Formation of singularities in nonlinear dispersive PDEs

Abstract

This contribution addresses the problem of singularity formation in nonlinear dispersive equations. Despite significant progress made in the last 20 years, for most even simplest canonical models our understanding of the question is far from being complete. The aim of this note is to give a selection of results and open questions illustrating the present state of the problem in the context of some basic model equations, mostly of Schrödinger type, such as the semilinear Schrödinger and Schrödinger map equations, putting an emphasis on the role of solitons in the mechanisms of singularity formation.