Recovery of the Nonlinearity From the Modified Scattering Map

We consider a class of one-dimensional nonlinear Schrodinger equations of the form$$ \begin{align*} & (i\partial_{t}+\Delta)u = [1+a]|u|<^>{2} u. \end{align*}$$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 $a$.