一类孤子方程族及其多个Hamilton结构

本文建立了一个含11个位势的新的等谱问题,得到了一组新的Lax对,由此得到一类新的孤子方程族.该族是Liouville可积的,具有4-Hamilton结构,且循环算子的共轭算子是一个遗传对称算子.另外,为确切说明所得方程族是一个4-Hamilton结构,在附录中证明了所得的4个Hamilton算子的线性组合恒为Hamilton算子.     

Kaup-Newell族的非线性双可积耦合及其自相容源

本文基于新的非半单矩阵Lie代数,介绍了构造孤子族非线性双可积耦合的方法,由相应的变分恒等式给出了孤子族非线性双可积耦合的Hamilton结构.作为应用,给出Kaup-Newell族的非线性双可积耦合及其Hamilton结构.最后利用源生成理论建立新的公式,并导出带自相容源Kaup-Newell族的非线性双可积耦合方程.     

Broer-Kaup-Kupershmidt族的非线性双可积耦合及其自相容源

基于新的非半单矩阵李代数,介绍了构造孤子族非线性双可积耦合的方法,由相应的变分恒等式给出了孤子族非线性双可积耦合的Hamilton结构.作为应用,给出了Broer-Kaup-Kupershmidt族的非线性双可积耦合及其Hamilton结构.最后指出了文献中的一些错误,利用源生成理论建立了新的公式,并导出了带自相容源Broer-Kaup-Kupershmidt族的非线性双可积耦合方程.     

拟Hamilton系统的随机平均

为拟Hamilton系统(小阻尼与弱随机激励作用下的Hamilton系统)提出了随机平均法,并就可积与不可积、共振与非共振情形进行了讨论.指出,现有的标准随机平均法与能量包线随机平均法为提出的拟Hamilton系统随机平均法的特殊情形.数例结果表明本方法是有效的.     

一族Liouville可积系及其约束流的Lax表示、Darboux变换

利用屠规彰格式求出了一族Liouville可积系,通过高阶位势特征函数约束将可积系分解成x部分和t_n部分可积Hamilton系统,求出了该系统的Lax表示及三类Darboux变换。     

一个新非线性可积晶格族和它们的可积辛映射

该文引入一个离散特征值问题,导出一族离散可积系,建立了它们的Hamilton结构,证明了它们Louville可积性.通过谱问题双非线性化,得到了一个可积辛映射与一族有限维完全可积系,最后给出了离散可积系统解的表示.     

一族Liouville可积的有限维Hamilton系统

本文生成了一族Liouville可积的Hamilton相流彼此可交换的有限维Hamilton系统,并且给出了一串对合的显式公共运动积分及其一组对合的显式生成元.     

一种构造可积Hamilton系统的新方法

建议了一种新的构造可积Hamilton系统的方法。对于给定的Poisson流形,本文利用Dirac-Poisson结构构造其上的新Poisson括号[1],进而获得了新的可积Hamilton系统。构造的Poisson括号一般是非线的,并且这种方法也不同于通常的方法[2～4]。本文还给出了两个实例。     

一维新的Hamilton可积方程

构造了Loop代数~A_{-1}的一个子代数,利用屠格式导出了一族新的可积孤子方程族,并且是Liouville可积系,具有双Hamilton结构。     

一族新的Lax可积系及其Liouville可积性

该文讨论了一个新的等谱特征值问题．按屠规彰格式导出了相应的Lax可积的非线性发展方程族，利用迹恒等式给出了它的Hamilton结构并且证明它是Liouville可积的．     

The binary nonlinearization of generalized Toda hierarchy by a special choice of parameters

A new discrete matrix spectral problem with two arbitrary constants is introduced. The corresponding 2-parameter hierarchy of integrable lattice equations, which can be reduced to the hierarchy of Toda lattice, is obtained by discrete zero curvature representation. Moreover, the Hamiltonian structure and a hereditary operators are deduced by applying the discrete trace identity. Finally, an integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonlinearization for the resulting hierarchy by a special choice of parameters.     

A unified expressing model of the AKNS hierarchy and the KN hierarchy, as well as its integrable coupling system

A new subalgebra of loop algebra Ã₁ is first constructed. Then a new Lax pair is presented, whose compatibility gives rise to a new Liouville integrable system(called a major result), possessing bi-Hamiltonian structures. It is remarkable that two symplectic operators obtained in this paper are directly constructed in terms of the recurrence relations. As reduction cases of the new integrable system obtained, the famous AKNS hierarchy and the KN hierarchy are obtained, respectively. Second, we prove a conjugate operator of a recurrence operator is a hereditary symmetry. Finally, we construct a high dimension loop algebra to obtain an integrable coupling system of the major result by making use of Tu scheme. In addition, we find the major result obtained is a unified expressing integrable model of both the AKNS and KN hierarchies, of course, we may also regard the major result as an expanding integrable model of the AKNS and KN hierarchies. Thus, we succeed to find an example of expanding integrable models being Liouville integrable.     

连续变量第二型Voltrra方程的p-均值可积性

本文主要是研究连续变量遗传系统V o lterra方程的第二型,即x(t h0)=η(t h0) F(t,(x(t),x(t-h1)…,x(t-hm))的p-均值可积性.同时举例说明了此方程的Lyapunov泛函的构造,以及利用Lyapunov泛函证明了例子的均方可积性.     

An integrable nonlinear diffusion hierarchy with a two-dimensional arbitrary function

By introducing the hereditary condition to a general first order differential strong symmetry operator, we obtain general 1+1 and 2+1 dimensional integrable nonlinear diffusion hierarchies with infinitely many symmetries and Lax pairs. For a special example the infinitely many nonlocal conservation laws and some explicit and implicit exact solutions are also given.     

On the relation between the full Kostant-Toda lattice and multiple orthogonal polynomials

The correspondence between a high-order non-symmetric difference operator with complex coefficients and the evolution of an operator defined by a Lax pair is established. The solution of the discrete dynamical system is studied, giving explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for our integrable systems. The method of investigation is based on the evolutions of the matrical moments.     

INTEGRABLE COUPLING OF THE TC HIERARCHY OF EQUATIONS

A new loop algebra and a new Lax pair are constructed, respectively. It follows that the integrable coupling of the TC hierarchy of equations, which is also an expanding integrable model, is obtained. Specially, the integrable coupling of the famous KdV equation is presented.     

Symmetries and Their Lie Algebra of a Variable Coefficient Korteweg-de Vries Hierarchy

Isospectral and non-isospectral hierarchies related to a variable coefficient Painlev′e integrable Korteweg-de Vries(Kd V for short) equation are derived. The hierarchies share a formal recursion operator which is not a rigorous recursion operator and contains t explicitly. By the hereditary strong symmetry property of the formal recursion operator, the authors construct two sets of symmetries and their Lie algebra for the isospectral variable coefficient Korteweg-de Vries(vc Kd V for short) hierarchy.     

可积模的权

本文定义了Ｋａｃ－Ｍｏｏｄｙ代数的一个新的可积模范畴，并且给出了一个可积模是否属于这个模范畴的一个判别准则．另外还详细研究了这个模范畴中的可积模的权系。特别我们定义了虚权和实权。还详细地计算了一些模的虚权和实权，还给出了双曲型广义Ｃａｒｔａｎ矩阵的新刻划．这使我们能够计算一些双曲型Ｋａｃ－Ｍｏｏｄｙ代数的可积模的权。