Partial Data Inverse Problems for the Nonlinear Schr\"odinger Equation
Abstract
In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient β(t, x) in the Schr\"odinger equation (i∂t + + q(t, x))u + β u2 = 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.
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