Binary Nonlinearization for AKNS-KN Coupling System

Abstract

The AKNS-KN coupling system is obtained on the base of zero curvature equation by enlarging the spectral equation. Under the Bargmann symmetry constraint, the AKNS-KN coupling system is decomposed into two integrable Hamiltonian systems with the corresponding variables x , [subscript] t n [/subscript] and the finite dimensional Hamiltonian systems are Liouville integrable.

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Xiangrong Wang 1 and Xiaoen Zhang 1 and Peiyi Zhao 2

Academic Editor:Yufeng Zhang

1, College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2, Shandong Provincial Academy of Education, Recruitment and Examination, Jinan 250011, China

Received 30 April 2014; Accepted 29 May 2014; 15 June 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Since the integrable coupling definition is proposed, we have got many integrable coupling systems. Furthermore, the exact solutions, Darboux transformation, and Hamiltonian structure of these coupling systems have been obtained [1-3]. In 2008, the AKNS-KN coupling system is obtained by the loop algebra whose Hamiltonian system is received with the variational identity [4]. In 1994, the binary nonlinearization method was put forward by Li and Ma [5], and then the technique of the binary nonlinearization has been successfully applied to many soliton equations, such as the AKNS hierarchy, the KdV hierarchy, and the super NLS-MKDV hierarchy, but there are few results on binary nonlinearization of the coupling system. In this paper, we design a proper spectrum equation and obtain the AKNS-KN coupling system under the zero curvature equation, but the recursive operator is different from the operator of [4].

This paper is organized as follows. In Section 2, we will consider the AKNS-KN coupling soliton hierarchy. Bargmann symmetry constraint for the AKNS-KN coupling system will be given in Section 3. Section 4 will be devoted to study the AKNS-KN coupling system by employing the binary nonlinearization technique which involves two sets of dependent variables x and _{t n} . We especially list the special cases, such as AKNS integrable coupling system and KN integrable coupling system.

2. The AKNS-KN Coupling System

We design a spectral problem _{[varphi] x} = U [varphi] , where the spectral operator is as follows: [figure omitted; refer to PDF] and set [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Under the zero curvature equation _{V x} = [ U V ] , we read [figure omitted; refer to PDF] Equation (4) is equivalent to [figure omitted; refer to PDF] where [figure omitted; refer to PDF] If we choose the initial conditions as [figure omitted; refer to PDF] then we can get all the other values according to (6). The first few sets are [figure omitted; refer to PDF] Let us associate (1) with the following problem: [figure omitted; refer to PDF] with [figure omitted; refer to PDF] The compatible condition of the spectral problem (1) and the auxiliary problem (10) is [figure omitted; refer to PDF]

After a direct calculation, we can get the AKNS-KN coupling system: [figure omitted; refer to PDF] Therefore, the AKNS-KN coupling system can be written as [figure omitted; refer to PDF] where the operator Φ is determined by (6) and the Hamiltonian operator is as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

To simplify (13), let _{u 1} = _{u 2} = _{u 5} = _{u 6} = 0 ; then the AKNS integrable coupling system (14) can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Furthermore, let _{u 1} = _{u 2} = _{u 5} = _{u 6} = _{u 7} = _{u 8} = 0 , and then _{u t n} = _{J 2} ^{Φ 2 n} ( _{u 4} , _{u 3} ^{) T} is the AKNS system, where [figure omitted; refer to PDF]

On the other hand, when _{u 3} = _{u 4} = _{u 7} = _{u 8} = 0 , then _{u t n} = _{J 3} ^{Φ 3 n} ( _{u 2} + _{u 6} , _{u 1} + _{u 5} , _{u 2} , _{u 1} ^{) T} is the KN integrable coupling system, where [figure omitted; refer to PDF]

When _{u 3} = _{u 4} = _{u 5} = _{u 6} = _{u 7} = _{u 8} = 0 , then _{u t n} = _{J 4} ^{Φ 4 n} ( _{u 2} , _{u 1} ^{) T} is the KN system, where [figure omitted; refer to PDF]

3. Bargmann Symmetry Constraint of AKNS-KN Coupling System

In order to get a Bargmann symmetry constraint, we can consider the Lax pairs and the adjoint Lax pairs of the AKNS-KN coupling system. The adjoint Lax pairs of the AKNS-KN coupling system are [figure omitted; refer to PDF] where T means the transpose of matrix and ψ = ( _{ψ 1} , _{ψ 2} , _{ψ 3} , _{ψ 4} ^{) T} . It follows from [figure omitted; refer to PDF] that [figure omitted; refer to PDF] where B = ∫ ... 2 ( Y9; _{ψ 1 ψ 1} YA; - Y9; _{ψ 2 ψ 2} YA; + Y9; _{ψ 3 ψ 3} YA; - Y9; _{ψ 4 ψ 4} YA; ) λ d x .

