Gross-Pitaevskii vortex motion with critically-scaled inhomogeneities

Abstract

We study the dynamics of vortices in an inhomogeneous Gross-Pitaevskii equation $i{\partial }_{t}u=\Delta u+\frac{1}{{\epsilon }^2}(p_{\epsilon }^2(x)-|u{|}^2)$. For a unique scaling regime $|p_{\epsilon }(x)-1|=O(|\log \epsilon {|}^{-1})$, 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.

35 pages, 1 figure, typos corrected and reference added. Dramatically improved exposition thanks to very thorough Referee Report