Soliton gas and nonlinear dispersive equations

The project will focus on integrable systems, in particular integrable partial differential equations. These equations have a particular set of solutions called solitons that interact in an elastic manner.
The main goal of the project is to study properties of solutions with infinite number of solitons both in the deterministic and stochastic setting and give a rigorous derivation and solution of the kinetic equations of solitons. These equations have received a lot of attention in recent years in the statistical mechanics of classical and quantum integrable systems. The proposal will mainly focus on the focusing nonlinear Schroedinger equation. A further goal is to consider generic initial data with infinite number of solitons. This setting introduces rich structural complexity due to the continuous spectrum of the Lax operator and the resulting dispersive radiation. This study will have as a byproduct the understanding of modulational instability of the nonlinear focusing Schroedinger equation under certain class of compact support perturbations.

The main goal of the project is to study properties of solutions of nonlinear wave equations with infinite number of solitons both in the deterministic and stochastic setting and provide quantitative and qualitative properties of solutions. This include universality profile of perturbation of modulationally unstable wave and the formation of rogue waves when a large set of solitons synchronise.