Partial Data Inverse Problems for the Nonlinear Schrödinger Equation

In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient $\beta (t,x)$ in the Schrödinger equation $(i{\mathrm{\partial }}_t+\mathrm{\Delta }+q(t,x))u+\beta u^2=0$, from the boundary Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial data problem, in which the DN-map is measured on a proper subset of the boundary. We show two results: a local uniqueness of the coefficient at the points where certain type of geometric optics (GO) solutions can reach; and a stability estimate based on the unique continuation property for the linear equation.