Recovery of Schrödinger 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, $$\mathrm{i}{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u=a(x)\mathrm{\mid }u{\mathrm{\mid }}^{2}u\mathrm{.}$$ 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 $a(x)$.