Soliton solutions associated with the third-order ordinary linear differential operator

Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation \(d^3\psi/dx^3+Q\,d\psi/dx+P\psi=k^3\psi,\) where \(P\) and \(Q\) are the potentials in the Schwar...

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Bibliographic Details
Published in:arXiv.org 2024-12
Main Authors: Aktosun, Tuncay, Choque-Rivero, Abdon E, Toledo, Ivan, Unlu, Mehmet
Format: Article
Language:English
Subjects:
Boussinesq equations
Constants
Differential equations
Inverse problems
Inverse scattering
Linear evolution equations
Nonlinear evolution equations
Operators (mathematics)
Solitary waves
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Summary:Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation \(d^3\psi/dx^3+Q\,d\psi/dx+P\psi=k^3\psi,\) where \(P\) and \(Q\) are the potentials in the Schwartz class and \(k^3\) is the spectral parameter. The input data set used to solve the relevant inverse problem consists of the bound-state poles of a transmission coefficient and the corresponding bound-state dependency constants. Using the time-evolved dependency constants, explicit solutions to the related integrable evolution equations are obtained. In the special cases of the Sawada--Kotera equation, the Kaup--Kupershmidt equation, and the bad Boussinesq equation, the method presented here explains the physical origin of the constants appearing in the relevant \(\mathbf N\)-soliton solutions algebraically constructed, but without any physical insight, by the bilinear method of Hirota.
ISSN:2331-8422