哈密顿系统的低维环面的保存性

本文给出了关于哈密顿系统低维环面的一个推广的KAM定理,它适用于同时存在法向频率和双曲法向分量的情况.其证明基于尤建功的一个定理的光滑性表述及法向双曲不变流形理论的应用.文中还给出了另外两种情况下的推广.     

KAM Type-Theorem for Lower Dimensional Tori in Random Hamiltonian Systems

In this paper, we study the persistence of lower dimensional tori for random Hamiltonian systems, which shows that majority of the unperturbed tori persist as Cantor fragments of lower dimensional ones under small perturbation. Using this result, we can describe the stability of the non-autonomous dynamic systems.     

Persistence of Hyperbolic Tori in Generalized Hamiltonian Systems

In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graff and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on sub manifolds.     

AKNS-KN孤子方程族的可积耦合与Hamilton结构

首先通过引入高维圈代数,在零曲率方程框架下得到了AKNS-KN孤子族(记为AKNS-KN-SH)的一个新的可积耦合系统;再由二次型恒等式得到了该系统的双-Hamilton结构形式.最后引进了一个新的Lie代数A_4,可通过建立其不同的圈代数与等价的列向量Lie代数,研究AKNS-KN-SH的多分量可积耦合系统及其Hamilton结构.     

Bifurcations of First Integrals in the Sokolov Case

We study the phase topology of a new Liouville integrable Hamiltonian system with an additional quartic integral (the Sokolov case).     

Classification of Flat Lagrangian Surfaces in Complex Lorentzian Plane

One of the most fundamental problems in the study of Lagrangian submanifolds from Riemannian geometric point of view is to classify Lagrangian immersions of real space forms into complex space forms. The main purpose of this paper is thus to classify flat Lagrangian surfaces in the Lorentzian complex plane C1^2. Our main result states that there are thirty-eight families of flat Lagrangian surfaces in C1^2. Conversely, every flat Lagrangian surface in C1^2 is locally congruent to one of the thirty-eight families.     

Persistence of lower dimensional tori of general types in Hamiltonian systems

This work is a generalization to a result of J. You (1999). We study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.

     

Hamiltonian Stationary Lagrangian Surfaces in Euclidean Four-Space and Related Minimization Problems. Part II: Some Minimization Results

We study some minimization problems for Hamiltonian stationaryLagrangian surfaces in R⁴. We show that the flat Lagrangian torusS ¹ × S ¹ minimizes the Willmore functional among Hamiltonianstationary tori of its isotopy class, which gives a new proof of thefact that it is area minimizing in the same class. Considering theLagrangian flat cylinder as a torus in some quotient space R⁴/v Z, we show that it is also area minimizing in its isotopy class.     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.     

On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations

We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a 2n-times continuously differentiable perturbation (n denotes the number of the degrees of freedom), provided that the moduli of continuity of the 2n-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the 2n-th partial derivatives of the perturbation are Hölder continuous.     

Nontrivial Solutions of Superquadratic Hamiltonian Systems with Lagrangian Boundary Conditions and the L-index Theory

In this paper, the authors study the existence of nontrivial solutions for the Hamiltonian systems z(t) = JH(t,z(t)) with Lagrangian boundary conditions, where H(t,z)=1/2((B)(t)z,z) (H)(t,z),(B)(t) is a semipositive symmetric continuous matrix and (H) satisfies a superquadratic condition at infinity. We also obtain a result about the L-index.     

Analytic intermediate dimensional elliptic tori for the planetary many-body problem

In our context,the planetary many-body problem consists of studying the motion of(n+1)-bodies under the mutual attraction of gravitation,where n planets move around a massive central body,the Sun.We establish the existence of real analytic lower dimensional elliptic invariant tori with intermediate dimension N lies between n and 3n-1 for the spatial planetary many-body problem.Based on a degenerate KolmogorovArnold-Moser(abbr.KAM)theorem proved by Bambusi et al.(2011),Berti and Biasco(2011),we manage to handle the difficulties caused by the degeneracy of this real analytic system.     

Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials

In this paper, under a similar but stronger condition than that of Ambrosetti and Rabinowitz we find a T-periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential for any T 〉 0; moreover, such a solution has T as its minimal period.     

Matrix Model and Stationary Problem in the Toda Chain

We analyze the stationary problem for the Toda chain and show that the arising geometric data exactly correspond to the multisupport solutions of the one-matrix model with a polynomial potential. We calculate the Hamiltonians and symplectic forms for the first nontrivial examples explicitly and perform the consistency checks. We formulate the corresponding quantum problem and discuss some of its properties and prospects. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 3–16, January, 2006.     

Compatible Dubrovin–Novikov Hamiltonian Operators,Lie Derivative,and Integrable Systems of Hydrodynamic Type

We prove that two Dubrovin–Novikov Hamiltonian operators are compatible if and only if one of these operators is the Lie derivative of the other operator along a certain vector field. We consider the class of flat manifolds, which correspond to arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators. Locally, these manifolds are defined by solutions of a system of nonlinear equations, which is integrable by the method of the inverse scattering problem. We construct the integrable hierarchies generated by arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators.     

Integral Geometry of Real Surfaces in the Complex Projective Plane

We give a Poincaré formula for any real surfaces in the complex projective plane which states that the mean value of the intersection numbers of two real surfaces is equal to the integral of some terms of their Kähler angles.     

An integrable coupling hierarchy of the Mkdv_integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy

A four-by-four matrix spectral problem is introduced, locality of solution of the related stationary zero curvature equation is proved. An integrable coupling hierarchy of the Mkdv_integrable systems is presented. The Hamiltonian structure of the resulting integrable coupling hierarchy is established by means of the variational identity. It is shown that the resulting integrable couplings are all Liouville integrable Hamiltonian systems. Ultimately, through the nonisospectral zero curvature representation, a nonisospectral integrable hierarchy associated with the resulting integrable couplings is constructed.     

Some Remarks on Left-Definite Hamiltonian Systems in the Regular Case

In the paper, a new method of constructing asymptotic solutions of differential equations on manifolds with singularities is presented. This method allows not only to widen essentially the space of asymptotics but also to obtain explicit formulas for asymptotic expansions, in particular, in the case when in a neighborhood of a singular point there exist strata of different dimensions.