Abstract and Applied Analysis

Volume 2014 (2014), Article ID 860935, 7 pages

http://dx.doi.org/10.1155/2014/860935

Research Article

Two Expanding Integrable Models of the Geng-Cao Hierarchy

Xiurong Guo,¹ Yufeng Zhang,² and Xuping Zhang¹

¹Department of Basic Courses, Shandong University of Science and Technology, Tai’an, Shandong 271019, China

²College of Science, China University of Mining and Technology, Xuzhou 221116, China

Received 21 November 2013; Accepted 13 January 2014; Published 27 February 2014

Academic Editor: Tiecheng Xia

Copyright © 2014 Xiurong Guo et al. 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.

As far as linear integrable couplings are concerned, one has obtained some rich and interesting results. In the paper, we will deduce two kinds of expanding integrable models of the Geng-Cao (GC) hierarchy by constructing different 6-dimensional Lie algebras. One expanding integrable model (actually, it is a nonlinear integrable coupling) reduces to a generalized Burgers equation and further reduces to the heat equation whose expanding nonlinear integrable model is generated. Another one is an expanding integrable model which is different from the first one. Finally, the Hamiltonian structures of the two expanding integrable models are obtained by employing the variational identity and the trace identity, respectively.

1. Introduction

Integrable couplings are a kind of expanding integrable models of some known integrable hierarchies of equations. Based on this theory, one has obtained some integrable couplings of the known integrable hierarchies [1–8]. These integrable couplings are all linear with respect to the coupled variables. That is, if we introduce an evolution equation , the coupled variable satisfying is linear in . The reason for this may be given by special Lie algebras. That is, such a Lie algebra can be decomposed into a sum of the two subalgebras and , which meets

If the subalgebra is not simple, then the integrable coupling

is linear with respect to the variable , which is obtained by introducing Lax pairs through the Lie algebra . However, it is more interesting to seek for nonlinear integrable couplings because most of the coupled dynamics from physics, mechanics, and so forth are nonlinear. Recently, Ma and Zhu [9] introduced a kind of Lie algebra to deduce the nonlinear integrable couplings of the nonlinear Schrödinger equation and so forth, where the Lie subalgebras are simple and are different from the above. Based on this, Zhang [10] proposed a simple and efficient method for generating nonlinear integrable couplings and obtained the nonlinear integrable couplings of the Giachetti-Johnson (GJ) hierarchy and the Yang hierarchy, respectively. In addition, Zhang and Hon [11] proposed another Lie algebra which is different from those in [9, 10] to deduce nonlinear integrable couplings. Wei and Xia [12] also obtained some nonlinear integrable couplings of the known integrable hierarchies.

In the paper, we want to start from a spectral problem proposed by Geng and Cao [13] to deduce an integrable hierarchy (called the GC hierarchy) under the frame of zero curvature equations by the Tu scheme [14] and obtain its new Hamiltonian structure. Then with the help of a 6-dimensional Lie algebra, a nonlinear expanding integrable model of the GC hierarchy is obtained, whose Hamiltonian structure is generated by the variational identity presented in [15]. The expanding integrable model can reduce to a generalized Burgers equation and further reduce to the heat equation. Another new 6-dimensional Lie algebra is constructed for which the second expanding integrable model is produced by using the Tu scheme whose Hamiltonian structure is derived from the trace identity proposed by Tu [14]. We shall find the two expanding integrable models of the GC hierarchy are different.

2. The GC Integrable Hierarchy and Its Hamiltonian Structure

We have known that

then one gets

It is well known that is a Lie algebra. A loop algebra of is given by

where

By using the loop algebra , introduce an isospectral problem [13]:

Set

where

The stationary equation admits that a solution for the is as follows:

which gives rise to

Set

from (10) and (11) we have

denote by

We have

The compatibility of the following Lax pair

gives rise to

where

is a Hamiltonian operator.

By the trace identity presented in [14], we have

Substituting the above results to the trace identity yields

where

Comparing the coefficients of of both sides in (20) gives

It is easy to see . Thus, we have

where

are Hamiltonian conserved densities of the Lax integrable hierarchy (17). Therefore, we get a Hamiltonian form of the hierarchy (17) as follows:

Let us consider the reduced cases of (17). When , we get that

Taking , one gets a generalized Burgers equation:

Remark 1. The Hamiltonian structure (23) is different from that in [14]. We call (17) the GC hierarchy.

3. The First Expanding Integrable Model of the GC Hierarchy

Zhang and Tam [16] proposed a few kinds of Lie algebras to deduce nonlinear integrable couplings. In the section we will choose one of them to investigate the nonlinear integrable coupling of the hierarchy (17).

Consider the following Lie algebra:

where

Define

It is easy to compute that

Set

we have

and are all simple Lie-subalgebras of the Lie algebra . The corresponding symmetric constant matrix appearing in the variational identity is that

A loop algebra corresponding to the Lie algebra is defined by

We use the loop algebra to introduce a Lax pair:

The stationary equation is equivalent to

from which we have

Set

we obtain from (37)

Note

a direct calculation yields

Therefore, zero curvature equation

admits that

Set , (44) reduces to the integrable hierarchy (17). When we take , we get an expanding nonlinear integrable model of the generalized Burgers equation (27) as follows:

Obviously, the coupled equations are nonlinear with respect to the coupled variables and . Therefore, the hierarchy (44) is a nonlinear expanding integrable model of the integrable system (17); actually, it is a nonlinear integrable coupling.

The nonlinear expanding integrable model (45) can be written as two parts, one is just right (27); another one is the latter two equations in (45), which can be regarded as a coupled nonlinear equation with variable coefficients , , and their derivatives in the variable , where the functions , satisfy (27). In particular, we take a trivial solution of (27) to be ; then (45) reduces to the following equations:

When we set , the above equations reduce to the well-known heat equation.

In order to deduce Hamiltonian structure of the nonlinear integrable coupling (44), we define a linear functional [11]:

where , .

It is easy to see that the Lie algebra is isomorphic to the Lie algebra if equipped with a commutator as follows:

Thus, under the Lie algebra , the Lax pair (36) can be written as

In terms of (48) and (49) we obtain that

Substituting the above results into the variational identity yields

where

Comparing the coefficients of on both sides in (51) gives

From (37) we have . Thus, we get that

where

Therefore, we obtain the Hamiltonian structure of the nonlinear integrable coupling (44) as follows:

where is obviously Hamiltonian.

4. The Second Expanding Integrable Model of the GC Hierarchy

In this section we construct a new 6-dimensional Lie algebra to discuss the second integrable coupling of the GC hierarchy. Set

It is easy to see that

If we set , , and , then we have that

Hence, the integrable couplings of the GC hierarchy cannot be generated by the Lie algebra as above under the frame of the Tu scheme. In what follows, we will deduce a nonlinear expanding integrable model of the GC hierarchy.

Set

where , .

Solving the stationary zero curvature equation

gives rise to

Let , , , and ; then one gets from (62) that

Noting , one infers that .

Set , by employing the zero curvature equation

we have

When , , (65) reduces to

It is remarkable that (66) is linear with respect to the variables , ; however, it is nonlinear.

Equation (60) can be written as

By computing that

thus, we have

Substituting the above consequences into the trace identity proposed by Tu [14] yields that

Comparing the coefficients of gives

Inserting the initial values in (62) gives . Therefore, we obtain that

where are conserved densities of the expanding integrable model (65). Thus, (65) can be written as the Hamiltonian structure

where , , is a Hamiltonian operator.

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundation of China (11371361) and the Natural Science Foundation of Shandong Province (ZR2012AQ011, ZR2012AQ015).

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