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

多维可积系统、微分差分方程和对称方法

• 批准号: EP/C527747/1
• 负责人: Vladimir Novikov
• 金额: $ 14.88万
• 依托单位: University of Kent
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2006
• 资助国家: 英国
• 起止时间: 2006 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FC527747%2F1
• 关键词: Multidimensional; integrable; systems; differential; difference

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

可积偏微分方程在现代数学中有着重要的意义，它与代数和微分几何有着重要的联系，在物理学中，它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、瞬子和量子群，都起源于可积系统理论。在可积系统理论中最重要和最具挑战性的问题之一是如何识别给定方程何时是可积的。本课题研究多维和微分-差分多项式可积系统。提出了微扰对称方法与数论方法相结合的方法来研究多维和微分-差分方程的可积性，并得到了这类可积系统的完整分类结果。本研究的目的如下：1。在2+1维中得到可积标量和耦合多项式齐次方程的一个全局（任意阶）分类。2。将微扰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。得到可积标量和耦合微分-差分方程的全局分类。AMS 2000学科分类：37K1 0, 37K35