Multidimensional integrable systems, differential-difference equations and the symmetry approach

可积偏微分方程在现代数学和物理学中有着重要的意义，在现代数学中，它们与代数和微分几何有着重要的联系，在物理学中，它们描述了许多重要的物理模型。现代数学物理中的许多概念，如孤子、瞬子、量子群等都起源于可积系统理论，而可积系统理论中最重要也是最具挑战性的问题之一就是如何识别一个给定的方程何时可积。这个项目致力于多维和微分-差分多项式可积系统。本文提出用摄动对称方法和数论方法相结合的方法来研究多维微分差分方程的可积性，并得到这类可积系统的完全分类结果。得到2+1维可积标量和耦合多项式齐次方程的全局（任意阶）分类。二.将扰动对称方法推广到高维偏微分方程.将符号方法推广到微分-差分方程.获得可积标量方程和耦合微分差分方程的整体分类。AMS 2000主题分类：37 K1 0，37 K35

Integrable partial differential equations are of great interest in modern mathematics, where they have important connections with algebra and differential geometry, and in physics, where they describe many important physical models. Many concepts of modern mathematical physics such as solitons, instantons and quantum groups have their origin in theory of integrable systems.One of the most important and challenging problems in the theory of integrable systems is how to recognize when a given equation is integrable. This project is devoted to multidimensional and differential-difference polynomial integrable systems. The combination of the Perturbative Symmetry Approach and number theoretical methods is proposed for studying integrability of multidimensional and differential-difference equations as well as for obtaining complete classification results of integrable systems of this type.The aims of the proposed research are the following:I. Obtain a global (at arbitrary order) classification of integrable scalar and coupled polynomial homogeneous equations in 2+1 dimensions. II. Extend the perturbative symmetry approach to higher dimensional partial differential equations.III. Extend the symbolic method to differential-difference equations.IV. Obtain a global classification of integrable scalar and coupled differential-difference equations.AMS 2000 subject classification: 37K1 0, 37K35