Abstract and Applied Analysis

Volume 2014 (2014), Article ID 357621, 6 pages

http://dx.doi.org/10.1155/2014/357621

Research Article

A New Approach for Generating the TX Hierarchy as well as Its Integrable Couplings

Guangming Wang

College of Teacher Education, Tianjin Normal University, Tianjin 300387, China

Received 20 January 2014; Accepted 3 May 2014; Published 11 May 2014

Academic Editor: Huanhe Dong

Copyright © 2014 Guangming Wang. 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.

Abstract.

Tu Guizhang and Xu Baozhi once introduced an isospectral problem by a loop algebra with degree being , for which an integrable hierarchy of evolution equations (called the TX hierarchy) was derived under the frame of zero curvature equations. In the paper, we present a loop algebra whose degrees are and to simply represent the above isospectral matrix and easily derive the TX hierarchy. Specially, through enlarging the loop algebra with 3 dimensions to 6 dimensions, we generate a new integrable coupling of the TX hierarchy and its corresponding Hamiltonian structure.

1. Introduction

Since the theory on integrable couplings was proposed [1, 2], some integrable couplings and properties were obtained, such as the results in [3–10]. Tu and Xu [11] employed loop algebra which is subalgebra of the loop algebra with degree being to obtain an integrable hierarchy, which is called by us the TX hierarchy, and its corresponding Hamiltonian structure. In the paper, we would like to extend the loop algebra with 3 dimensions into enlarged loop algebra with 6 dimensions so that an integrable coupling of the TX hierarchy can be derived, the Hamiltonian structure of which is also produced by making use of the variational identity [5].

In paper [7], the Lie algebra was once presented as follows:

where

along with commutative relations as follows:

where

form subalgebra of the Lie algebra , denoted by again; that is, . The corresponding loop algebra of was given by

Through the some integrable couplings were obtained, and some exact solutions of the reduced equations were also produced. In the paper, by redefining the degrees of the Lie algebra , we give the following loop algebra:

where

The commutative relations read that

2. The TX Hierarchy and Its Integrable Coupling

In the section, we want to investigate the TX hierarchy and its integrable coupling by employing the loop algebra under the frame of zero curvature equations. Then through the trace identity proposed by Tu [12] and the variational identity [5], we derive the Hamiltonian structure of the TX hierarchy and the Hamiltonian structure of the integrable coupling, respectively.

Obviously, the Lie algebra is isomorphic to the subalgebra of the Lie algebra , where , that is,

The resulting loop algebra is also isomorphic to , where

equipped with

Based on the above fact, the isospectral matrix presented in [11] can be written as

or

Set

or

It is easy to check that the stationary zero curvature equations

have the same solutions for . That is, starting from (16), we can derive all of the following recursion relations among :

which is equivalent to

Given some initial values ,  , (18) admits some explicit solutions as follows:

Denoting

we can obtain that

Thus, the compatibility condition of the Lax pair

gives rise to

and we call (23) the TX hierarchy.

When ,  , (23) reduces to

which is called the TX equation.

In what follows, we discuss the Hamiltonian structure of the TX hierarchy (23) by using the loop algebra . Equation (14) can be written as

where ,  .

A direct calculation gives that

Substituting these consequences into the trace identity yields

Comparing the coefficients of in (27), we have

By the previous initial values, we see that . Thus, we get

Therefore, the TX hierarchy (23) can be written as

In order to derive the integrable coupling of the TX hierarchy, we introduce the following Lax matrices:

According to the Tu scheme [12], the stationary zero curvature equation

leads to the following recursion relations:

plus (17), which are equivalent to

plus (18).

If we set ,  , which are the initial values, then we have from (18) and (34) that

Denoting

then we have

Hence, the zero curvature equation

gives that

When , (39) reduces to the TX hierarchy. According to the theory on integrable couplings, (39) is a kind of integrable coupling of the TX hierarchy.

We consider a reduced case of (39). Set ; we get an integrable coupling of the TX equation (24):

When we set , the above equation reduces to (24). In addition, if we set , the above equation reduces to two different heat equations.

Next, we discuss the Hamiltonian structure of the integrable coupling (39), that is, the TX hierarchy. For this reason, we need to construct Lie algebra which is isomorphic to the Lie algebra . Consider the linear space . Define an operation on as follows:

where

It can be verified that becomes Lie algebra if equipped with the commutator (41). In addition, we assume a linear map

We can prove that is an isomorphism between and . Therefore, (31) can be written as

where ,  .

In order to make use of the variational identity to derive the Hamiltonian structure of (39), we need to solve a matrix equation as follows:

where is a constant matrix independent of and .

From (45), we can obtain that

from which we construct a linear functional

where is the corresponding loop algebra of the Lie algebra . A direct calculation gives, by using (44) and (47), that

Substituting the above consequences into the variational identity leads to

Comparing the coefficients of in (49), we get that

It can be determined that in terms of the initial values of (18) and (34). Thus, we have

Therefore, the integrable coupling (39) can be written as the Hamiltonian form

Obviously, is a Hamiltonian operator. To our knowledge, (52) is completely new consequence.

Conflict of Interests

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

References

B. Fuchssteiner, “Coupling of completely integrable system: the perturbation bundle,” in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, P. A. Clarkson, Ed., NATO ASI Series, pp. 125–138, Kluwer Academic, Dodrecht, The Netherlands, 1993.

W.-X. Ma, “Integrable couplings of soliton equations by perturbation I. A general theory and application to the KdV equation,” Methods and Applications of Analysis, vol. 7, p. 21, 2000.

W.-X. Ma, “Integrable couplings of vector AKNS soliton equations,” Journal of Mathematical Physics, vol. 46, no. 3, pp. 1–19, 2005.

W.-X. Ma, X.-X. Xu, and Y. Zhang, “Semi-direct sums of Lie algebras and continuous integrable couplings,” Physics Letters A: General, Atomic and Solid State Physics, vol. 351, no. 3, pp. 125–130, 2006.

W.-X. Ma and M. Chen, “Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras,” Journal of Physics A: Mathematical and General, vol. 39, no. 34, pp. 10787–10801, 2006.

Y. F. Zhang and H. Q. Zhang, “A direct method for integrable couplings of TD hierarchy,” Journal of Mathematical Physics, vol. 43, no. 1, pp. 466–472, 2002.

Y. F. Zhang and J. Liu, “Induced Lie algebras of a six-dimensional matrix Lie algebra,” Communications in Theoretical Physics, vol. 50, no. 2, pp. 289–294, 2008.

Y. F. Zhang, “Lie algebras for constructing nonlinear integrable couplings,” Communications in Theoretical Physics, vol. 56, no. 5, pp. 805–812, 2011.

T. C. Xia, F. C. You, and D. Y. Chen, “A generalized cubic Volterra lattice hierarchy and its integrable couplings system,” Chaos, Solitons and Fractals, vol. 27, no. 1, pp. 153–158, 2006.

H. H. Dong and X. Q. Liang, “A new multi-component hierarchy and its integrable expanding model,” Chaos, Solitons and Fractals, vol. 38, no. 2, pp. 548–555, 2008.

G. Z. Tu and B. Z. Xu, “The trace identity, a powerful tool for constructing the Hamiltonian structures of integrable systems,” Chinese Annals of Mathematics B, vol. 17, p. 497, 1996.

G. Z. Tu, “The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems,” Journal of Mathematical Physics, vol. 30, no. 2, pp. 330–338, 1989.