Recovery of Schr\"odinger nonlinearity from the large-data scattering behavior

In this article, we study scattering and inverse scattering problems for 3D nonlinear Schrödinger equations of the form, $$ \mathrm{i}\partial_t u + \Delta u = a(x)|u|^2u.$$ For $a(x)$ in a suitable class, such equations are global well-posed and admit a large-data scattering theory.

In this work, we prove that the large-data scattering behavior uniquely determines the inhomogeneity $a(x)$.