Abstract

This work studies the inverse fixed energy scattering problem for the generalised nonlinear Schrödinger operators. We prove that in a three-dimensional case the unknown compactly supported generalised nonlinear potential (with some restriction for this potential) from \(L^{2}\) space can be uniquely determined by the scattering data with fixed positive energy (meaning that we have the knowledge of the scattering amplitude with fixed non-zero spectral parameter). The results are based on the new estimates for the Faddeev’s Green function in \(L^{∞}\). These results may have applications in nonlinear optics for the saturation model. In particular, the constant coefficients of this model can be uniquely reconstructed by the scattering data with fixed energy.