An integrable decomposition of the Manakov equation

An integrable decomposition of the Manakov equation

Shouting Chen^I; Ruguang Zhou^II,*

^ISchool of Mathematics and Physical Sciences, Xuzhou Institute of Technology, Xuzhou, 221018, China

^IISchool of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116, China E-mail: [e-mail]

ABSTRACT

An integrable decomposition of the Manakov equation is presented. A pair of new finite-dimensional integrable Hamiltonian systems which constitute the integrable decomposition of the Manakov equation are obtained.

Mathematical subject classification: 37K10.

Key words: binary nonlinearization of the spectral problem, the 4-component AKNS equation, the Manakov equation, integrable decomposition.

1 Introduction

The Manakov equation

where q₁, q₂ are potentials, is nothing but the 2-component vector nonlinear Schrödinger equation and sometimes is referred to as the coupled nonlinear Schrödinger equation. Manakov first examined equation (1) as an asymptotic model for the propagation of the electric field in a waveguide [1]. Subsequently, the system (1) was derived as a key model for lightwave propagation in optical fibers [2]. The system admits vector-soliton solutions, the solution collision is elastic and the dynamics of soliton interactions can be explicitly computed [3].

The nonlinearization of spectral problem (NSP), which was put forward by Cao [4, 5], is a powerful tool to study integrable systems in (1+1)-dimensions. With it we can generate new finite-dimensional integrable Hamiltonian systems, decompose (1+1) dimensional integrable systems into a pair of compatible finite-dimensional integrable Hamiltonian systems, and thus construct explicit or numerical solutions of the (1+1) dimensional integrable systems. During the past two decades, many powerful techniques of generalizations of the NSP havebeen obtained. For example, the binary NSP, which was presented for the first time by Ma and Strampp [6], has been studied [7, 8]. After that the higher-order symmetry constraint method [9] and binary nonlinearization of spectral problems under higher-order symmetry constraints [10] were discussed. The nonlinearization and binary nonlinearization of the discrete eigenvalue problem were introduced [11, 12], respectively. And then Ma and Zhou proposed the adjoint symmetry constraint method [13, 14]. The adjoint symmetry constraint can be used to solve the multicomponent AKNS equations associated with degenerate spectral problems. Because the spectral matrix of the Manakov equation is degenerate, general methods of the NSPs are impossible for the success of making integrable decompositions. Therefore, the introduction of the adjoint symmetry is very crucial for the integrable decomposition of the Manakov equation.

In this paper, we will present an integrable decomposition of the Manakov equation. As is well known, equation (1) can be reduced from the 4-component AKNS equation by imposing the reality condition r = -q^†. Therefore, we can make the integrable decomposition of equation (1) through the following procedure: couple the spectral problems of equation (1) with their complex conjugates and then reduce these problems to that of the 4-component AKNS equation by introducing new variables. The paper is organized as follows. Firstly, we recall the integrable decomposition of the 4-component AKNS equation with the help of the adjoint symmetry constraints and the binary NSP. Secondly, on the basis of section 2, we study the integrable decomposition of the Manakov equation and finally obtain a pair of finite-dimensional completely integrable Hamiltonian systems.

2 An integrable decomposition of the 4-component AKNS equation

In Ref. [15] the multi-wave interaction equations associated with the 3 × 3 matrix AKNS spectral problem were decomposed into finite-dimensional Liouville integrable Hamiltonian systems by carrying out binary Bargmann symmetry constraints. As a special case, in this section, following [13, 15] we review the integrable decomposition of the 4-component AKNS equation.

2.1 The 4-component AKNS hierarchy of equations

It is well known that the 4-component AKNS hierarchy of equations is associated with the following spectral problem

where λ is a spectral parameter and q₁, q₂, r₁, r₂ are four potentials,

Φ = (ϕ₁, ϕ₂, ϕ₃)^T, u = (q₁, q₂, r₁, r₂)^T, q = (q₁, q₂), r = (r₁, r₂)^T.

From the adjoint representation equation

with

and then through a tedious and straightforward calculation, we work out

where 1 < i, j < 2.

Consider the following auxiliary problem

From the compatibility condition of (2) and (4)

we obtain the 4-component AKNS hierarchy of equations

The first nontrivial equation or the 4-component AKNS equation is

which admits spectral problem (2) and

where

2.2 Adjoint symmetry constraint

In order to preform the binary NSPs of the 4-component AKNS hierarchy, we take N distinct eigenvalue parameters λ_j, 1 < j < N, and N copies of the 3 × 3 matrix spectral problem for AKNS hierarchy (2) and their adjoint spectral problems as follows

which can be written as the following compact form

where

ϕ_j = (ϕ_j1, ..., ϕ_jN)^T, ψ_j = (ψ_j1, ..., ψ_jN)^T , A = diag(λ₁, ..., λ_N).

Following [13], we consider the following Bargmann adjoint symmetry constraint

where < ·,· > refers to the standard inner product of the Euclidian space and γ₁, γ₂, γ₃ are distinct parameters.

2.3 Completely integrable Hamiltonian system

Substituting (11) and (12) into (10), then we get the finite-dimensional Hamiltonian system

where

Under the control of (11), (12) and (13), we can get the following Hamiltonian system

where

Hamiltonian systems (13) and (14) allow Lax representations

respectively, where

λ₁,λ₂,...,λ_N are N distinct parameters, and U(, λ), ⁽²⁾(, λ) are the constraint spectral matrices generated from U, V⁽²⁾ under constraint (11) and (12).

