This talk will discuss problems related to the kinetic theory of solitons. This problem was initially studied in the 1970s by V. Zakharov, but a rigorous mathematical frame work for understanding the kinetic theory of solitons is a work in progress.
I will introduce the idea of nonlinear soliton waves that are localized (in one direction) when they are far apart, but strongly interact when close. I will discuss various mathematical frameworks for solving deterministic models of many interacting solitons, and discuss some ideas on how to use this framework to tackle statistical problems.
I will also discuss how recent results with K. McLaughlin, D. Zakharov and V. Zakharov relate to the formation of periodic and quasi-periodic patterns in a soliton gas wave field as well as high amplitude solitons that only persist for a finite amount of time. These phenomena are due to a more general phenomenon V. Zakharov has called integrable turbulence.