Nonlinear Waves and Dispersive Equations

Abstract

Nonlinear dispersive equations describe nonlinear wave phenomena that arise in many physical systems, for instance, in the context of general relativity, quantum mechanics, or water waves. Linear dispersive equations have solutions that spread out and decay while keeping a constant $L^2$-norm. The interaction with nonlinear effects leads to a rich variety of behaviors, including finite-time blow-up, soliton formation, and scattering. These equations are connected to many branches of mathematics, such as integrable systems, harmonic analysis, geometry, and probability.