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In this paper we prove the uniqueness and stability in determining atime-dependent nonlinear coefficient $\beta(t, x)$ in the Schr\"odingerequation $(i\partial_t + \Delta + q(t, x))u + \beta u^2 = 0$, from the boundaryDirichlet-to-Neumann (DN) map. In particular, we are interested in the partialdata problem, in which the DN-map is measured on a proper subset of theboundary. We show two results: a local uniqueness of the coefficient at thepoints where certain type of geometric optics (GO) solutions can reach; and astability estimate based on the unique continuation property for the linearequation.

Partial Data Inverse Problems for the Nonlinear Schr\"odinger Equation