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

We study an inverse problem related to the dynamical Schr{ö}dinger equation in a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linear Schrödinger 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 $\eps$-expansion. By taking into account the fact that the terms $\Oh(|\nabla u(t,x)|^3)$ 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.

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