Periodic orbits of Gross–Pitaevskii in the disc with vortices following point vortex flow

Abstract

We prove the existence of non-constant time periodic vortex solutions to the Gross–Pitaevskii equations for small but fixed $$\varepsilon > 0.$$ The vortices of these solutions follow periodic orbits to the point vortex system of ordinary differential equations for all time. The construction uses two approaches—constrained minimization techniques adapted from Gelantalis and Sternberg (J Math Phys 53:083701, 2012) and topological minimax techniques adapted from Lin and Lin (Sel Math New Ser 3:99–113, 1997), applied to a formulation of the problem within a rotational ansatz.