Isospectral kinetic equation for solitons: integrability, exact solutions and physical applications

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The idea of introducing statistical description into soliton theory has two well established physical premises: a) natural wave phenomena are often so complex that they must be described statistically; b) integrable wave equations capture important qualitative and quantitative features of nonlinear wave propagation in dispersive media. By bringing together these two premises, one arrives at the challenging problem of an adequate mathematical description of the behaviour of disordered soliton systems with large number of degrees of freedom. Although the first works in this direction had been published in early 1970s, only recently a substantial progress has been achieved by considering a special thermodynamic type limit for the nonlinear modulation equations associated with the (integrable) dynamics preserving spectral parameters of the interacting waves. In the thermodynamic limit, the nonlinear interacting modes transform into randomly distributed localised states (solitons) and the modulation system assumes the form of a nonlinear kinetic equation for a soliton gas. This new kinetic equation has nontrivial mathematical structure (which is drastically different from Bolzmann's kinetic equation) and a potential for various physical applications. Both are virtually unexplored. This project is set to establish main mathematical properties of the isospectral kinetic equation for solitons and to explore its possible applications to fluid dynamics and nonlinear optics. The connections with some actively studied mathematical objects such as hydrodynamic chains and two-dimensional dispersionless hierarchies will be studied. Exact solutions will be constructed and their physical implications will be investigated. The results of the project will be of considerable interest for the nonlinear wave community in general as the isospectral kinetic equation for solitons represents a universal mathematical model applicable in different physical contexts.