Exact Solutions of Soliton Equations and Dynamical Systems

Abstract: This report investigates how one can construct soliton solutions to the following solitonequations: the Korteweg-de Vries equation, the Benjamin-Ono equation, and the spinBenjamin-Ono equation by making rational pole ansätze, which are ansätze that dependon eponymous pole parameters moving in the complex plane. In doing so we demonstratea connection between these soliton solutions and the class of integrable dynamical systemsknown as Calogero-Moser systems. We find that the ansätze solves the given solitonequation if the motion of the poles is governed by an integrable dynamical system relatedto the Calogero-Moser systems. Additionally, we numerically implement and analyse thesoliton solutions constructed for the Korteweg-de Vries and Benjamin-Ono equations.