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Structure of partial difference equations with continuous symmetries and conservation laws

具有连续对称性和守恒定律的偏差分方程的结构

基本信息

批准号：
EP/I038675/1
负责人：
A Mikhailov
金额：
$ 32.58万
依托单位：
University of Leeds
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI038675%2F1
关键词：
Structure partial difference equations continuous

项目摘要

This is a Mathematics proposal in the broad area of Integrable Systems with a focus on difference equations. Many natural phenomena have a discrete nature or can be modelled in terms of difference equations. Any simulation of a continuous phenomena on a digital computers requires an appropriate discretisation. Difference equations have a wide range of practical applications from Fundamental Physics to Engineering. The theory of difference equations is considerably less developed than the classical theory of differential equations. Broadly speaking our research project aims to reduce the gap between them and to explore new features that are not available in the case of differential equations. A reformulation of the theory of difference equations in terms of difference algebra will enable us to use a variety of new methods and provide a rigorous framework. From the other side, non-trivial examples originated from the applied theory of difference equations could serve as a basis for further development and new concepts in difference algebra. It is difficult to overestimate the importance of continuous symmetries and local conservation laws in the theory and applications of differential equations. Often they carry the most valuable information about the model and are more important than exact solutions. In the project we will find a sequence of necessary conditions for the existence of ahigh order symmetry (or a conservation law) for a given system of difference equations. Continuous symmetries and conservation laws can serve as a characteristic property for the class of integrable systems. Symmetries ofintegrable partial differential equations can be generated by recursion (or Lenard) operators. We propose to develop an interesting and rather non-trivial extended analogue of Lenard's scheme.Together with solutions of clearly set problems our project is poised to invade an uncharted territory of difference equations with approximate symmetries. To study properties and algebraic structures associated with approximately integrable equations will be a new and challenging direction of research.

这是在可积系统的广泛领域中的一个数学建议，重点是差分方程组。许多自然现象都是离散的，或者可以用差分方程式来模拟。在数字计算机上对连续现象的任何模拟都需要适当的离散化。差分方程组从基础物理到工程都有广泛的实际应用。与经典的微分方程组理论相比，差分方程组理论的发展要慢得多。总的来说，我们的研究项目旨在缩小它们之间的差距，并探索在微分方程式的情况下没有的新功能。用差分代数重新表述差分方程组理论将使我们能够使用各种新方法，并提供一个严格的框架。另一方面，源于差分方程组应用理论的非平凡例子可以作为差分代数的进一步发展和新概念的基础。连续对称性和局部守恒律在微分方程组的理论和应用中的重要性怎么估计都不为过。通常，它们携带着关于模型的最有价值的信息，并且比确切的解决方案更重要。在这个项目中，我们将找到给定差分方程组存在高阶对称性(或守恒律)的一系列必要条件。连续对称性和守恒律可以作为这类可积系统的一个特征性质。可积偏微分方程的对称性可以由递归(或Lenard)算子生成。我们建议开发一个有趣且非平凡的Lenard方案的扩展类比。随着明确设置问题的解决方案，我们的项目准备入侵具有近似对称性的差分方程组的未知领域。研究与近似可积方程相关的性质和代数结构将是一个新的具有挑战性的研究方向。