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).
From: Parveen Kumar [view email]
[v1]
Mon, 15 Jun 2026 17:50:14 UTC (21 KB)
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