Increasing stability in the inverse scattering problem for a nonlinear Schrödinger equation with multiple frequencies

This paper is concerned with the inverse scattering problem of determining the unknown coefficients for a nonlinear two-dimensional Schrödinger equation. We establish for the first time the increasing stability of the inverse scattering problem from the multi-frequency far-field pattern for nonlinear equations. To achieve this goal, we prove the existence of a holomorphic region and an upper bound for the solution with respect to the complex wavenumber, which also leads to the well-posedness of the direct scattering problem. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the unknown coefficients, where the latter decreases as the upper bound of the frequency increases.