To analyze the integrability of (13) and (14), we define a symplectic structure

The corresponding Poisson bracket is given by

Furthermore, it has been shown that Lax matrix (16) satisfies an r-matrix relation [13]. Therefore, we can get 3N conserved integrals

A direct check shows that

F_ik, 1 <i< 3, 1 < k < N,

are in involution and functionally independent. Thus Hamiltonian systems (13) and (14) are completely integrable in the sense of Liouville. Bringing these together, we arrive at the following proposition.

Proposition 1. If ϕ₁, ϕ₂, ϕ₃, ψ₁, ψ₂, ψ₃satisfy both (13) and (14), then

solve the 4-component AKNS equation (6).

3 An integrable decomposition of the Manakov equation

This section is devoted to an integrable decomposition of the Manakov equation. It is known that equation (1) may be reduced from the 4-component AKNS equation by imposing the reality condition r = -q^†. Moreover, the Manakov equation associates with the following spectral problems

and the adjoint spectral problems

where U(q, r, λ), V⁽²⁾(q, r, λ) are defined by (2), (8).

On the basis of the reality condition r = -q^† we have the following lemma.

Lemma 1. Let r = -q^†,(ϕ₁, ϕ₂, ϕ₃)^T solves (22), (23), and (ψ₁, ψ₂, ψ₃)^T solves (24), (25),

(i) If parameter λ is nonreal complex, then solves (22) and (23) with parameter λ*. In the same time, solves (24) and (25) with parameter λ*.

(ii) If parameter λ is real, then solves the adjoint spectral problems (24) and (25) with parameter λ.

Now let us carry out the binary NSPs of the Manakov equation. We take N parameters: λ₁, ..., λ_r, λ_r+1, ..., λ_N, where λ₁, ..., λ_r are distinct nonreal complex numbers, λ_i≠ , 1 < i, j < r, and λ_r+1, ..., λ_N are N - r distinct real numbers.

Guided by Lemma 1, we introduce new variables and notation

where

and then we can get the following vector form of spectral problem (22) and (24) coupling with their complex conjugates

which is just a compact form of special 3 × 3 matrix spectral problems for AKNS hierarchy and their adjoint spectral problems. Therefore, we only need to modify the procedure of the binary NSPs of the 4-component AKNS hierarchyby replacing variables ϕ₁, ϕ₂, ϕ₃, ψ₁, ψ₂, ψ₃ with such that the reality condition r = -q^† holds.

Now we consider the constraint of the Manakov equation

which solves

r = -q^†, 1 <j< 2.

Substituting (29) into (28), then we get the finite-dimensional Hamiltonian system

where

Similarly, we can get the temporal Hamiltonian system

where

Hamiltonian systems (30) and (31) are completely integrable systems. In fact, first of all, it is not difficult to check that (30) and (31) allow the following Lax matrix

Introducing symplectic structure

The corresponding Poisson bracket is given by

By a direct check, we come to a conclusion that L(λ) satisfies the r-matrix relation

{L₁(λ), L₂(µ)} = [r₁₂(λ, µ), L₁(λ)] + [r₂₁(λ, µ), L₂(µ)],

where L₁(λ) = L(λ)⊗I, L₂(µ) = I⊗L(µ), I is the 3 × 3 unit matrix,

and e_ij is the matrix with the element 1 at the (i, j) position and zeros elsewhere.

We can expand (19), (20), (21) as follows

where 1 < k < N + r,

It can be checked that , 1 < k < N + r, are 3(N + r) integrals in involution. To establish the completely integrability of Hamiltonian systems (30) and (31), it is essential to prove the functional independence of the above 3(N + r) integrals by making use of a small epsilon technique [16].

In fact, let P₀ be a point of ^3(N+r) satisfying = ε, 1 < i < 3, 1 < s < N + r, where ε is a small constant, and , , 1 < i < 3, 1 < s < N + r, are defined by (26). At this point P₀, by a direct calculation, we have

where

Since the Jacobian is a polynomial function of ϕ_is, ψ_is, , , 1 < i < 3, 1 < s < N + r, it is not zero over a dense open subset of ^3(N+r). Therefore, the functions , 1 < i < 3, 1 < k < N + r, are functionally independent over that dense open subset of ^3(N+r).

Moreover, a direct computation shows that the following proposition holds.

Theorem 1. If , 1 < j < 3, satisfy both (30) and (31), then

solve the Manakov equation (1). This means that (30) and (31) constitute an integrable decomposition of the Manakov equation.

4 Concluding remarks

In this paper, we have presented the integrable decomposition of the Manakov equation. The main idea of this method is to couple the spectral problems of the Manakov equation with their complex conjugates and then reduce these problems to that of 4-component AKNS equation by introducing new variables. Finally, the integrable decompositions of the Manakov equation are obtained by applying the corresponding results of the 4-component AKNS equation.

As indicated in [17], usually it is very complicated to deal with the soliton equation with reduction conditions or reality conditions. In [18, 19], one of the author (Zhou) proposed an approach to construct integrable decompositions of soliton equations with reality conditions such as the nonlinear Schrödinger equation, real-valued mKdV equation and the nonlinear derivative Schrödinger equation. But the approach can only be used to the cases that all the eigenvalue parameters λ₁, ..., λ_N are nonreal complex. In the present paper, such restrictions are released by using binary nonlinearizations of spectral problem, instead of mono-nonlinearization of spectral problem. Viewing from (33), we can easily find that whether the eigenvalue parameters are real or not leads tocompletely different symplectic structures.

Acknowledgments. This work has been supported by the National Natural Science Foundation of China under Grant No. 10871165.

Received: 23/IV/10.

Accepted: 10/XI/11.

#CAM-208/10.

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*

Corresponding author.

Publication Dates

• Publication in this collection
  26 Apr 2012
• Date of issue
  2012

History

• Received
  23 Apr 2010
• Accepted
  10 Nov 2011