Exact solutions for discrete and continuous nonlinear systems

This is a Mathematics proposal in the broad area of Integrable Systems with a focus on exact solutions of nonlinear systems. The area of Integrable Systems started with the remarkable discovery of solitary waves on shallow water, known as solitons, which has changed the paradigm and our understanding of nonlinear phenomena in general. As a mathematical concept, solitons first appeared about 50 years ago when analytical solutions for the Korteweg-de Vries equation, describing shallow water waves, were explicitly constructed by the inverse scattering method. This method were soon applied for many systems, which are important for applications, such as the Nonlinear Schrodinger equation (non-linear optics, modulation instability), sine-Gordon equation (non-linear optics, superconductive Josephson junctions, low-frequency collective motion in proteins and DNA), Heisenberg and Landau-Lifshitz models (in the theory of magnetism) and many others.

Over the last decade surprising connections of the soliton theory for the Kadomtsev-Petviashvili (KP) equation with cluster algebras and enumerative geometry have been discovered. The KP equation is used to model shallow water waves on a surface. Its soliton solutions form web structures. Kodama and Williams described them in terms of totally positive Grassmanians. Recently, for systems of partial differential and differential-difference equations we have developed a method for construction of exact solutions based on symmetries of the Lax representations and discovered new classes of solutions, which represent nonlinear wave fronts propagating with constant velocity. It has become clear that the world of solitons is much richer than it has been anticipated. Equipped with this new methodology, we are well prepared to tackle the problem of construction, description and visualisation of exact solutions for basic systems of partial differential and differential-difference equations.

The aims of this proposal are highly ambitious. The proposal is divided into four parts corresponding to the four objectives listed above. Each objective can be achieved independently, though they are closely related.

The proposed project is of an intradisciplinary nature. It relates rather abstract constructions from Kac-Moody algebras, Schubert decompositions of Grassmanians, vector field representations of Verma modules with very concrete problems in the applied theory of partial differential and differential-difference equations. It will stimulate the interaction of researchers across the traditional boarders of well established disciplines, which will guarantee a lasting impact on Pure and Applied Mathematics and impact broadly on culture and science.

It is feasible that the findings of this project can be used for enhancing the capacity and quality of fibre optics telecommunication systems. The spectral transform method, which is a part of this research proposal, will play a crucial role in future fiber optics technology aiming to support the bandwidth-hungry on-line services, such as cloud computing, on-demand HD video streams, on-line business analytics and content sharing. It is hard to overstate the impact that optical fibre communications has made on the economy, public and government services, society, and almost all aspects of our lives. Its importance has already been recognised and supported by the EPSRC (Programme Grant UNLOC EP/J017582/1). According to K.Steiglitz, vector solitons can be used for design of logic gates and soliton computations. This is another possible, although less understood area which could have a considerable economic and strategic impact in the future.


Apart from an obvious impact on the PDRAs, there will be much broader educational impact in the UK mathematical community. The framework and methods developed in our project will form a core material for a new (MAGIC and/or LTCC) graduate course on exact solutions for nonlinear systems. The research finding on visualisation of exact solutions in this project is the suitable material for outreach. We will develop new activities for it with the support of our outreach officers.