Title: Inverse scattering method for nonlinear Klein--Gordon equation coupled with a scalar field

Abstract: In the present work, a novel class of negative order Ablowitz-Kaup- Newell-Segur (AKNS) nonlinear evolution equations are obtained by applying the Lax hierarchy of the generalized Zakharov-Shabat (ZS) system. The inverse scattering problem on the whole axis is examined in the case where the ZS system consists of two equations and admits a real 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 are obtained by using the inverse scattering method.