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