Partial data inverse problems for the nonlinear magnetic Schr\"odinger equation

TL;DR

This paper investigates the inverse problem for nonlinear magnetic Schrödinger equations, demonstrating that boundary measurements can uniquely determine various coefficients, including electric, magnetic, and nonlinear terms, even with partial boundary data.

Contribution

It establishes uniqueness results for recovering time-dependent coefficients in nonlinear magnetic Schrödinger equations from partial boundary data.

Findings

Boundary data determines all coefficients in the nonlinear case.

Unique recovery of coefficients in the linear magnetic Schrödinger equation.

Analysis of both forward and inverse problems for time-dependent equations.

Abstract

In this paper, we study the partial data inverse problem for nonlinear magnetic Schr\"odinger equations. We show that the knowledge of the Dirichlet-to-Neumann map, measured on an arbitrary part of the boundary, determines the time-dependent linear coefficients, electric and magnetic potentials, and nonlinear coefficients, provided that the divergence of the magnetic potential is given. Additionally, we also investigate both the forward and inverse problems for the linear magnetic Schr\"odinger equation with a time-dependent leading term. In particular, all coefficients are uniquely recovered from boundary data.