Coefficient Determination for Non-Linear Schr\"odinger Equations on manifolds

TL;DR

This paper investigates an inverse problem for a nonlinear Schrödinger equation on manifolds, demonstrating that boundary measurements uniquely determine unknown coefficients in the equation, including potential and nonlinearity parameters.

Contribution

It establishes unique identifiability of time-dependent coefficients in nonlinear Schrödinger equations on Euclidean and Riemannian manifolds using boundary measurements.

Findings

Unique determination of potential V(t,x) from boundary data.

Identification of nonlinearity coefficient β(t,x) in the equation.

Applicability to Gross-Pitaevskii equation with cubic non-linearity.

Abstract

We consider an inverse problem of recovering the unknown coefficients $\beta(t,x)$ and $V(t,x)$ appearing in a time-dependent nonlinear Schr\"odinger equation $ (\mathrm{i} \partial_t +\Delta +V)u + \beta u^2=0$ in $(0,T) \times M$, on Euclidean geometry as well as on Riemannian geometry. We consider measurements in $\Omega \subset M$ that is a neighborhood of the boundary of $M$ and the source-to-solution map $ L_{\beta, V}$ that maps a source $f$ supported in $ \Omega\times (0,T) $ to the restriction of the solution $u$ in $ \Omega\times (0,T) $. We show that the map $L_{\beta, V}$ uniquely determines the time-dependent potential and the coefficient of the non-linearity, for the above non-linear Schr\"odinger equation and for the Gross-Pitaevskii equation, with a cubic non-linear term $\beta |u|^2 \, u$, that is encountered in quantum physics.