2016/09/30 by Peter A. Clarkson, Peter A Clarkson, Ellen Dowie · 2 citations
Mathematics · Physics and Astronomy · #Boussinesq approximation (buoyancy) #Inverse #Inverse problem #Inverse scattering problem #Inverse scattering transform #Korteweg–de Vries equation #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Rational function #Rogue wave #math-ph #math.AP #math.MP #nlin.SI
paper · pdf · doi:10.1093/imatrm/tnx003
published as Transactions of Mathematics and its Applications, 1 (2017) · 20 pages, 10 figures
openalex created_date 2016/09/16 · openalex publication_date 2017/01/01 · arxiv created 2017/07/29 · arxiv updated 2017/11/07 · openalex updated_date 2026/08/05
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 Schrödinger (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.