Inverse initial data for nonlinear Schr\"odinger equation via Carleman estimates and the contraction principle

TL;DR

This paper introduces a novel method combining Legendre polynomial expansion and Carleman estimates to reconstruct initial data for nonlinear Schrödinger equations from boundary measurements, with proven stability and numerical validation.

Contribution

It develops a new approach using a Carleman-based contraction principle and spectral reduction for inverse problems in nonlinear Schrödinger equations, including stability analysis and numerical experiments.

Findings

The method accurately reconstructs initial wave fields in two dimensions.

It remains stable under noisy boundary data.

Numerical experiments confirm effectiveness across various geometries.

Abstract

We study an inverse initial-data problem for a nonlinear Schr\"odinger equation in which the initial wave field is reconstructed from lateral measurements. Our approach combines a Legendre-polynomial-exponential-time dimensional reduction with a Carleman-based contraction principle. First, we expand the solution in a weighted Legendre basis in time and truncate the expansion to obtain a coupled nonlinear elliptic system for the spatial coefficients. Next, we solve this reduced system by constructing a contraction map on a suitable admissible set. This contraction map admits a unique fixed point, which is the limit of the corresponding Picard iteration. We also establish a stability estimate showing that this fixed point remains close to the exact reduced solution in the noisy-data case. Finally, we present numerical experiments in two space dimensions for several different geometries and nonlinear exponents. The numerical results show that the proposed method accurately reconstructs the main features of the initial wave field and remains stable even when the boundary data contain noise.