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

Abstract

We study an inverse problem related to the dynamical Schrödinger equation in a bounded domain of R n , n ≥ 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 ε-expansion. By taking into account the fact that the terms O(|∇u(t, x)| 3 ) are negligible in this context, we shall reconstruct the time-dependent coefficients such as electric potential and vectorvalued function associated with quadratic nonlinearity from the knowledge of input-output map using the geometric optics solution and Fourier inversion. H r,s (Σ) = L 2 (0, T ; H r (∂Ω)) ∩ H s (0, T ; L 2 (∂Ω))2020 Mathematics Subject Classification. 35R30.