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We consider an inverse problem of recovering the unknown coefficients$\beta(t,x)$ and $V(t,x)$ appearing in a time-dependent nonlinear Schr\"odingerequation $ (\mathrm{i} \partial_t +\Delta +V)u + \beta u^2=0$ in $(0,T) \timesM$, on Euclidean geometry as well as on Riemannian geometry. We considermeasurements 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 determinesthe time-dependent potential and the coefficient of the non-linearity, for theabove 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 quantumphysics.

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