A few new higher-dimensional Lie algebras and two types of coupling integrable couplings of the AKNS hierarchy and the KN hierarchy

Abstract

Four higher-dimensional Lie algebras are introduced. With the help of their different loop algebras and the block matrices of Lax pairs for the zero curvature representations of two given integrable couplings, the two types of coupling integrable couplings of the AKNS hierarchy and the KN hierarchy are worked out, respectively, which fill up the consequences obtained by Ma and Gao (2009) [9]. The coupling integrable couplings of the AKNS hierarchy obtained in the paper again reduce to the coupling integrable couplings of the nonlinear Schrödinger equation and the modified KdV (mKdV) equation, which are different from the resulting results given by Ma and Gao.

Introduction

The notion on integrable couplings was introduced when study of Virasoro symmetric algebras [1], [2]. Integrable couplings have much richer mathematical structures than scalar integrable equations [1], [2], [3], [4], [5], [6], [7], [8], [9]. Moreover, the study of integrable couplings generalizes the symmetric problem and provides clues towards complete classification of integrable equations [1], [2], [9], [10]. Therefore, it is important to investigate integrable couplings in soliton theory.

Consider an integrable evolution equation

$$u_t=K(u)\text{,}$$

where u is a column vector of dependent variables. Suppose it has a zero curvature representation [9], [10], [11]

$$U_t-V_x+[U\text{,}V]=0\text{,}$$

where the Lax pair, U and V, belongs to a matrix loop algebra. Given two integrable couplings of the integrable equation [1], [2], [9]:

$${\overline{u}}_{1\text{,}t}={\overline{K}}_1\left({\overline{u}}_1\right)=\left(\begin{array}{c}\hfill K(u)\hfill \\ \hfill S(u\text{,}v)\hfill \end{array}\right)\text{,}{\overline{u}}_1=\left(\begin{array}{c}\hfill u\hfill \\ \hfill v\hfill \end{array}\right)\text{,}$$

$${\overline{u}}_{2\text{,}t}={\overline{K}}_2\left({\overline{u}}_2\right)=\left(\begin{array}{c}\hfill K(u)\hfill \\ \hfill T(u\text{,}w)\hfill \end{array}\right)\text{,}{\overline{u}}_2=\left(\begin{array}{c}\hfill u\hfill \\ \hfill w\hfill \end{array}\right)\text{.}$$

A new bigger system is formed as follows:

$${\hat{u}}_t=\hat{K}\left(\hat{u}\right)=\left(\begin{array}{c}\hfill K(u)\hfill \\ \hfill S(u\text{,}v)\hfill \\ \hfill T(u\text{,}w)\hfill \end{array}\right)\text{,}\hat{u}=\left(\begin{array}{c}\hfill u\hfill \\ \hfill v\hfill \\ \hfill w\hfill \end{array}\right)\text{.}$$

We call (5) coupling integrable couplings of (3), (4). The paper [9] points that (5) is a degenerate system, since the second and third dependent variables are separate. In order to deduce the coupling integrable couplings (5), Ma and Gao [9] gave two types of classifications according to the Lax pairs for the zero curvature representations of the integrable couplings (3), (4).

Type one:

$${\overline{U}}_i\left({\overline{u}}_i\right)=\left(\begin{array}{cc}\hfill U(u)\hfill & \hfill U_i\left({\overline{u}}_i\right)\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\text{,}{\overline{V}}_i\left({\overline{u}}_i\right)=\left(\begin{array}{cc}\hfill V(u)\hfill & \hfill V_i\left({\overline{u}}_i\right)\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\text{,}i=1\text{,}2\text{,}$$

where ${\overline{u}}_1=(u^T\text{,}{v}^T{)}^T\text{,}{\overline{u}}_2=(u^T\text{,}{w}^T{)}^T$. The enlarged zero curvature equations [9]:

$${\overline{U}}_{i\text{,}t}-{\overline{V}}_{i\text{,}x}+\left[{\overline{U}}_i\text{,}{\overline{V}}_i\right]=0\text{,}i=1\text{,}2$$

equivalently give rise to

$$\left\{\begin{array}{c}{U}_t-V_x+[U\text{,}V]=0\text{,}\hfill \\ U_{i\text{,}t}-V_{i\text{,}x}+[U\text{,}{V}_i]=0\text{,}i=1\text{,}2\text{,}\hfill \end{array}\right.$$

which exactly present the integrable couplings (3), (4), respectively. As for the zero curvature representation of the coupled system (5) of two integrable couplings, there is the following theorem [9]:

