Speaker: Professor Peter Clarkson
Affiliation: School of Mathematics, Statistics and Actuarial Science, University of Kent, UK
Venue: Building 8 Room G25
Date: Thursday 10th November 2016
Time: 14.30-15.30 pm
Title: Rational solutions of integrable equations and applications to rogue waves
Abstract: In this talk I shall discuss rational solutions of three integrable equations, the Boussinesq equation, the focusing nonlinear Schr\"odinger (NLS) equation and the Kadomtsev-Petviashvili I (KPI) equation. The Boussinesq equation was introduced by Boussinesq in 1871 to describe the propagation of long waves in shallow water and is a soliton equation solvable by the inverse scattering method. These rational solutions, which are algebraically decaying and depend on two arbitrary parameters, are expressed in terms of special polynomials that are derived through a bilinear equation, have a similar appearance to rogue-wave solutions of the NLS equation and have an interesting structure. Further rational solutions of the KPI equation are derived in three ways, from rational solutions of the NLS equation, from rational solutions of the Boussinesq equation and from the linear scattering problem for the KPI equation. It is shown that these three families of rational solutions of the KPI equation are fundamentally different.