The Liouville integrable coupling system of the m-AKNS hierarchy and its Hamiltonian structure

In this paper a type of 9-dimensional vector loop algebra F is constructed, which is devoted to establish an isospectral problem. It follows that a Liouville integrable coupling system of the m-AKNS hierarchy is obtained by employing the Tu scheme, whose Hamiltonian structure is worked out by making use of constructed quadratic identity. The method given in the paper can be used to obtain many other integrable couplings and their Hamiltonian structures.

A NEW HIERARCHY OF LAX AND LIOUVILLE INTEGRABLE EVOLUTION EQUATIONS ASSOCIATED WITH AN ISOSPECTRAL PROBLEM IN THE LOOP ALGEBRA■

在这篇论文，有五个潜力的一个 isospectral 问题在环代数学 A_2 被调查以便有五任意的功能的进化方程的一个新层次被获得。然后由修理五任意的功能是某些功能并且使用踪迹身份，进化方程的层次的概括 Hamiltonian 结构被给。方程的这个层次是 Liouville integrable，这被显示出。最后， isospectral 问题的一些特殊情况也被给。

High-Order Binary Symmetry Constraints of a Liouville Integrable Hierarchy and Its Integrable Couplings

<Abstract>A 3×3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra _2.Furthermore,high-order binary symmetry constraints of the corresponding hierarchy are obtained by using the binary nonlinearization method.Finally,according to another new subalgebra of loop algebra _2, its integrable couplings are established.

Derivation of Expanded Isospectral-Nonisospectral Integrable Hierarchies via the Column-vector Loop Algebra

A scheme for generating nonisospectral integrable hierarchies is introduced.Based on the method,we deduce a nonisospectral hierarchy of soliton equations by considering a linear spectral problem.It follows that the corresponding expanded isospectral and nonisospectral integrable hierarchies are deduced based on a 6 dimensional complex linear space ■.By reducing these integrable hierarchies,we obtain the expanded isospectral and nonisospectral derivative nonlinear Schr?dinger equation.By using the trace identity,the biHamiltonian structure of these two hierarchies are also obtained.Moreover,some symmetries and conserved quantities of the resulting hierarchy are discussed.

A Multi-component Matrix Loop Algebra and Its Application

A set of multi-component matrix Lie algebra is constructed. It follows that a type of new loop algebra AM-1 is presented. An isospectral problem is established. Integrable multi-component hierarchy is obtained by Tu pattern, which possesses tri-Hamiltonian structures. Furthermore, it can be reduced to the well-known AKNS hierarchy and BPT hierarchy. Therefore, the major result of this paper can be regarded as a unified expression integrable model of the AKNS hierarchy and the BPT hierarchy.

New Matrix Loop Algebra and Its Application

A new matrix Lie algebra and its corresponding Loop algebra are constructed firstly,as its application,the multi-component TC equation hierarchy is obtained,then by use of trace identity the Hamiltonian structure of the above system is presented.Finally,the integrable couplings of the obtained system is worked out by the expanding matrix Loop algebra.

Integrable Couplings of Classical-Boussinesq Hierarchy and Its Hamiltonian Structure

由使用谎言代数学， classical-Boussinesq 层次的 integrable couplings 被获得。然后， classical-Boussinesq 的 integrable couplings 的 Hamiltonian 结构被二次形式的身份获得。

Two Types of Expanding Lie Algebra and New Expanding Integrable Systems

From a new Lie algebra proposed by Zhang, two expanding Lie algebras and its corresponding loop algebrasare obtained.Two expanding integrable systems are produced with the help of the generalized zero curvature equation.One of them has complex Hamiltion structure with the help of generalized Tu formula (GTM).

一族Liouville可积系及其双Hamilton结构

本文得到了一族Liouville可积系含有6个因变量，具有双Hamilton结构，L*=J2J是一个遗传对称。另外，它可约化为Liouville可积的BPT族与MKdV族。

一个新的方程族及其Liouville可积性

从等谱问题出发,基于Loop代数 A1的基的个数与换位运算,利用屠规彰格式得到了一族方程及其Hamilton结构,证明了该方程是Liouville可积的,作为该系统的约化,得到了著名的Schr¨odinger方程,广义Mkdv方程,热传导方程和耦合的Burgers方程.

