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)

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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
DOI/Identification number: 10.1093/imatrm/tnx003
Subjects: Q Science
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Peter A Clarkson
Date Deposited: 18 Oct 2017 09:45 UTC
Last Modified: 29 May 2019 19:43 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/64073 (The current URI for this page, for reference purposes)
Clarkson, Peter: https://orcid.org/0000-0002-8777-5284
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