Inverse scattering method via the Gel'fand-Levitan-Marchenko equation for some negative-order nonlinear wave equations

Abstract

A class of negative-order Ablowitz-Kaup-Newell-Segur nonlinear evolution equations are obtained by applying the Lax hierarchy of a first-order linear system of three equations. The inverse scattering problem on the line is examined in the cases where the linear system becomes the classical Zakharov-Shabat system with real antisymmetric and real symmetric potentials. Referring to these results, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document}-soliton solutions for the integro-differential version of the nonlinear Klein-Gordon equation coupled to a scalar field and the negative-order modified Korteweg-de Vries equation are obtained by using the inverse scattering method via the Gel'fand-Levitan-Marchenko equation.