A novel approach to integrability of semi-discrete systems

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Physical phenomena are often described in terms of nonlinear differential/difference equations.In general it is impossible to obtain exact analytic solutions to most nonlinear systems and the best one can do is to use approximate, asymptotic or numerical methods. Meanwhile, there exists a remarkable class of nonlinear systems called integrable systems. Integrable systems can be analysed meticulously, possess rich hidden algebraic structures and often have geometric realisations. The interest in integrable systems is surging. The range of their applications and unexpected connections with other areas of Mathematics is growing fast. Therefore, the central problems are to determine whether a given equation is integrable, and ultimately to make a complete classification of integrable systems. In this project we propose to test new methods in the theory of integrable systems inspired by recent developments. We propose to re-formulate the problem of integrability in rigorous terms of difference algebra. Our novel idea is to study non-local extensions of the corresponding difference field in order to achieve a better flexibility in the application of formal pseudo--difference series and aiming to find universal necessary integrability conditions. We are going to develop a symbolic representation of the objects involved to re-cast the problem in terms of symmetric Laurent polynomials (similar ideas proved to be successful in the differential case, but have not been developed in the difference setting). We aim to make a progress in a long standing problem, whose solution would have a lasting impact on the development of Mathematics, Mathematical Physics, Numerical Analysis and far beyond. We also will attempt to extend these new methods to non-commutative (free associative and quantum) settings to explore uncharted terrain of non-commutative integrable systems, their Poisson structures and quanisations. A recently emerged new approach to quantisation is based on the study of dynamical systems for functions with values in a free associative algebra and certain invariant differential ideals of the algebra. We aim to extend the theory semi-discrete integrable systems to free associative and quantised algebra domains and to link it with non-commutative algebraic geometry, quantum and statistical mechanics. This small research project is a spring-board or feasibility studies for the future full scale research projects suitable for standard mode applications.