On inverse scattering for the two-dimensional nonlinear Klein–Gordon equation

The inverse scattering problem for the two-dimensional nonlinear Klein–Gordon equation [Formula: see text] is studied. We assume that the unknown nonlinearity [Formula: see text] of the equation satisfies [Formula: see text], [Formula: see text] ([Formula: see text]) and [Formula: see text] ([Formula: see text]) for any [Formula: see text] Here, [Formula: see text] is a positive constant. We establish a reconstruction formula of [Formula: see text] ([Formula: see text]) by the knowledge of the scattering operator for the equation. As an application, we also give an expression for higher-order Gâteaux differentials of the scattering operator at 0.