ÃÛÌÇÖ±²¥

Rational solutions of three integrable equations and applications to rogue waves

In this talk I shall discuss rational solutions of three integrable equations, the Boussinesq equation, the focusing nonlinear Schrödinger (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. Conservation laws and integral relations associated with rational solutions of the Boussinesq equation will also be discussed. 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.

References:
1. A Ankiewicz, A P Bassom, P A Clarkson and E Dowie, Conservation laws and integral relations for the Boussinesq equation, Stud. Appl. Math., 139 (2017) 104-128
2. P A Clarkson and E Dowie, Rational solutions of the Boussinesq equation and applications to rogue waves, Transactions of Mathematics and its Applications, 1 (2017) tnx003