According to the zero boundary conditions _{lim ... | x | [arrow right] ∞} [varphi] = _{lim ... | x | [arrow right] ∞} ψ = 0 , we can get [figure omitted; refer to PDF] where Φ and δ λ / δ u are given by (6) and (26).

Now, let us discuss the spatial systems [figure omitted; refer to PDF] and the temporal systems [figure omitted; refer to PDF] where 1 ...4; j ...4; N and _{λ 1} , _{λ 2} , ... , _{λ N} are N distinct parameters. From [6] and [7], the expression of the potential u can be easily calculated: [figure omitted; refer to PDF]

4. Binary Nonlinearization of AKNS-KN Coupling System

In order to perform binary nonlinearization of AKNS-KN coupling system, let us substitute (30) into the Lax pairs and adjoint Lax pairs (28) and (29); then we can get the following nonlinearized spatial Lax pairs and the adjoint Lax pairs: [figure omitted; refer to PDF] where P ~ ( U ) means an expression of P ( U ) under the constraint equation (30). Clearly, (28) can be written: [figure omitted; refer to PDF] where Λ = diag ... ( _{λ 1 λ 2} , ... , _{λ N} ) . When n = 1 , the coupling system equation is exactly system equation with _{t 1} = x . Obviously, system (28) can be written in the following form: [figure omitted; refer to PDF] where the Hamiltonian form is the following: [figure omitted; refer to PDF] When _{u 3} = _{u 4} = _{u 7} = _{u 8} = 0 , then (33) can be written as [figure omitted; refer to PDF] and the Hamiltonian system is given by [figure omitted; refer to PDF] Furthermore, when _{u 3} = _{u 4} = _{u 5} = _{u 6} = _{u 7} = _{u 8} = 0 , then (33) deduces to [figure omitted; refer to PDF] and the Hamiltonian form is given by [figure omitted; refer to PDF]

When _{u 1} = _{u 2} = _{u 5} = _{u 6} = 0 , (33) is equivalent to the following: [figure omitted; refer to PDF] and the Hamiltonian system is given by [figure omitted; refer to PDF]

When _{u 1} = _{u 2} = _{u 5} = _{u 6} = _{u 7} = _{u 8} = 0 , (33) is equivalent to the following system: [figure omitted; refer to PDF] and the Hamiltonian system is given by [figure omitted; refer to PDF] When n = 2 , the coupling of system (29) is as follows: [figure omitted; refer to PDF] where _{u ~ 1 , x} , _{u ~ 2 , x} , _{u ~ 3 , x} , _{u ~ 4 , x} are given by [figure omitted; refer to PDF] Then we can get the following Hamiltonian form of (33): [figure omitted; refer to PDF] where the Hamiltonian system is given by [figure omitted; refer to PDF]

In what follows, we want to prove that (28) is a completed integrable Hamiltonian system in the Liouville sense. In addition, we want prove that (29) is also completed integrable system. From (29) and (5), we can obtain the following form: [figure omitted; refer to PDF]

Next, we can check that (31) is a Hamiltonian system. From (48), we know that coadjoint equation _{V x} = [ U V ] remains true. Furthermore, we know that ^{V ~ x 2} = [ U ~ , V ~ ] is also true. Let F = str [ ^{V ~ 2} ] ; it is clear that _{F x} = 0 . Let F = _{∑ n ...5; 0} ... _{F n} ^{λ - 2 n} ; we can get the following formulas of integrable of motion: [figure omitted; refer to PDF]

After a direct calculation, we have [figure omitted; refer to PDF] which means that the AKNS-KN coupling system is a Hamiltonian system. In order to prove that nonlinearized system is completely integrable in the Liouville sense, we choose the following Poisson bracket: [figure omitted; refer to PDF]

At this time, we still have the equality _{V ~ t n} = [ U ~ , V ~ ] . After a discussion, we know that F is also a generating function of the motion for equation, which makes [figure omitted; refer to PDF]

In addition, similar to the method in [6], we know that [figure omitted; refer to PDF] is integrable of motion for (45) and (46). It is easy to see that the 4N functions are involution in pairs.

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 11271007), Nature Science Foundation of Shandong Province of China (no. ZR2013AQ017), SDUST Research Fund (no. 2012KYTD105), and Open Fund of the Key Laboratory of Ocean Circulation and Waves, Chinese Academy of Science (no. KLOCAW1401).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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