Reconstruction of time-dependent coefficients in a semilinear dynamical Schr{ö}dinger equation

Abstract

In the present manuscript, we study an inverse problem related to a semilinear dynamical Schr{ö}dinger equation with lower order terms, in a bounded domain of $\Rb^{1+n},n\geq 2$. Our focus is on determination of the time-dependent coefficients appearing in the aforementioned equation, from the boundary measurements of the solutions. More precisely, we establish the {pointwise reconstruction} formulae for determining the time-dependent coefficients of linear and nonlinear terms from the knowledge of Dirichlet-to-Neumann map. Since the concerned non-linear Schrödinger equation possesses a trivial solution, we linearize the equation around the trivial solution and use the asymptotic solutions (\textit{with concentrated amplitudes}) of the linearized problem for reconstructing the aforementioned coefficients. To be more specific, we use first-order linearization to reconstruct vector and scalar potentials associated with the coefficients of linear terms and the higher-order linearization technique is used to reconstruct coefficients of nonlinearity. The nonlinear equation considered in this manuscript can be seen as a generalization of the Gross-Pitaevskii equation (GPE), which is employed to describe the dynamics of dilute Bose-Einstein condensates (BEC).