Inverse Born approximation for the generalized nonlinear Schrödinger operator in two dimensions

Abstract

This work deals with the inverse scattering problems for two nonlinear Schrödinger operators in two dimensions. One operator has a finite combinations of any power-like nonlinearity and another has a saturation-like nonlinearity. Both these operators can be met in nonlinear optics. The coefficients of these operators are assumed to be the real-valued functions from LlocP(R2) with certain behaviour at infinity. We prove Saito's formula for both operators which implies the uniqueness result and a representation formula for a sum of the unknown coefficients in the sense of tempered distributions. What is more, we prove that the leading order singularities of the sum can be obtained exactly by the inverse Born approximation method from general scattering data at arbitrarily large energies. Especially, we show for the functions in LlocP(R2), for certain values of p, that the approximation agrees with the true sum up to the functions from the Sobolev spaces H8(R2). In particular, for the sum being the characteristic function of a smooth bounded domain this domain is uniquely determined by this scattering data.