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

TL;DR

This paper establishes uniqueness and stability in recovering a time-dependent nonlinear coefficient in the Schrödinger equation from partial boundary measurements, advancing inverse problem theory for nonlinear PDEs.

Contribution

It proves the first local uniqueness and stability results for partial data inverse problems involving a nonlinear Schrödinger equation with a time-dependent coefficient.

Findings

Uniqueness of the nonlinear coefficient from partial boundary data.

Stability estimates based on unique continuation.

Applicability to geometric optics solutions.

Abstract

In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient $\beta(t, x)$ in the Schr\"odinger equation $(i\partial_t + \Delta + q(t, x))u + \beta u^2 = 0$, 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.