Rational solutions of the Boussinesq equation and applications to rogue waves

Clarkson, Peter, Dowie, Ellen (2017) Rational solutions of the Boussinesq equation and applications to rogue waves. Transactions of Mathematics and Its Applications, 1 (1). ISSN 2398-4945. E-ISSN 2398-4945. (doi:10.1093/imatrm/tnx003)

PDF - Author's Accepted Manuscript

 Preview
Official URL
https://doi.org/10.1093/imatrm/tnx003

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. Further the rational solutions have an interesting structure as they are comprised of a linear combination of four independent solutions of the bilinear equation. 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 these two families of rational solutions of the KPI equation are fundamentally different and a unifying framework is found which incorporates both families of solutions.

Item Type: Article 10.1093/imatrm/tnx003 Q Science Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Peter A Clarkson 18 Oct 2017 09:45 UTC 29 May 2019 19:43 UTC https://kar.kent.ac.uk/id/eprint/64073 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-8777-5284