一类新的圈代数和它的Liouville可积演化方程族

基于L ie代数An-1的推广,构造了一类新的圈代数,并设计了一个新的谱问题。然后,利用屠格式获得了一个新的可积系统,并推导出它相应的非线性演化方程族,最后,证明了该演化方程族在L iouv ille意义下是可积的。

一族Liouville可积系及其Hamilton结构

指出了获得可积系的方法一般是在零曲率框架Ut-V(n)x+[U,V(n)]=0中进行,所选取的V的形式为V=∑m≥0Vmλ-m,Vm=ambmcm-am.构造了一个等谱问题,特别是对于该等谱问题,选取的V中既含有谱位势,又含有a,b,c的关于x的导数项,得到了一个新的Lax对.利用屠格式获得了一族新的Liouville可积系,具有Hamil ton结构.

An integrable Hamiltonian hierarchy and associated integrable couplings system

This paper establishes a new isospectral problem. By making use of the Tu scheme, a new intcgrablc system is obtained. It gives integrable couplings of the system obtained. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.

Two new discrete integrable systems

In this paper,we focus on the construction of new（1＋1）-dimensional discrete integrable systems according to a subalgebra of loop algebra A 1.By designing two new（1＋1）-dimensional discrete spectral problems,two new discrete integrable systems are obtained,namely,a 2-field lattice hierarchy and a 3-field lattice hierarchy.When deriving the two new discrete integrable systems,we find the generalized relativistic Toda lattice hierarchy and the generalized modified Toda lattice hierarchy.Moreover,we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity.

两类新的Lie代数和它们的Liouville可积演化方程族(英文)

把 n − 1 , 新谎言代数学的二种类型被得出，进化方程的相关层次的 integrability 在 Liouville 的意义被证明。

在Loop代数 _3框架内获得可积耦合的一类方法

本文首先构造了Loop代数A3的一个特殊子代数G,使和其二个子代数G1,G2满足关系G=G1(?)G2。利用G1构造一个等谱问题;再利用屠格式得到一个具有双Hamilton结构的新的孤子方程族。其次,利用G1的基元的线性组合,得到了一个另一类Loop代数G1,由此再利用屠格式获得第二类新的可积系。最后,由可积耦合理论,得到第一个孤子族的可积耦合系统。类似地,构造了Loop代数A3的子代数G1的一个扩展Loop代数G,获得第二孤子族的可积耦合。

Loop代数_2的一个新子代数及其有关的一族新的可积系

构造了Loop代数 A2的一个新的子代数,由此设计了一个等谱问题,利用屠格式获得一类新的Liouville可积系,且具有双Hamilton结构.作为其约化,得到了一族非线性广义Schrodinger方程.

Loop代数_2的一个新子代数及其有关的一族新的可积系

构造了Loop代数A^2的一个新的子代数,由此设计了一个等谱问题,利用屠格式获得一类新的Liouville可积系,且具有双Hamilton结构.作为其约化,得到了一族非线性广义Schrodinger方程.

一个新的loop代数及相应的一族可积系

构造了一个复数 loop代数 A*1 ,由此设计了一个复的Lax对,根据其相容性得到了一族新的可积系.再利用迹恒等式,求出了该可积系的分解Hamilton结构.作为约化情形,获得了一个类似于AKNS族的可积系统.

Loop Algebras and Bi-integrable Couplings

A class of non-semisimple matrix loop algebras consisting of triangular block matrices is introduced and used to generate bi-integrable couplings of soliton equations from zero curvature equations.The variational identities under non-degenerate,symmetric and ad-invariant bilinear forms are used to furnish Hamiltonian structures of the resulting bi-integrable couplings.A special case of the suggested loop algebras yields nonlinear bi-integrable Hamiltonian couplings for the AKNS soliton hierarchy.