Inverse Scattering for the Nonlinear Schrödinger Equation with Cubic Convolution Nonlinearity

Abstract

In this paper it will be shown that a potential $V(x)$ and a constant $\lambda$ are uniquely determined from the scattering operator $S$ associated with the nonlinear Schrödinger equation \[ i\frac{\partial u}{\partial t}+(-\Delta+V)u+\lambda(|x|^{-\sigma}*|u|^{2})u=0 , \] and the corresponding unperturbed equation \[ i\frac{\partial u}{\partial t}-\Delta u=0 . \]