Stable Determination of Coefficients in Nonlinear Dynamical Schr\"odinger Equations by Carleman Estimates

TL;DR

This paper develops a method to stably determine coefficients in nonlinear dynamical Schrödinger equations using Carleman estimates, with results applicable to partial boundary measurements and under certain regularity conditions.

Contribution

It introduces a novel approach combining high-order linearization and Carleman estimates to achieve stable coefficient recovery in nonlinear Schrödinger equations.

Findings

Stable and unique determination of coefficients near the boundary.

Extension to arbitrary boundary measurement subsets under stronger assumptions.

Application of Carleman estimates to nonlinear inverse problems.

Abstract

We consider the inverse problem of recovering stationary coefficients in a class of dynamical Schr\"odinger equations with locally analytic nonlinear terms. Upon treating the well-posedness for small initial data and trivial boundary data, we proceed to establish stable and unique determination provided knowledge of the coefficients near the boundary and the measured Neumann data of the solution. We discuss both the case of measurement on a subset of the boundary large enough to satisfy a certain geometrical condition and, under stronger assumptions on the regularity and size of the coefficients, the case of measurement on arbitrary subsets of the boundary. Our argument relies on high-order linearization and Carleman estimates for the linear Schr\"odinger equation.