Nonconservative Reflectionless Inverse Scattering and Soliton Solutions of an Associated Nonlinear Evolution System

J. Math. Sci. Univ. Tokyo
Vol. 28 (2021), No. 4, Page 651-712.

Kamimura, Yutaka
Nonconservative Reflectionless Inverse Scattering and Soliton Solutions of an Associated Nonlinear Evolution System
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Abstract:
A nonconservative, reflectionless inverse scattering problem is discussed on an energy dependent Schrödinger equation. A scattering transform from the potential of the equation to the reflectionless scattering data is completely characterized by a function induced from a Gelfand-Levitan-Marchenko equation, with an expression of the inverse scattering transform in terms of the function. Based upon the inverse scattering theory, we establish an inverse scattering method by which $N$-soliton solutions of a nonlinear evolution system (Boussinesq system) are constructed.

Keywords: Inverse scattering theory, reflectionless potential, energy dependent Schrödinger equation, nonlinear evolution system, soliton solution, Boussinesq system.

Mathematics Subject Classification (2010): 35Q53, 37K10, 37K40, 81U40.
Received: 2021-02-12