Coefficient Determination for Non-Linear Schrödinger Equations on manifolds

Abstract

We consider an inverse problem of recovering the unknown coefficients $\beta (t,x)$ and $V(t,x)$ appearing in a time-dependent nonlinear Schrödinger 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 Schrödinger equation and for the Gross-Pitaevskii equation, with a cubic non-linear term $\beta |u{|}^{2}u$, that is encountered in quantum physics.

31 pages