两族可积的Hamilton方程

把屠规彰格式应用于等谱问题得到了两族可积的Ｈａｍｉｌｔｏｎ方程．得到后一族需要对屠格式进行变更，这为扩大屠格式的应用范围指出了一条途径．     

Special Quasigraded Lie Algebras and Integrable Hamiltonian Systems

We construct a family of special quasigraded Lie algebras of functions of one complex variables with values in finite-dimensional Lie algebra , labeled by the special 2-cocycles F on . The main property of the constructed Lie algebras is that they admit Kostant-Adler-Symes scheme. Using them we obtain new integrable finite-dimensional Hamiltonian systems and new hierarchies of soliton equations.     

The Mirror Systems of Integrable Equations

We provide an algorithm to convert integrable equations to regular systems near noncharacteristic, movable singularity manifolds of solutions. We illustrate how the algorithm is equivalent to the Painlevé test. We also use thealgorithm to prove the convergence of the Laurent series obtained from the Painlevé test.     

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.     

Vertical Cohomologies and Their Application to Completely Integrable Hamiltonian Systems

Some functorial and topological properties of vertical cohomologies and their application to completely integrable Hamiltonian systems are studied.     

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。     

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 Nonlinear Integrable Couplings of Yang Equations Hierarchy and Its Hamiltonian Structure

Based on a kind of non-semisimple Lie algebras, we establish a way to construct nonlinear continuous integrable couplings. Variational identities over the associated loop algebras are used to furnish Hamiltonian structures of the resulting continuous couplings. As an illustrative example of the scheme is given nonlinear continuous integrable couplings of the Yang hierarchy.     

新的Liouville完全可积的广义Burgers方程族及其Hamilton结构

基于屠格式,从一个新的等谱问题,本文获得了一族广义Ｂｕｒｇｅｒｓ 方程及其Ｈａｍ ｉｌｔｏｎ 结构．最后证明了该族方程是Ｌｉｏｕｖｉｌｌｅ 完全可积的,并且有无穷多个彼此对合的公共守恒密度     

可积的与Hamilton形式的NLS-MKdV方程族

本文基于ｌｏｏｐ代数Ａ２的一个特殊子代数，设计了一个等谱问题，应用屠规彰格式计算出一族具有Ｈａｍｉｌｔｏｎ结构的可积系．此族含有非线性Ｓｃｈｒｄｉｎｇｅｒ方程与修正的ＫｄＶ方程，称之为ＮＬＳ ＭＫｄＶ方程族     

Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification

The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.     

Topological Obstacles to the Realizability of Integrable Hamiltonian Systems by Billiards

Doklady Mathematics - We introduce the following classes of integrable billiards: elementary billiards, topological billiards, billiard books, billiards with a potential, with a magnetic field, and...     

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。     

Integrable Systems of Linear Ordinary Differential Equations of General Order

In this paper, we define sufficient conditions for the solvability in quadratures of systems of linear ordinary differential equations of higher orders with variable coefficients. The method of variation of constants for equations of higher orders and for systems of equations of first order is generalized to the case of such systems. Results are obtained by reducing the systems under consideration to matrix equations of higher orders studied by methods of the theory of Lie algebras.     

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.     

The Isoenergetic KAM-Type Theorem at Resonant Case for Nearly Integrable Hamiltonian Systems

In this paper, we study the persistence of resonant invariant tori on energy surfaces for nearly integrable Hamiltonian systems under the usual R$\ddot{u}$ssmann nondegenerate condition. By a quasilinear iterative scheme, we prove the following things: (1) The majority of resonant tori on a given energy surface will be persisted under R$\ddot{u}$ssmann nondegenerate condition. (2) The maximal number about the preserved frequency components on a perturbed torus is characterized by the smaller of the maximal rank of the Hessian matrices of the unperturbed system and the nondegeneracy of resonance. (3) If unperturbed systems admit subisoenergetic nondegeneracy on an energy surface, then the majority of the unperturbed resonant tori on the energy surface will be persisted and give rise to a family of perturbed tori with the same energy, whose frequency ratios among respective ''nondegenerate'' components are preserved.     

Finite-Dimensional Integrable Hamiltonian Systems Related to the Nonlinear Schrödinger Equation

A complete treatment of the binary nonlinearizations of spectral problems of the nonlinear Schrödinger (NLS) equation with the choice of distinct eigenvalue parameters is presented. Two kinds of constraints between the potentials and the eigenfunctions of the NLS equation are considered. From the first constraint, a pair of new finite-dimensional completely integrable Hamiltonian systems which constitute an integrable decomposition of the NLS equation are obtained. From the second constraint, a novel finite-dimensional integrable Hamiltonian system, which includes the system of multiple three-wave interaction as a special case, is obtained. It is found that the eigenvalue parameters real or not can lead to completely different symplectic structures of the restricted NLS flows. In addition, a relationship between the binary restricted Ablowitz–Kaup–Newell–Segur flows and the restricted NLS flows is revealed.     

Integrable Quasilinear Equations

We develop a classification scheme for integrable third-order scalar evolution equations using the symmetry approach to integrability. We use this scheme to study quasilinear equations of a particular type and prove that several equations that were suspected to be integrable can be reduced to the well-known Korteweg–de Vries and Krichever–Novikov equations via a Miura-type differential substitution.     

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     

Stability of Nearly Integrable, Degenerate Hamiltonian Systems with Two Degrees of Freedom

We consider the problem of the stability of action variables in properly degenerate, nearly integrable Hamiltonian systems and prove, in particular, stability results for systems with two degrees of freedom. An application of such results to celestial mechanics is presented.