Numerical reconstruction of Schr\"odinger equations with quadratic nonlinearities

TL;DR

This paper presents a numerical method to reconstruct potentials in 2D semilinear elliptic PDEs with quadratic nonlinearities using higher order linearization and Fourier analysis, achieving accurate results for various test cases.

Contribution

It introduces a novel numerical framework combining higher order linearization and Fourier data inversion for potential reconstruction in nonlinear PDEs.

Findings

Accurate reconstruction of smooth potentials.

Effective handling of discontinuous potentials.

Validated through numerical experiments.

Abstract

We introduce a numerical framework for reconstructing the potential in two dimensional semilinear elliptic PDEs with power type nonlinearities from the nonlinear Dirichlet to Neumann map. By applying higher order linearization method, we compute the Fourier data of the unknown potential and then invert it to recover $q$. Numerical experiments show accurate reconstructions for both smooth and discontinuous test cases.