Theorem 1

Assume that two integrable couplings (3), (4) have the zero curvature representations (7) with Lax pairs ${\overline{U}}_i\text{and}{\overline{V}}_i\text{,}i=1\text{,}2$, being defined by Eq. (6). Then the coupled system (5) has the following enlarged zero curvature representation:

$${\widehat{U}}_t-{\widehat{V}}_x+\left[\widehat{U}\text{,}\widehat{V}\right]=0$$

with the Lax pair being defined by

$$\widehat{U}\left(\hat{u}\right)=\left(\begin{array}{ccc}\hfill U(u)\hfill & \hfill 0\hfill & \hfill U_1(u\text{,}v)\hfill \\ \hfill 0\hfill & \hfill U(u)\hfill & \hfill U_2(u\text{,}w)\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\text{,}\widehat{V}\left(\hat{u}\right)=\left(\begin{array}{ccc}\hfill V(u)\hfill & \hfill 0\hfill & \hfill V_1(u\text{,}v)\hfill \\ \hfill 0\hfill & \hfill V(u)\hfill & \hfill V_2(u\text{,}w)\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\text{,}$$

where $\hat{u}=(u^T\text{,}{v}^T\text{,}{w}^T{)}^T\text{,}\widehat{U}\in \hat{g}\text{,}\widehat{V}\in \hat{g}$, here $\hat{g}$ is a Lie algebra consisting of square matrices of the following block form [9]:

$$\widehat{Q}=\left(\begin{array}{ccc}\hfill Q\hfill & \hfill 0\hfill & \hfill Q_1\hfill \\ \hfill 0\hfill & \hfill Q\hfill & \hfill Q_2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\text{,}$$

here Q is the same size square sub-matrix as U and V, and $Q_1\text{and}{Q}_2$ are the same size square sub-matrices as $U_1\text{and}{U}_2$. $\hat{g}$ has two sub-Lie algebras

$${\hat{g}}_1=\left\{\left.\widehat{Q}\right|Q_2=0\right\}\text{,}{\hat{g}}_2=\left\{\left.\widehat{Q}\right|Q_1=0\right\}\text{.}$$

The two given integrable couplings (3), (4) are generated from those two sub-Lie algebras ${\overline{g}}_i\text{,}i=1\text{,}2$, respectively.

Type two:

$${\overline{U}}_i\left({\overline{u}}_i\right)=\left(\begin{array}{cc}\hfill U(u)\hfill & \hfill U_i\left({\overline{u}}_i\right)\hfill \\ \hfill 0\hfill & \hfill U(u)\hfill \end{array}\right)\text{,}{\overline{V}}_i\left({\overline{u}}_i\right)=\left(\begin{array}{cc}\hfill V(u)\hfill & \hfill V_i\left({\overline{u}}_i\right)\hfill \\ \hfill 0\hfill & \hfill V(u)\hfill \end{array}\right)\text{,}i=1\text{,}2\text{.}$$

The enlarged zero curvature equations

$${\overline{U}}_{i\text{,}t}-{\overline{V}}_{i\text{,}x}+\left[{\overline{U}}_i\text{,}{\overline{V}}_i\right]=0\text{,}i=1\text{,}2$$

equivalently yields [9]:

$$\left\{\begin{array}{c}{U}_t-V_x+[U\text{,}V]=0\text{,}\hfill \\ U_{i\text{,}t}-V_{i\text{,}x}+[U\text{,}{V}_i]+[U_i\text{,}V]=0\text{,}i=1\text{,}2\text{,}\hfill \end{array}\right.$$

which exactly present the integrable couplings (3), (4), respectively. The associated Lie algebra $\hat{g}$ consists of the following block form:

$$\widehat{P}=\left(\begin{array}{ccc}\hfill P\hfill & \hfill 0\hfill & \hfill P_1\hfill \\ \hfill 0\hfill & \hfill P\hfill & \hfill P_2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill P\hfill \end{array}\right)\text{,}$$

where $P\text{,}{P}_1\text{and}{P}_2$ are the same size square sub-matrices as U and V. Similarly, Ma and Gao proposed the following theorem [9]:

Theorem 2

Assume that two integrable couplings (3), (4) have the zero curvature representation (14) with Lax pairs ${\overline{U}}_i\text{and}{\overline{V}}_i\text{,}i=1\text{,}2$, being defined by Eq. (13). Then the coupled system (5) has an enlarged zero curvature representation

$${\widehat{U}}_t-{\widehat{V}}_x+\left[\widehat{U}\text{,}\widehat{V}\right]=0\text{,}$$

with the Lax pair being defined by

$$\widehat{U}\left(\hat{u}\right)=\left(\begin{array}{ccc}\hfill U(u)\hfill & \hfill 0\hfill & \hfill U_1(u\text{,}v)\hfill \\ \hfill 0\hfill & \hfill U(u)\hfill & \hfill U_2(u\text{,}w)\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill U(u)\hfill \end{array}\right)\text{,}\widehat{V}\left(\hat{u}\right)=\left(\begin{array}{ccc}\hfill V(u)\hfill & \hfill 0\hfill & \hfill V_1(u\text{,}v)\hfill \\ \hfill 0\hfill & \hfill V(u)\hfill & \hfill V_2(u\text{,}w)\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill V(u)\hfill \end{array}\right)\text{,}$$

where $\hat{u}={\left(u^T\text{,}{v}^T\text{,}{w}^T\right)}^T$. As application examples of the above types, Ma and Gao [9] discussed the coupling integrable couplings of the nonlinear Schrödinger equations.

In this paper, we base on the two types of Lax pairs for the curvature representations of the integrable couplings (3), (4), we construct four kinds of higher-dimensional Lie algebras and employ the Tu scheme [12], [13] to deduce the first type and the second type coupling integrable couplings of the AKNS equation hierarchy and the KN equation hierarchy, respectively. Moreover, we obtain two types of coupling integrable couplings of the nonlinear Schrödinger equations and the mKdV equation as the reduced cases, which enrich some results presented in [9].