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We study an inverse problem related to the dynamical Schr{\"o}dinger equationin a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linearSchr\"odinger equation possesses a trivial solution, we linearize the equationaround the trivial solution. Demonstrating the well-posedness of the directproblem under appropriate conditions on initial and boundary data, it isobserved that the solution admits $\eps$-expansion. By taking into account thefact that the terms $\Oh(|\nabla u(t,x)|^3)$ are negligible in this context, weshall reconstruct the time-dependent coefficients such as electric potentialand vector-valued function associated with quadratic nonlinearity from theknowledge of input-output map using the geometric optics solution and Fourierinversion.

Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{\"o}dinger equation