Structure of partial difference equations with continuous symmetries and conservation laws

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 a
high 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 of
integrable 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.

The interdisciplinary and cross-disciplinary nature of the project relating abstract concepts from difference algebra to notions and problems from the applied theory of difference equations represents a new kind of approach, the success of which will guarantee a long term impact across Mathematics and its applications and impact broadly on culture and science.

A reformulation of the applied theory of difference equations in the rigorous framework of difference algebra will enable both mathematical disciplines to benefit from each other. The applied theory will get a solid foundation and rigorous methods from difference algebra including the difference Galois theory. Difference algebra will get an important set of non-trivial examples, new concepts and naturally formulated problems. This undoubtedly will have impact on the both mathematical disciplines and will lead to further convergence of applied and pure research in difference equations.

The derivation of explicit integrability conditions will have immediate academic impact opening the door for classification of integrable difference equations as well as for testing for integrability.

Research on difference algebra is well developed in the USA, Europe, China and Russia but in the UK it is represented mostly by the Model theory and Logic. Meanwhile the UK is one of the world leaders in integrable systems research. Our project will draw attention to difference algebra and its methods. It will naturally motivate research in this area. Our
ambitious aim is to improve the landscape of mathematics in the UK, to make the UK competitive in a wide spectrum of research related to difference algebra and its applications.

Apart from an obvious impact on the PGRS and PDRA, there will be much broader educational impact on the whole integrable community in the UK and beyond. We are planning to prepare a graduate course "algebraic theory of integrable systems" for the MAGIC and/or LTCC courses (supported by the EPSRC).