  The paper studies the inverse problem of reconstructuring the coefficient
$\beta(t,x)$ of the non-linear term and the potential $V(t,x)$ of a non-linear
Schr\"odinger equation in time-domain, $( i \frac{\partial}{\partial t} +
\Delta + V) u + \beta u^2 = f$ in $(0,T)\times M$, where $M\subset
\mathbb{R}^n$ is a convex and compact set with smooth boundary. We consider
measurements in a neighborhood $\Omega\subset M$ 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 a
second-order non-linear Schr\"odinger equation and for the Gross-Pitaevskii
equation, with a cubic non-linear term $\beta |u|^2u$, that is encountered in
quantum physics.
