Orateur
Description
Semiclassical soliton ensembles (SSE) in the small dispersion limit are initially coherent collections of many solitons that well-approximate some initial profile. Evolving forward in time, the profile will eventually undergo wave breaking, shedding the solitons and generating a dispersive shock wave. We study this phenomenon for two PDE. The first SSE, for the intermediate long wave equation, is constructed to approximate general smooth Klaus-Shaw initial data. We first conduct a heuristic WKB approximation to determine the approximate scattering data and then rigorously study the inverse scattering problem using the methods of Lax and Levermore. We show the initial condition is recovered in the limit and the solution up until wave breaking approaches that of Invicid Burgers' equation in an $L^2$ sense. The second SSE is the $\mathrm{sech}^2$ initial condition for the Korteweg-de Vries equation. Inverse scattering is done via a Riemann-Hilbert problem and the method of nonlinear steepest descent is employed. This project is joint work with K. Schmidt (University of Central Florida) and R. Buckingham (University of Cincinnati).