In this talk we will use analytical solutions to completely integrable nonlinear dispersive equations for nonlinear long waves (modeling tsunamis, solitons, undular bores, and internal waves for example) and their perturbations as a framework for understanding recent results from mathematics, physics, and engineering on nonlinear waves. We will then introduce the Zakharov-Manakov dressing method, weak wave kinetic equations, and soliton wave kinetic equations (describing solitons gasses and soliton condensates) using this framework. A soliton wave kinetic equation for the KP equation would be a novel breakthrough in mathematics, physics and engineering. Vladimir Zakharov conjectured that a weak wave kinetic equation described the dynamics of the wave action (or the evolution of a wave spectrum) for the KP 1 equation. A proof of this conjecture would be an important addition to the rapidly growing literature on wave kinetic equations in mathematics.

This is joint work with Brandon Young, Solomon Yim, Peter Prins, and Sander Wahls.