Soliton Interactions with Dispersive Wave Background

Abstract

The Korteweg – de Vries (KdV) equation is a classical model for describing long surface gravity waves propagating in dispersive media. It is known to possess many families of exact analytic solutions, including solitons, which due to their distinct physical nature, are of particular interest to physicists and mathematicians alike. The propagation of solitons on the background of large-scale waves is a fundamental problem, with applications in fluid dynamics, nonlinear optics and condensed matter physics. This thesis centers around construction and analysis of a soliton as it interacts with either a rarefaction wave (RW) or a modulated dispersive shock wave. Using the Darboux transformation for the KdV equation, we construct and analyze exact solutions describing the dynamic interaction of a soliton and a dispersive mean field.