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

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 $N$-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.