Structure of partial difference equations with continuous symmetries and conservation laws

这是一个数学建议在可积系统的广泛领域，重点是差分方程。许多自然现象具有离散的性质，或者可以用差分方程来建模。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。差分方程具有广泛的实际应用，从基础物理到工程。差分方程理论远不如经典的微分方程理论发达。总的来说，我们的研究项目旨在缩小它们之间的差距，并探索微分方程中不可用的新特征。用差分代数重新表述差分方程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，源于差分方程应用理论的非平凡例子可以作为差分代数进一步发展和新概念的基础。在微分方程的理论和应用中，连续对称性和局部守恒律的重要性怎么评价都不过分。它们通常携带着关于模型的最有价值的信息，比精确的解决方案更重要。在该项目中，我们将找到一个序列的必要条件，为一个给定的系统的差分方程的高阶对称性（或守恒定律）的存在。连续对称性和守恒律可以作为可积系统的特征性质。可积偏微分方程的对称性可以由递归（或Lenard）算子生成。我们建议发展Lenard格式的一个有趣的、相当不平凡的扩展模拟。连同明确设定的问题的解决方案，我们的项目准备入侵一个未知的领域差分方程近似对称。研究近似可积方程的性质和代数结构将是一个新的、具有挑战性的研究方向。

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.