一列Liouville可积的有限维Hamilton系统族

本文直接递归地生成了一列Liouville可积的有限维Hamilton系统族，给出了其一串对合的公共运动积分和一组对合的生成元．     

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。     

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.     

一族可积Hamilton方程

本文利用屠规彰格式,导出了一族新的可积系,包含４个未知函数,具有双Ｈａｍｉｌｔｏｎ结构,且以ＴＣ族为特例。     

一族离散的可积Hamilton方程与可积的辛映射

该文讨论一个新的离散特征值问题，导出了相应的离散的Hamilton系统的保谱族，并且证明了它们是Liouville可积系。通过谱问题的双非线性化，导出一个新的可积的辛映射 。      

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.     

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。     

从无穷维可积Hamilton系统到有限维可积Hamilton系统的一个一般途径

本文综述了作者关于将无穷维可积Ｈａｍｉｌｔｏｎ系统（ＩＤＩＨＳ）分解为两个可交换的有限维可积Ｈａｍｉｌｔｏｎ系统（ＦＤＩＨＳ）的一般途径方面能做的工作，该途径提出了联系势和特征函数的一般约束（包括高阶约束），提供了从ＩＤＩＨＳ到ＦＤＩＨＳ的一般方法；在零曲率表示理论框架内，统一处理了一族ＩＤＩＨＳ的分解；证明了在一般约束下，一族ＩＤＩＨＳ中的每一个都可以分解为两个可交换的ｘ－和ｔｎ－ＦＤＩＨＳ；建     

一类孤立子系统的Hamilton结构及Liouville可积性

本文给出了一个2×2谱问题及其相应的孤子族,并利用此孤子族的Lenard算子对的性质,证明了该系统是具有Bi-Hamilton结构和Multi-Hamilton结构的广义Hamilton系统,进一步给出其Liouville可积性的证明.此外,值得提出的是此系统可约化为广义TD族、TD族和广义C-KdV族、C-KdV族等,并得到了该孤子族的Hamilton泛函与守恒密度之问的一一对应关系.     

一平面可积三次非Hamilton系统的Abel积分

本文讨论一平面可积三次非Hamilton系统在n次多项式扰动下Abel积分零点个数上确界，得到的结论是该Abel积分的零点个数的上确界为n。     

线性Hamilton系数的几个辛格式

本文利用二次Pad啨逼近的几个结果构造了线性Ｈａｍｉｌｔｏｎ系数的几个辛格式     

The Reduction Method in the Theory of Lie-Algebraically Integrable Oscillatory Hamiltonian Systems

We study the problem of the complete integrability of nonlinear oscillatory dynamical systems connected, in particular, both with the Cartan decomposition of a Lie algebra is the Lie algebra of a fixed subgroup with respect to an involution : G G on the Lie group G, and with a Poisson action of special type on a symplectic matrix manifold.     

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.     

KAM-Type Theorem on Resonant Surfaces for Nearly Integrable Hamiltonian Systems

Summary. In this paper, we consider analytic perturbations of an integrable Hamiltonian system in a given resonant surface. It is proved that, for most frequencies on the resonant surface, the resonant torus foliated by nonresonant lower dimensional tori is not destroyed completely and that there are some lower dimensional tori which survive the perturbation if the Hamiltonian satisfies a certain nondegenerate condition. The surviving tori might be elliptic, hyperbolic, or of mixed type. This shows that there are many orbits in the resonant zone which are regular as in the case of integrable systems. This behavior might serve as an obstacle to Arnold diffusion. The persistence of hyperbolic lower dimensional tori has been considered by many authors [5], [6], [15], [16], mainly for multiplicity one resonant case. To deal with the mechanisms of the destruction of the resonant tori of higher multiplicity into nonhyperbolic lower dimensional tori, we have to deal with some small coefficient matrices that are the generalization of small divisors. Received December 18, 1997; revised December 30, 1998; accepted June 21, 1999     

A New Approach to the Parameterization Method for Lagrangian Tori of Hamiltonian Systems

We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorov’s non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equation of the parameterization of the torus by three conditions which are altogether equivalent to invariance. We construct a quasi-Newton method by solving, approximately, the linearization of the functional equations defined by these three conditions around an approximate solution. Instead of dealing with the invariance error as a single source of error, we consider three different errors that take account of the Lagrangian character of the torus and the preservation of both energy and frequency. The condition of convergence reflects at which level contributes each of these errors to the total error of the parameterization. We do not require the system to be nearly integrable or to be written in action-angle variables. For nearly integrable Hamiltonians, the Lebesgue measure of the holes between invariant tori predicted by this parameterization result is of \({\mathcal {O}}(\varepsilon ^{1/2})\), where \(\varepsilon \) is the size of the perturbation. This estimate coincides with the one provided by the KAM theorem.     

Hamilton图的一个新的充分条件

设G是一个n阶3-连通1-坚韧图,以4(G)表示G的四元独立点集的次和的最小值,(G)为G的连通度,证明若     

New Local Conditions for a Graph to be Hamiltonian

For a vertex w of a graph G the ball of radius 2 centered at w is the subgraph of G induced by the set M₂(w) of all vertices whose distance from w does not exceed 2. We prove the following theorem: Let G be a connected graph where every ball of radius 2 is 2-connected and d(u)+d(v)≥|M₂(w)|−1 for every induced path uwv. Then either G is hamiltonian or for some p≥2 where ∨ denotes join. As a corollary we obtain the following local analogue of a theorem of Nash-Williams: A connected r-regular graph G is hamiltonian if every ball of radius 2 is 2-connected and for each vertex w of G. Supported by the Swedish Research Council (VR)