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Families of semi-rational solutions to the Kadomtsev-Petviashvili I equation. (English) Zbl 1508.35127

Summary: In this work we derive families of explicit breather solutions of any order to the Kadomtsev-Petviashvili equation (KPI) and the Boussinesq equation. We employ the Hirota bilinear method combined with the KP hierarchy reduction method to determine these solutions. By taking a long wave limit of breather solutions, two types of semi-rational solutions to the KPI equation are constructed via using the determinant expression. The first type of semi-rational solutions only consists of breathers of arbitrary order and lumps of arbitrary order in the \((x, y)\)-plane. The second type of semi-rational solutions comprises of solitons of arbitrary order, breathers of arbitrary order and lumps of arbitrary order in the \((x,y)\)-plane.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35Q55 NLS equations (nonlinear Schrödinger equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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