Abstract

In this talk I shall discuss rational solutions of the Boussinesq equation, the focusing nonlinear Schr”odinger (NLS) equation and the Kadomtsev-Petviashvili I (KPI) equation, which are all soliton equations solvable by the inverse scattering.

The Boussinesq equation was introduced by Boussinesq in 1871 to describe the propagation of long waves in shallow water. Rational solutions of the Boussinesq equation, 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 focusing NLS equation and have an interesting structure. Conservation laws and integral relations associated with rational solutions of the Boussinesq equation will also be discussed.

Rational solutions of the KPI equation will be derived in three ways: from rational solutions of the NLS equation; from rational solutions of the Boussinesq equation; and from the spectral problem for the KPI equation. It'll be shown that these three families of rational solutions are fundamentally different.