Increasing stability of a linearized inverse boundary value problem for a nonlinear Schr\"odinger equation on transversally anisotropic manifolds

TL;DR

This paper demonstrates that the stability of recovering a nonlinear potential in a Schrödinger equation improves as the wavenumber increases, using complex geometric optics solutions on transversally anisotropic manifolds.

Contribution

It introduces a method to achieve increasing stability in inverse boundary value problems for nonlinear Schrödinger equations at high wavenumbers.

Findings

Stability improves with larger wavenumber values.

Calibration of CGO solutions is key to stability enhancement.

Results apply to transversally anisotropic manifolds.

Abstract

We consider the problem of recovering a nonlinear potential function in a nonlinear Schr\"odinger equation on transversally anisotropic manifolds from the linearized Dirichlet-to-Neumann map at a large wavenumber. By calibrating the complex geometric optics (CGO) solutions according to the wavenumber, we prove the increasing stability of recovering the coefficient of a cubic term as the wavenumber becomes large.