Inverse scattering method via 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 the first order linear system of three equations. The inverse scattering problem on the whole axis are examined in the case where linear system becomes the classical Zakharov--Shabat system consists of two equations and admits a real symmetric and real anti-symmetric potential. Referring to these results, the N--soliton solutions for the integro-differential version of the nonlinear Klein--Gordon equation coupled with a scalar field (CKG) and negative order modified Korteweg-de Vries (nmKdV) equation are obtained by using the inverse scattering method via the Gel'fand--Levitan--Marchenko equation.