Study of special solutions to dispersive equations – ESSED

This project is about nonlinear dispersive partial differential equations (PDEs) coming from physics, more specifically quantum mechanics, statistical physics, quantum field theory, or wave turbulence. The idea behind the whole project is that there exist special solutions to some PDEs around which the PDE can be reduced to a simpler model in a certain asymptotic. It divides into 3 axes : the study of coherent states, the dynamics of multisolitons, and the evolution of random
initial data. In the study of coherent states, the asymptotic is semiclassical, and the reduced model is the classical system associated to the PDE, which is an ordinary differential equation (ODE). In studying the evolution of a multisoliton, one would like to be able to study only the evolution of the parameters of the solitons, which is again an ODE. In weak turbulence, the asymptotic is as time goes to infinity and the nonlinearity to 0 in a certain regime. The reduced model is the second-order resonances system. In every respect, we try to obtain a simpler equation, or prove that our starting PDE converges in some sense to a simpler equation, or prove the opposite, that there is some additional effect to take into account and identify where this effect comes from and what impact it has.
The tools and general strategy come from the study of dispersive PDEs and imply either dispersive estimates or conservation laws (in a large sense). For this, the members of the team all have a strong background in dispersive PDEs, but also each has each their specificities. The team is composed of young researchers, three maître de conférences and one chargé de recherche, two of them are in Paris 13, one is in Bordeaux and one in Besançon. The realization of this project would be an opportunity to communicate and start new collaborations, between the members of the team, as many meetings and two workshops are planned, and start or continue international collaborations.
The estimated cost of the project is 75600 euros, with 36000 euros for personal travels and invitations, 30000 for the organization of two workshops, 4000 for material, and 5600 as administration costs.
The different outcomes of this project are a better understanding of various aspects of the asymptotic regime of some naturally arising PDEs, the beginning of new collaborations between the members of the team, the strengthening or initiating of national and international collaborations thanks to travels and the organization of workshops, which would imply national and international diffusion and visibility of the project.