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

We study an inverse problem related to the dynamical Schr{"o}dinger equation in a bounded domain of ${\text{Rb}}^n,n\ge 2$. Since the concerned non-linear Schr"odinger equation possesses a trivial solution, we linearize the equation around the trivial solution. Demonstrating the well-posedness of the direct problem under appropriate conditions on initial and boundary data, it is observed that the solution admits $\text{eps}$-expansion. By taking into account the fact that the terms $\text{Oh}(\mathrm{\mid }\mathrm{\nabla }u(t,x){\mathrm{\mid }}^3)$ are negligible in this context, we shall reconstruct the time-dependent coefficients such as electric potential and vector-valued function associated with quadratic nonlinearity from the knowledge of input-output map using the geometric optics solution and Fourier inversion.