Neumann型约束Tu流的r矩阵和分离变量

本文发展了适用于Neumann型约束流的r矩阵方法．研究了Neumann型约束Tu流．获得了Neumann型约束Tu流的Lax表示；证明了在Dirac括号下这个Lax算子满足r矩阵关系，从而再次证明了Neumann型约束Tu流是Liouville完全可积的．另外，还利用r矩阵关系，找到了这个Neumann型约束流的一组正则变量，从而把Neumann型约束Tu流线性化．     

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。     

具有Lenard递推结构之发展方程族的Lax表示

该文对具有Ｌｅｎａｒｄ递推结构的发展方程族，通过转换速推结构为一个算子方程给出了Ｌａｘ表示的一种理论描述．这种描述可应用于各种１＋１维可积系统族Ｌａｘ表示的寻求之中，本文仅就ＫｄＶ可积族和Ａｎｔｏｎｏｗｉｃｚ—Ｆｏｒｄｙ可积族详细阐述了Ｌａｘ表示的具体构造过程．     

一个3×3谱问题产生的约束系统的动力r-矩阵

给出一个3×3谱问题产生的Harry—Dym型方程族的约束系统的Lax表示，动力r矩阵及Poisson结构，并给出3N个守恒积分．从而利用一般，r-矩阵理论证明了该约束系统在Liouville意义下的完全可积性．     

两类超Tu族的自相容源和守恒律

基于两类不同的Lie超代数和超迹恒等式, 建立了两类超可积Tu族的自相容源方程. 另外, 还建立了两类超可积Tu族的无穷守恒律. 特别地, 费米变量在超可积系统里面起了重要作用, 它不同于一般的可积系统.     

耦合Burgers方程的Painleve分析和相关性质

为研究耦合Burgers方程的可积性,利用WTC测试方法,给出了第一类Burgers方程的Painleve性质和第二类Burgers方程的条件Painleve性质．进而得到了第一类方程的变量分离解和第二类方程的（N2＋3N＋6/2）-参数Lie点对称群．     

一族新的可积系及其双约束流的Hamilton正则表示

基于一个新的等谱问题，按屠格式导出了一族新的可积系，具有双Hamilton结构，通过建立双对称约束，得到了该方程族的两组约束流，并将其化为正则的Hamilton系统。     

耦合Burgers方程的Painleve分析和相关性质

为研究耦合Burgers方程的可积性，利用WTC测试方法，给出了第一类Burgers方程的Painleve性质和第二类Burgers方程的条件Painleve性质．进而得到了第一类方程的变量分离解和第二类方程的（N2＋3N＋6/2）-参数Lie点对称群．     

Lax表示的变形与Hamilton方程族的Lax表示

本文首先给出了构造演化方程族的Ｌａｘ表示的马文秀方法的一种变形，后对这一方法作了改进，使之适用于Ｈａｍｉｌｔｏｎ形式的方程族．作为应用，得到了具有非等谱Ｌａｘ表示的杨方程族．     

Guo族约束流的经典Poisson结构、Lax表示、r-矩阵和Liouville可积性

基于伴随表示,通过引入Ｊａｃｏｂｉ－Ｏｓｔｒｏｇｒａｄｓｋｙ坐标,获得了Ｇｕｏ族约束流的Ｌａｘ表示,Ｐｏｉｓｓｏｎ结构和ｒ－矩阵,最后,借助Ｐｏｉｓｓｏｎ结构和ｒ－矩阵,说明了Ｇｕｏ族约束 是Ｌｉｏｕｖｉｌｌｅ可积的。     

Commutativity of the extended KP flows

The commutativity problem of the extended KP hierarchy is analyzed. The compatibility equation of two extended KP flows is constructed, together with its Lax representations involving two extended Lax operators. The resulting theory shows that the extended KP hierarchy is a natural generalization of the KP flows, but does not commute unlike the constrained KP hierarchy. A few particular examples are computed, along with their Lax pairs.     

