Inverse source scattering problem for a nonlinear Schr\"odinger equation

Abstract

We study an inverse source scattering problem for the Schr\"odinger equation with a quadratic nonlinearity. In general, uniqueness of inverse source problems can not be guaranteed at a fixed energy. Therefore, additional information is required for the source in order to obtain a unique solution. By adding reference point sources, we show that a general source function could be uniquely determined from boundary measurements at a fixed wavenumber. This method does not apply to inverse source problems of linear equations since it uses the nonlinearity as a tool. The proof utilizes the method of linearization to reduce the nonlinear inverse source scattering problem to the inverse potential scattering problem of the linear Schr\"odinger equation.