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

TL;DR

This paper addresses the inverse problem of reconstructing time-dependent coefficients in a quasilinear Schrödinger equation by linearizing around the trivial solution and using geometric optics solutions and Fourier inversion techniques.

Contribution

It introduces a method to reconstruct coefficients in a nonlinear Schrödinger equation from input-output data by linearization and approximation techniques.

Findings

Successfully reconstructs time-dependent coefficients from boundary measurements.

Demonstrates well-posedness of the linearized inverse problem.

Employs geometric optics solutions and Fourier inversion for reconstruction.

Abstract

We study an inverse problem related to the dynamical Schr{\"o}dinger equation in a bounded domain of $\Rb^n,n\geq 2$. 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 $\eps$-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.