非线性户田晶格方程的延拓结构（英文）

本文研究户田晶格方程的延拓结构.利用非交换微分和求差分微分方程延拓结构的方法,得到户田晶格方程的拉克斯对.     

某些微分差分方程的周期解

本文给出了微分差分方程及 x′(t)=-g(x(t))F(x(t),x(t-1))x′(t)=-g(x(t))F(x(t),x(t-1),…,x(t-n))存在非平凡周期解的充分条件。     

构造非线性微分差分方程精确解的一种方法

把最近提出的G′/G展开法推广到了非线性微分差分方程,利用该方法成功构造了一种修正的Volterra链和Toda链的双曲函数、三角函数以及有理函数三类涉及任意参数的行波解,当这些参数取特殊值时,可得这两个方程的扭状孤立波解、奇异行波解以及三角函数状的周期波解等.研究结果表明,该算法探讨非线性微分差分方程精确解十分有效、简洁.     

关于微分差分方程的边值问题

本文考虑含小参数ε>0且自变量具有固定时滞1的微分差分方程边值问题(?)其中L[y(x,ε)]=εy″(x,ε)-a(x,ε)y′(x,ε)-b(x,ε)y(x,ε),R[y(x,ε)]=A(x,ε)y′(x-1,ε)+B(x,ε)y(x-1,ε)+f(x,ε),T 是一正数,10下讨论了边值问题解的存在性、唯一性和区间-1≤x≤T 上当ε→0~+时解的一致有效估计.     

某些一阶非齐次中立型微分差分方程的振动性

该文讨论了 d/dt[x(t) cx(t-τ)] P(t)x(t-σ) f(t)=0,t≥t_0一阶非齐次中立型微分差分方程的振动性.得到了一些方程振动的充分条件,推广了某些齐次方程的振动结果.     

一类微分差分方程的周期解

泛函微分方程周期解的存在性问题是重要而困难的。文[1—3]分别用Kaplan—Yorke方法研究了含一个滞量的微分差分方程的周期解问题。文[4]用Kaplan-Yorke方法研究了含二个滞量的微分差分方程周期解的存在性问题。本文研究微分差分方程     

一类差分方程的全局吸引性

考虑非自治差分方程xn 1=xnexp(rn 1-xn/1 λxn),n∈N的全局吸引性．这里{rn}是非负实数列，λ≥0，我们获得了方程每一解在初始条件下趋于1的充分条件．     

几类三阶非线性微分差分方程平凡解的稳定性

本文讨论几类三阶非线性微分差分方程平凡解的稳定性,为了作出恰当的泛函,我们使用J.A.Walker、L.G.Clark的积分法并加以改进。从而得到若干充分性条件。这些条件简单,易于实现。     

Necessary Covariance Conditions for a One-Field Lax Pair

We study the covariance with respect to Darboux transformations of polynomial differential and difference operators with coefficients given by functions of one basic field. In the scalar (Abelian) case, the functional dependence is established by equating the Frechet differential (the first term of the Taylor series on the prolonged space) to the Darboux transform; a Lax pair for the Boussinesq equation is considered. For a pair of generalized Zakharov-Shabat problems (with differential and shift operators) with operator coefficients, we construct a set of integrable nonlinear equations together with explicit dressing formulas. Non-Abelian special functions are fixed as the fields of the covariant pairs. We introduce a difference Lax pair, a combined gauge-Darboux transformation, and solutions of the Nahm equations.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 122–132, July, 2005.     

主曲率满足四次函数关系的Weingarten曲面的基本方程和LAX对

讨论三维欧氏空间中两个主曲率满足四次函数关系的Weingarten曲面的可积性,得到其Gauss-Codazzi方程的完全分类,并给出相应的Lax对.     

Boundary Conditions for Multidimensional Integrable Equations

We suggest an efficient method for finding boundary conditions compatible with integrability for multidimensional integrable equations of Kadomtsev-Petviashvili type. It is observed in all known examples that imposing an integrable boundary condition at a point results in an additional involution for the t-operator of the Lax pair. The converse is also likely to be true: if constraints imposed on the coefficients of the t-operator of the L-A pair result in a broader group of involutions of the t-operator, then these constraints determine integrable boundary conditions.New examples of boundary conditions are found for the Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili equations.     

获得非线性演化方程Backlund变换的一种新的途径

本文给出一种求非线性演化方程Backlund变换的方法,应用于非线性演化方程时,得到了与WTC方法一致的Backlund变换,避开了WTC方法涉及到的递推关系和截尾的讨论.     

On the Liapunov Second Method for Difference Equations

We study various kinds of stability of a constant solution of an autonomous difference equation by using auxiliary functions like in the Liapunov Second Method. We establish new general principles and we improve several classical results.     

谱问题、发展方程族及Lax表示的等价性

讨论了四对谱问题、相应的保谱方程族以及它们Ｌａｘ表示的等价住，并利用这种等价关系可导出一些方程的Ｌａｘ表示．     

A Variational Complex for Difference Equations

An analogue of the Poincaré lemma for exact forms on a lattice is stated and proved. Using this result as a starting-point, a variational complex for difference equations is constructed and is proved to be locally exact. The proof uses homotopy maps, which enable one to calculate Lagrangians for discrete Euler–Lagrange systems. Furthermore, such maps lead to a systematic technique for deriving conservation laws of a given system of difference equations (whether or not it is an Euler–Lagrange system).     

Ergodic isospectral theory of the Lax pairs of Euler equations with harmonic analysis flavor

Isospectral theory of the Lax pairs of both 3D and 2D Euler equations of inviscid fluids is developed. Eigenfunctions are represented through an ergodic integral. The Koopman group and mean ergodic theorem are utilized. Further harmonic analysis results on the ergodic integral are introduced. The ergodic integral is a limit of the oscillatory integral of the first kind.

     

Bilinear Bäcklund transformation and Lax pair for a coupled Ramani equation

In this paper, a coupled Ramani equation is proposed. The bilinear Bäcklund transformation and Lax pair for this equation are derived starting from its bilinear form. Multisoliton solutions to the system can also be obtained.     

非线性时滞差分方程的振动性

考虑非线性时滞差分方程x_{n+1}-x_n+p_nf(x_{n-l_1},x_{n-l_2}x_{n-l_m})=0, n=0,1,2, 获得了方程所有解振动的充分条件, 推广并改进了现有文献中的结果.