde{z}}_{1}) + λ_{1}^{+} η_{02} cosh ({\tilde{z}}_{2})]}{(μ_{1}^{2} + ν_{1}^{2}) (η_{01}^{2} {cosh}^{2} ({\tilde{z}}_{1}) + η_{02}^{2} {cosh}^{2} ({\tilde{z}}_{2})) + 2 η_{10} η_{20} (μ_{1}^{2} - ν_{1}^{2}) cosh ({\tilde{z}}_{1}) cosh ({\tilde{z}}_{2})}, \end{matrix}

(5.55)

where

{\tilde{z}}_{k} = 2 z_{k} + ζ_{0 k}

and

{\tilde{ϕ}}_{k} = ϕ_{k} + arg n_{10, k}

. In addition

\begin{matrix} q_{5} (x, t) = - 16 ν_{1} | n_{10, 1} | | n_{10, 4} | e^{- 2 i {\tilde{ϕ}}_{1}} \times \\ \times \frac{[ν_{1} μ_{1} (a_{1} η_{01} cosh ({\tilde{z}}_{2}) - a_{2} η_{02} sinh ({\tilde{z}}_{2})) - i λ_{1}^{-} a_{1} (λ_{1}^{-} η_{01} cosh ({\tilde{z}}_{1}) + λ_{1}^{+} η_{02} cosh ({\tilde{z}}_{2})))]}{μ_{1} ((μ_{1}^{2} + ν_{1}^{2}) (η_{01}^{2} {cosh}^{2} ({\tilde{z}}_{1}) + η_{02}^{2} {cosh}^{2} ({\tilde{z}}_{2})) + 2)} \end{matrix}

Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

Abstract

1. Introduction

2. From the Lax representation to the RHP

2.1. N-waves according to Manakov and Zakharov

2.2. MNLS equations according to Manakov, Fordy and Kulish

2.3. Generic Lax representation

2.4. Jost solutions and FAS of L

2.5. The time-dependence of T ( t , λ )

3. RHP and integrable NLEE

3.1. Uniqueness of the regular solution of RHP

3.2. Zakharov-Shabat theorem

4. Mikhailov’s reduction groups and the contours of RHP

4.1. General theory

4.2. Involutive reductions

4.3. Z h reduction groups

4.4. D h reduction groups

5. Parametrizing the RHP with canonical normalization

5.1. Generic parametrization of the RHP with canonical normalization

5.2. The family of N-wave equations with cubic nonlinearities

5.3. The main idea of the dressing method

5.4. Dressing of N-wave equations: two involutions

2.5. The time-dependence of $T (t, λ)$

4.3. $Z_{h}$ reduction groups

4.4. $D_{h}$ reduction groups