Exact solutions for discrete and continuous nonlinear systems

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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.