An inverse problem from condense matter physics

TL;DR

This paper addresses reconstructing anisotropic background potentials in quantum fluids by analyzing vortex dipole trajectories, proving uniqueness, deriving formulas, and providing numerical examples.

Contribution

It introduces a novel inverse problem approach for reconstructing background potentials from vortex dynamics in nonlinear Schrödinger equations.

Findings

Reconstruction of background potential is unique under certain conditions.

Derived an approximate formula for reconstructing the potential.

Numerical examples demonstrate the effectiveness of the method.

Abstract

We consider the problem of reconstructing the features of a weak anisotropic background potential by the trajectories of vortex dipoles in a nonlinear Gross-Pitaevskii equation. At leading order, the dynamics of vortex dipoles are given by a Hamiltonian system. If the background potential is sufficiently smooth and flat, the background can be reconstructed using ideas from the boundary and the lens rigidity problems. We prove that reconstructions are unique, derive an approximate reconstruction formula, and present numerical examples.