Rational solutions of the Boussinesq equation and applications to rogue waves

TL;DR

This paper investigates rational solutions of the Boussinesq equation, their algebraic structure, and applications to rogue waves, also comparing solutions of the KPI equation derived from different methods.

Contribution

It introduces new rational solutions of the Boussinesq equation expressed via special polynomials and explores their relation to rogue waves and solutions of the KPI equation.

Findings

Rational solutions depend on two arbitrary parameters.

Solutions are expressed through special polynomials derived from a bilinear equation.

Two distinct families of rational solutions for the KPI equation are identified.

Abstract

We study rational solutions of the Boussinesq equation, which 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 focusing nonlinear Schr\"odinger (NLS) equation and have an interesting structure. Further rational solutions of the Kadomtsev-Petviashvili I (KPI) equation are derived in two ways, from rational solutions of the NLS equation and from rational solutions of the Boussinesq equation. It is shown that the two families of rational solutions of the KPI equation are fundamentally different.