A few Lie algebras and their applications for generating integrable hierarchies of evolution types

Abstract

A Lie algebra consisting of 3 × 3 matrices is introduced, whose induced Lie algebra by using an inverted linear transformation is obtained as well. As for application examples, we obtain a unified integrable model of the integrable couplings of the AKNS hierarchy, the D-AKNS hierarchy and the TD hierarchy as well as their induced integrable hierarchies. These integrable couplings are different from those results obtained before. However, the Hamiltonian structures of the integrable couplings cannot be obtained by using the quadratic-form identity or the variational identity. For solving the problem, we construct a higher-dimensional subalgebra R and its reduced algebra Q of the Lie algebra A₂ by decomposing the induced Lie algebra and then again making some linear combinations. The subalgebras of the Lie algebras R and Q do not satisfy the relation ($G=G_1\oplus G_2\text{,}[G_1\text{,}{G}_2]\subset G_2$), but we can deduce integrable couplings, which indicates that the above condition is not necessary to generate integrable couplings. As for application example, an expanding integrable model of the AKNS hierarchy is obtained whose Hamiltonian structure is generated by the trace identity. Finally, we give another Lie algebras which can be decomposed into two simple Lie subalgebras for which a nonlinear integrable coupling of the classical Boussinesq–Burgers (CBB) hierarchy is obtained.

Introduction

Since the notion of integrable couplings was proposed [1], [2], the integrable couplings of some well-known integrable soliton hierarchies were obtained. For example, Ma [2], [3] took use of the perturbation method to obtain the integrable couplings of the KdV system. Based on this, Guo and Zhang [4] proposed a simple and straightforward method for generating integrable couplings by introducing a kind of Lie algebra and the corresponding loop algebra which consists of square matrices. From which, the integrable couplings of the AKNS hierarchy was derived [4]. By following the method, the integrable couplings of other known integrable soliton hierarchies, such as the TD hierarchy [5] and the KN hierarchy [4]. were also worked out [6], [7]. Although the trace identity proposed by Tu [5], [8] is a powerful tool for deducing Hamiltonian structures of soliton hierarchies, however, it fails to generate Hamiltonian structures of integrable couplings. In order to overcome the problem, Guo and Zhang [9] proposed the quadratic-form identity, which is a very useful means to produce Hamiltonian structures of integrable couplings. By using the identity, we have obtained the integrable couplings of the AKNS hierarchy, the KN hierarchy and so on [9], [10]. Ma and Chen [11] generalized the quadratic-form identity and built the more general variational identity for which a kind of integrable couplings of the AKNS hierarchy was obtained. It is a more general tool that the variational identity and its extensive applications were reported in one hour by Ma at WCNA 2008 [12].

When deducing integrable couplings of some soliton equations with help of Lie algebras, we need the Lie algebra G satisfy the condition, i.e.,

$$G=G_1\oplus G_2\text{,}[G_1\text{,}{G}_2]\subset G_2\text{,}$$

where G₁ and G₂ are two subalgebras of the Lie algebra G, the symbol ⊕ represents a direct sum or semi-direct sum. In addition, Ma et al. [13] classified the spectral matrices expressed by Lie algebras as follows:

$$U_1=\left(\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill o\hfill & \hfill o\hfill \end{array}\right)\text{,}{U}_2=\left(\begin{array}{cc}\hfill A\hfill & \hfill B_1\hfill \\ \hfill o\hfill & \hfill B_2\hfill \end{array}\right)\text{,}{U}_3=\left(\begin{array}{cccc}\hfill U\hfill & \hfill U_{a_1}\hfill & \hfill \dots \hfill & \hfill U_{a_v}\hfill \\ \hfill o\hfill & \hfill U\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill U_{a_1}\hfill \\ \hfill o\hfill & \hfill \dots \hfill & \hfill o\hfill & \hfill U\hfill \end{array}\right)\text{.}$$

In the paper, we want to construct a 3 × 3 matrix Lie algebra H and its two subalgebra H₁ and H₂, which satisfy the condition (1). However, the corresponding spectral matrix do not belong to any one in (2). We shall employ the Lie algebra H and its loop algebra $\stackrel{\sim }{H}$ to deduce a type of soliton hierarchy which can be reduced to the integrable couplings of the well-known AKNS hierarchy, the D-AKNS hierarchy and the TD hierarchy. That is to say, We provide a unified integrable model of the integrable couplings of the AKNS hierarchy, D-AKNS hierarchy and the TD hierarchy. Besides, an induced Lie algebra G of the Lie algebra H is obtained by using an inverted linear transformation. As application examples, we can obtain an integrable system which is also reduced to a few integrable couplings of the AKNS hierarchy, the D-AKNS hierarchy. These integrable couplings are far cry from those obtained before. We shall see below that the Hamiltonian structures of the integrable couplings cannot be obtained by using the trace identity and the variational identity. For overcoming the problem, we construct a new subalgebra denoted by R of the Lie algebra A₂ by decomposition the basis of the Lie algebra G and then making some linear combination. Especially, the Lie algebra R reduces to the Lie algebra Q given by Fordy and Gibbons [20] which was once used to generate the modified Boussinesq equation and the nonlinear Klein–Gordon equation. From this, we see the importance of the Lie algebras presented in the paper. By using the Lie algebra Q, we obtain a type of integrable coupling of the AKNS hierarchy, specially we produce its Hamiltonian structure by using the trace identity. The feature of the Lie algebras Q and R presents that their subalgebras cannot meet the relation (1), but we can use them to deduce integrable couplings as usual. Similarly, we can deduce the integrable couplings of the D-AKNS hierarchy and the TD hierarchy, and so on. Finally, based on the known Lie algebras, we introduce another Lie algebra which possesses two simple Lie subalgebras. As its application, we obtain a nonlinear integrable coupling of the CBB equation hierarchy.