Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra

By using a six-dimensional matrix Lie algebra [Y.F. Zhang and Y. Wang, Phys. Lett. A 360 （2006） 92], three induced Lie algebras are constructed. One of them is obtained by extending Lie bracket, the others are higher-dimensional complex Lie algebras constructed by using linear transformations. The equivalent Lie algebras of the later two with multi-component forms are obtained as well. As their applications, we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.     

推广的一类Lie代数及其相关的一族可积系统

对已知的Lie代数An-1作直接推广得到一类新的Lie代数gl(n,C).为应用方便,本文只考虑Lie代数gl(3,C)情形.构造了gl(3,C)的一个子代数,通过对阶数的规定,得到了一类新的loop代数.作为其应用,设计了一个新的等谱问题,得到了一个新的Lax对.利用屠格式获得了一族新的可积系统,具有双Hamilton结构,且是Liouville可积系.作为该方程族的约化情形,得到了新的耦合广义Schrdinger方程.关键词：Lie代数可积系Hamilton结构     

A New Liouville Integrable Hamiltonian System

With the help of a Lie algebra, an isospectral Lax pair is introduced for which a new Liouville integrable hierarchy of evolution equations is generated. Its Hamiltonian structure is also worked out by use of the quadratic-form identity.     

Three New Integrable Hierarchies of Equations

A general Lie algebra Vs and the corresponding loop algebra Vx are constructed, from which the linear isospectral Lax pairs are established, whose compatibility presents the zero curvature equation. As its application, a new Lax integrable hierarchy containing two parameters is worked out. It is not Liouville-integrable, however, its two reduced systems are Liouville-integrable, whose Hamiltonian structures are derived by making use of the quadratic-form identity and the γ formula （i.e. the computational formula on the constant γ appeared in the trace identity and the quadratic-form identity）.     

Two Integrable Couplings of Modified Tu Hierarchy

Two different integrable couplings of the modified Tu hierarchy are obtained under the zero curvature equation by using two higher dimension Lie algebras. Furthermore, a complex Hamiltonian structures of the second integrable couplings is presented by taking use of the variational identity.     

A Type of New Loop Algebra and a Generalized Tu Formula

A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a kind of zero curvature equation, which permits Lax integrable hierarchies of soliton equations. To aim at generating Hamiltonian structures of such soliton-equation hierarchies, a beautiful Killing-Cartan form, a generalized trace functional of matrices, is given, for which a generalized Tu formula （GTF） is obtained, while the trace identity proposed by Tu Guizhang [J. Math. Phys. 30 （1989） 330] is a special case of the GTF. The computing formula on the constant γ to be determined appearing in the GTF is worked out, which ensures the exact and simple computation on it. Finally, we take two examples to reveal the applications of the theory presented in the article. In details, the first example reveals a new Liouville-integrable hierarchy of soliton equations along with two potential functions and Hamiltonian structure. To obtain the second integrable hierarchy of soliton equations, a higher-dimensional loop algebra is first constructed. Thus, the second example shows another new Liouville integrable hierarchy with 5-potential component functions and bi- Hamiltonian structure. The approach presented in the paper may be extensively used to generate other new integrable soliton-equation hierarchies with multi-Hamiltonian structures.     

The Double Integrable Couplings of Tu Hierarchy

Two types of Lie algebras are presented, from which two integrable couplings associated with the Tu isospectral problem are obtained, respectively. One of them possesses the Hamiltonian structure generated by a linear isomorphism and the quadratic-form identity. An approach for working out the double integrable couplings of the same integrable system is presented in the paper.     

A Lie algebra structure on variation vector fields along curves in 2-dimensional space forms

A Lie algebra structure on variation vector fields along an immersed curve in a 2-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar filament equation are finally discussed from this point of view.     

New Matrix Lie Algebra and Its Application

A set of new matrix Lie algebra and its corresponding loop algebra are constructed. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equation is generated. As its reduction cases, the multi-component Tu hierarchy is given. Finally, the multi-component integrable coupling system of Tu hierarchy is presented through enlarging matrix spectral problem.     

完全可积的非线性方程建立哈密顿理论的一般方法和对SG方程应用

完全可积的非线性方程的单式矩阵的泊松括号已知可以表为对x的积分,指出被积函数一定 可以表为约斯特解对的直积的线性组合的微分,并可由直积矩阵相应元的对比确定组合系数 .从而解决了建立非线性方程哈密顿理论的一般方法.由于实验室系中的SG方程,相应的表述 异常复杂,所以以它为例来说明方法的实质.同时由于现有的相关工作违反了泊松括号同时 性的要求,给出了必要的改正.关键词：非线性方程哈密顿理论孤子     

Hamiltonian Formalism of the Derivative Nonlinear Schrodinger Equation

A particular form of poisson bracket is introduced for the derivative nonlinear Schrodinger (DNLS) equation.And its Hamiltonian formalism is developed by a linear combination method. Action-angle variables are found.