Hamiltonian Equations for Scale-Invariant Waves

We use a Hamiltonian formalism to derive equations for weakly nonlinear scale-invariant waves. We apply the results to nonlinear surface waves in elasticity and magnetohydrodynamics that satisfy nonlocal generalizations of the inviscid Burgers' equation.     

Instability of Standing Waves for Hamiltonian Wave Equations

This paper deals with the standing wave for a Hamiltonian nonlinear wave equation which can be viewed as a representative of the class of equations of interest. On the one hand, by proving a compactness lemma and solving a variational problem, the existence of the standing wave with ground state for the aforementioned equation is proved. On the other hand, the authors derive the instability of the standing wave by applying the potential well argument, the concavity method and an invariant region under the solution flow of the Cauchy problem for the equation under study, and the invariance of the region aforementioned can be shown by introducing an auxiliary functional and a supplementary constrained variational problem.     

Kinetic Equations and Integrable Hamiltonian Systems

A survey of interrelations between kinetic equations and integrable systems is presented. We discuss common origin of special classes of solutions of the Boltzmann kinetic equation for Maxwellian particles and special solutions for integrable evolution equations. The thermodynamic limit and soliton kinetic equation for the integrable Korteweg-de Vries equation are considered. The existence of decaying and degenerate dispersion laws and the appearance of additional integrals of motion for the interacting waves is discussed. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 731–741, June, 2005.     

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．     

Morse-type index theory for flows and periodic solutions for Hamiltonian Equations

An index theory for flows is presented which extends the classical Morse theory for gradient flows on compact manifolds. The theory is used to prove a Morse-type existence statement for periodic solutions of a time-dependent (periodic in time) and asymptotically linear Hamiltonian equation.     

Convergence of the Equations for a Maxwell Fluid

The equations for the flow of a viscoelastic fluid of the Maxwell type are analyzed in a linear approximation. First, we establish that the solution depends continuously on changes in the relaxation time. Next, we investigate how the solution to the linearized Maxwell system converges to the solution to Stokes flow as the relaxation time tends to zero. Convergence in different measures is examined and specific a priori bounds are derived.     

Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations

In this paper we study optimal control problems governed by semilinear parabolic equations. We obtain necessary optimality conditions in the form of an exact Pontryagin's minimum principle for distributed and boundary controls (which can be unbounded) and bounded initial controls. These optimality conditions are obtained thanks to new regularity results for linear and nonlinear parabolic equations. Accepted 17 March 1997     

A Continuous-Stage Modified Leap-Frog Scheme for High-Dimensional Semi-Linear Hamiltonian Wave Equations

Among the typical time integrations for PDEs, Leap-frog scheme is the well-known method which can easily be used. A most welcome feature of the Leap-frog scheme is that it has very simple scheme and is easy to be implemented. The main purpose of this paper is to propose and analyze an improved Leap-frog scheme, the so-called continuous-stage modified Leap-frog scheme for high-dimensional semi-linear Hamiltonian wave equations. To this end, under the assumption of periodic boundary conditions, we begin with the formulation of the nonlinear Hamiltonian equation as an abstract second-order ordinary differential equation (ODE) and its operator-variation-of-constants formula (the Duhamel Principle). Then the continuous-stage modified Leap-frog scheme is formulated. Accordingly, the convergence, energy preservation, symplecticity conservation and long-time behaviour of explicit schemes are rigorously analysed. Numerical results demonstrate the remarkable advantage and efficiency of the improved Leap-frog scheme compared with the existing mostly used numerical schemes in the literature.     

Non-Oscillatory Solutions for Higher Order Dynamic Equations

A new non-oscillatory theory is presented for higher order delayand neutral equations. The results include both singular andnonsingular problems.     

一些数学物理问题中的Hamilton方程

讨论了新的一系列在数学物理方程中微分方程的Hamilton正则表示,其中包括变系数2阶对称方程的Hamilton系统,关于常系数的4阶对称方程新的非齐次Hamilton表示,MKdV方程以及KP方程的正则表示.     

Perturbations of Dynamic Equations

We give conditions on the coefficient matrix for certain perturbed linear dynamic equations on time scales ensuring that there exists a bounded solution (which is explicitly given) to which all other solutions converge, and similarly conditions ensuring a bounded solution from which all other solutions diverge. We also consider periodic time scales and corresponding linear dynamic equations with periodic coefficients and prove similar statements about periodic solutions to which all other solutions converge or from which all other solutions diverge.     

轨道运动方程数值解的一种加速算法

针对卫星轨道受大气阻力摄动的运动方程,提出了一种数值加速算法,该算法实现简单、计算量小、精度高,适合于各类卫星轨道的方程的求解.     

时标动力方程的稳定性分析

近来，一种称之为时标动力系统的新理论得到了蓬勃的发展. 该文给出并建立了时标动力系统的各种稳定性概念和Lyapunov稳定性准则(稳定, 一致稳定, 渐近稳定, 指数渐近稳定等等).     

Dynamic Game Problems of Approach for Fractional-Order Equations

We propose a general method for the solution of game problems of approach for dynamic systems with Volterra evolution. This method is based on the method of decision functions and uses the apparatus of the theory of set-valued mappings. Game problems for systems with Riemann–Liouville fractional derivatives and regularized Dzhrbashyan–Nersesyan derivatives (fractal games) are studied in more detail on the basis of matrix Mittag-Leffler functions introduced in this paper.