Аннотация
A class of negative order Ablowitz–Kaup–Newell–Segur nonlinear evolution equations are obtained by applying the \nLax hierarchy of the first order linear system of three equations. The inverse scattering problem on the whole axis is \nexamined in the case where linear system becomes the classical Zakharov–Shabat system consists of two equations and \nadmits a real anti-symmetric potential. Referring to these results, the N-soliton solutions for the integro-differential \nversion of the nonlinear Klein–Gordon equation coupled with a scalar field are obtained by using the inverse scattering \nmethod via the Riemann–Hilbert problem.