关于(0,1)的对称和规范矩阵

给出更多的对于给定行和向量存在（０,１）对称矩阵的等价条件,同时也讨论了（０,１）规范矩阵的情形．     

一个演化方程族约束系统的r-矩阵

本文给出一个演化方程约束系统的Ｌａｘ表示,ｒ－矩阵,经典Ｐｏｉｓ－ｓｏｎ结构与可积性的证明。     

一个类似于KN族的可积系及其可积耦合

本文选用loop代数A₁的一个子代数,建立了一个线笥等谱问题,导出了一个类似KN族的可积方程族．通过建立求可积耦合的一种简便直接方法,求出了该方程族的可积耦合．这种方法也适用于其它方程族。     

关于Riemann可积的一个注记

鉴于R iem ann积分对区间的分割和取点的双重任意性要求给应用带来的困难,本文中我们将选择“等距分割”这种特殊的分割方式来定义函数的可积性,即所谓“等分可积”.从表面上看,等分可积性要弱于R iem ann可积性,但本文将证明这两种可积性是等价的.文中还给出了等分可积准则.     

关于一个可积的广义Hamilton方程族

本文利用ｒ－矩阵生成了一个广义的Ｈａｍｉｌｔｏｎ方程族，并证明了它是广义可积的，然后讨论了它和（４）中Ｌｉｏｖｖｉｌｌｅ可积的新的广义Ｈａｍｉｌｔｏｎ方程族之间的关系。     

一个经典的可积Neumann系统和Boussinesq族Lax对的非线性化

本文在位势与特征函数之间的Neumann约束条件下,经典Boussinesq族的Lax对被非线性化成为自然相容的Lax系统;而且,其为Liouville完全可积的Hamiltonian系统,同时获得了Boussinesq方程解的对合表示。     

非奇H矩阵与M—矩阵的等价表征

引进了局部α－对角占优矩阵的概念,得到了非奇Ｈ矩阵与Ｍ－矩阵的等价表征和判定准则,推广和改进了「１～３」的相应结果。     

关于M—矩阵等价表征的注记

「２」指出了「１」的某些错误,并给出了修正的结果。本继续「２」的讨论,给出了Ｍ－矩阵等价表征的进一步结果。     

一类非线性演化方程族的可积耦合

Starting from a Tu Guizhang‘s isospectral‘problem, a Lax pair is obtained by means of Tu scheme ( we call it Tu Lax pair ). By applying a gauge transformation between matrices, the Tu Lax pair is changed to its equivalent Lax pair with the traces of spectral matrices being zero, whose compatibility gives rise to a type of Tu hierarchy of equations. By making use of a high order loop algebra constructed by us, an integrable coupling system of the Tu hierarchy of equations are presented. Especially, as reduction cases, the integrable couplings of the celebrated AKNS hierarchy, TD hierarchy and Levi hierarchy are given at the same time.     

一个新的Liouville可积系统及其Lax表示，Bi—Hamilton结构

从一个特征值问题出发,首先推导一族非线性发展方程,其中包括著名MKdV方程做为特殊约化,进一步证明这族方程在Liounille意义下可积并具有Bi-Hamilton结构,而在位执函数和特征函数之间的一定约束下,特征值问题被非线性化为一完全可积的有限维Hamilton系统。     

AKNS族与Geng族的规范等价的约束流

先由ＡＫＮＳ族出发用规范变换建立了Ｇｅｎｇ族的约束流及它的第一组守恒积分和Ｌａｘ表示,再用其Ｌａｘ表示建立了Ｇｅｎｇ族约束流的ｒ－矩阵。并且还给出了Ｇｅｎｇ族约束流的第二组守恒积分及其对合性的证明。     

一族包含任意多个未知函数的可积方程组

基于 Lie代数 A1,建立了一个等谱问题 ,由相应的零曲率方程 ,导出一族可积方程组 ,其中未知函数个数可以是任意的     

Integrable Hierarchy, 3×3 Constrained Systems, and Parametric Solutions

This paper provides a new integrable hierarchy. The DP equation: m _t +um _x +3mu _x =0, m=u–u _xx , proposed recently by Degasperis and Procesi, is the first member in the negative order hierarchy while the first equation in the positive order hierarchy is: m _t =4(m ^–2/3) _x –5(m ^–2/3) _xxx +(m ^–2/3) _xxxxx . The whole hierarchy is shown Lax-integrable through solving a key matrix equation. To obtain the parametric solutions for the whole hierarchy, we separately discuss the negative order and the positive order hierarchies. For the negative order hierarchy, its 3×3 Lax pairs and corresponding adjoint representations are cast in Liouville-integrable Hamiltonian canonical systems under the Dirac–Poisson bracket defined on a symplectic submanifold of R ^6N . Based on the integrability of those finite-dimensional canonical Hamiltonian systems we give the parametric solutions of all equations in the negative order hierarchy. In particular, we obtain the parametric solution of the DP equation. Moreover, for the positive order hierarchy, we consider a different constraint and process a procedure similar to the negative case to obtain the parametric solutions of the positive order hierarchy. In a special case, we give the parametric solution of the 5th-order PDE m _t =4(m ^–2/3) _x –5(m ^–2/3) _xxx +(m ^–2/3) _xxxxx . Finally, we discuss the stationary solutions of the 5th-order PDE, which may be included in the parametric solution.     

一个新的可积方程族及其Hamiltonian结构

基于一个带有三个位势函数新的等谱问题,本文得到了一个带有任意函数的新的Lax可积族.当位势选取特殊函数时,得到了著名的Schrodinger方程,广义MkdV方程,热传导方程和耦合的Burgers方程及其Hamiltonian结构,并证明方程是Liouville可积的.     

新的耦合mKdV方程族及其Liouville可积的无限维Hamilton结构

根据第Ⅱ屠格式,从一个特征值问题出发,本文推得了一族新的耦合ｍＫｄＶ方程,然后用迹恒等式人出了其无限维Ｈａｍｉｌｔｏｎ结构。最后证明了该Ｈａｍｉｌｔｏｎ方程族是Ｌｉｏｕｖｉｌｌｅ可积的,并且有无穷多个彼此对合的公共守恒密度。     

Weierstrass Representations for Surfaces in 4D Spaces and Their Integrable Deformations via DS Hierarchy

Generalized Weierstrass representations for generic surfaces conformally immersed into four-dimensional Euclidean and pseudo-Euclidean spaces of different signatures are presented. Integrable deformations of surfaces in these spaces generated by the Davey–Stewartson hierarchy of integrable equations are proposed. The Willmore functional of a surface is invariant under such deformations.     

一类可积非Hamilton系统的Abel积分的零点估计

讨论了一类含参可积非Hamilton系统在一般二次多项式扰动下的Abel积分的零点,得出了不同参数范围下的Abel积分的零点数目的估计.     

Generalized Fractional Trace Variational Identity and A New Fractional Integrable Couplings of Soliton Hierarchy

Based on fractional isospectral problems and general bilinear forms, the generalized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable couplings of a fractional soliton hierarchy are derived from a fractional zero-curvature equation. Finally, we obtain the fractional Hamiltonian structures of the fractional integrable couplings of the soliton hierarchy.     

A （2＋1）-Dimensional Dispersive Long Wave Hierarchy and its Integrable Couplings

Under the frame of the (2 1)-dimensional zero curvature equation and Tu model, the (2 1)-dimensional dispersive long wave hierarchy is obtained. Furthermore, the loop algebra is expanded into a larger one. Moreover, a class of integrable coupling system for dispersive long wave hierarchy and (2 1)-dimensional multi-component integrable system will be investigated.