Abstract

Solitons are the remarkably stable solitary wave solutions to certain nonlinear evolution equations, such as the nonlinear Schroedinger equation (NLS) or the Korteveg-de Vries equation (KdV). They also enjoy a remarkable sociological stability by being of interest to applied mathematicians, PDE experts, algebraic geometers, and representation theorists. The particle-like behaviour of solitons is visible when external potentials are added to the original equations. That means that in addition to self-interaction modeled by the nonlinearity, an external field is applied to the solitary waves. That can result in different kinds of phenomena involving one or more solitons. I will describe results obtained in collaboration with Justin Holmer, and with Justin Holmer and Galina Perelman, on solitons for NLS and mKdV interacting with slowly varying external fields (semiclassical regime), and with highly localized impurities (delta function potentials). The mathematical results are strikingly confirmed in numerical experiments which also suggest many open questions.