Recovery of the nonlinearity from the modified scattering map

We consider a class of one-dimensional nonlinear Schrödinger equations of the form $$(i{\mathrm{\partial }}_t+\mathrm{\Delta })u=[1+a]\mathrm{\mid }u{\mathrm{\mid }}^{2}u\mathrm{.}$$ For suitable localized functions $a$, such equations admit a small-data modified scattering theory, which incorporates the standard logarithmic phase correction. In this work, we prove that the small-data modified scattering behavior uniquely determines the inhomogeneity $a$.