Gross-Pitaevskii vortex motion with critically-scaled inhomogeneities

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.