Fields Institute Workshop on Nonlinear Wave Dynamics

Carleton University, August 20-22 2008


Walid Abou-Salem

University of Toronto

Limiting dynamics of solitons in random potentials

I discuss recent progress in rigorously understanding the effective dynamics of solitons for the generalized nonlinear Schroedinger equation in a random (time-dependent) potential. If the random potential is almost surely space-adiabatic, then the long-time dynamics of the center of the soliton is almost surely described by Hamilton's equations for a classical particle in the random potential. Furthermore, one has sufficient control on the dynamics that allows for studying its limiting behavior in the weak-coupling/space-adiabatic limit. Under certain mixing hypotheses for the potential, the momentum of the center of mass of the soliton converges in law to a diffusion process on a sphere of constant energy. In three and higher dimensions, the trajectory of the center of mass of the soliton converges to a spatial Brownian motion. (This is joint work with Catherine Sulem.)