Binary nonlinearization of spectral problems of the perturbation AKNS systems

The method of nonlinearization of spectral problems is extended to the perturbation AKNS systems, and a new kind of finite-dimensional Hamiltonian systems is obtained. It is shown that the obtained Hamiltonian systems are just the perturbation systems of the well-known constrained AKNS flows and thus their Liouville integrability is established by restoring from the Liouville integrability of the constrained AKNS flows. As a byproduct, the process of binary nonlinearization of spectral problems and the process of perturbation of soliton equations commute in the case of the AKNS hierarchy.     

双约束孤立子流的可积形变

提出了基于Lax矩阵的构造双约束孤立子流的可积形变的新方法.作为应用,导出了双约束KdV流和双约束mKdV流的可积形变,并给出了这些形变的Lax表示、r-矩阵和守恒积分.     

由伴随坐标得到的Dirac族的可积约束流

引入伴随坐标建立了Dirac族的某些非正则高阶约束流及其对应的Lax表示和r-矩阵,并证明这些约束流在Liouville意义下是完全可积的．     

ADJOINT SYMMETRY CONSTRAINTS OF MULTICOMPONENT AKNS EQUATIONS

A soliton hierarchy of multicomponent AKNS equations is generated from an arbitraryorder matrix spectral problem,along with its bi-Hamiltonian formulation.Adjoint symmetryconstraints are presented to manipulate binary nonlinearization for the associated arbitraryorder matrix spectral problem.The resulting spatial and temporal constrained fiows are shownto provide integrable decompositions of the multicomponent AKNS equations.     

The Application of Neumann Type Systems for Solving Integrable Nonlinear Evolution Equations

An algorithm to obtain finite‐gap solutions of integrable nonlinear evolution equations (INLEEs) is provided by using the Neumann type systems in the framework of algebraic geometry. From the nonlinearization of Lax pairs, some INLEEs in 1+1 and 2+1 dimensions are reduced into a class of new Neumann type systems separating the spatial and temporal variables of INLEEs over a symplectic submanifold (M, ω²) . Based on the Lax representations of INLEEs, we deduce the Lax–Moser matrix for those Neumann type systems that yield the integrals of motion, elliptic variables, and a hyperelliptic curve of Riemann surface. Then, we attain the Liouville integrability for a hierarchy of Neumann type systems in view of a Lax equation on (M, ω²) and a set of quasi‐Abel–Jacobi variables. We also specify the relationship between Neumann type systems and INLEEs, where the involutive solutions of Neumann type systems give rise to the finite parametric solutions of INLEEs and the Neumann map cuts out a finite dimensional invariant subspace for INLEEs. Under the Abel–Jacobi variables, the Neumann type flows, the 1+1, and 2+1 dimensional flows are integrated with Abel–Jacobi solutions; as a result, the finite‐gap solutions expressed by Riemann theta functions for some 1+1 and 2+1 dimensional INLEEs are achieved through the Jacobi inversion with the aid of the Riemann theorem.     

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.     

New hierarchy of integrable system bi-Hamiltonian structure and constrained flows

We have considered the hierarchy of integrable systems associated with the unstable nonlinear Schrodinger equation. The spectral gradient approach and the trace identity are used to derive the bi-Hamiltonian structure of the system. The bi-Hamiltonian property and the square eigenfunctions determined via the spectral gradient approach are then used to construct constrained flows, which is also proved to be derivable from a rational Lax operator. This new Lax operator of the constrained flows is seen to generate the classical r-matrix. Lastly it is also explicitly demonstrated that the different integrals of motion of the constrained flows Poisson commute.     

The Bargmann Symmetry Constraint and Binary Nonlinearization of the Super Dirac Systems

An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super Dirac systems. Under the obtained symmetry constraint, the n-th flow of the super Dirac hierarchy is decomposed into two super finite-diinensional integrable Hamiltonian systems, defined over the super- symmetry manifold R^4N{2N with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.