Structure of partial difference equations with continuous symmetries and conservation laws

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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.