Gross-Pitaevskii vortex motion with critically-scaled inhomogeneities

TL;DR

This paper investigates vortex dynamics in an inhomogeneous Gross-Pitaevskii equation under a critical scaling regime, revealing interactions between vortices and background inhomogeneities, with implications for related parabolic and elliptic problems.

Contribution

It introduces a novel critical scaling regime for inhomogeneities in the Gross-Pitaevskii equation and analyzes vortex interactions within this regime.

Findings

Vortices interact with background inhomogeneities and each other.

Results extend to associated parabolic and elliptic problems.

Critical scaling regime identified for inhomogeneity effects.

Abstract

We study the dynamics of vortices in an inhomogeneous Gross-Pitaevskii equation $i \partial_t u = \Delta u + {1\over \varepsilon^2} (p_\varepsilon^2(x) - |u|^2)$. For a unique scaling regime $|p_\varepsilon(x) - 1 | = O(|\log \varepsilon|^{-1})$, it is shown that vortices can interact both with the background perturbation and with each other. Results for associated parabolic and elliptic problems are discussed.