Existence of vortex-free solutions in the Painlevé boundary layer of a Bose–Einstein condensate

Abstract

In recent experiments on Bose–Einstein condensates (BEC), it has been observed that when a laser beam, modelled by a cylinder C along the z direction, is translated in the x direction along the condensate, there is no dissipation at small velocity. This is related to the existence of vortex-free solutions of Δ u−2i c∂ x u+( z−| u| 2) u=0, where u is a complex-valued function, (x,y,z)∈ R 3⧹ C , and c is the velocity of the laser. One particularity of BEC is its inhomogeneity, so that, away from the cylinder, the wave function u varies in z like the solution p( z) of the Painlevé equation p″+( z− p 2) p=0 and in particular vanishes near the boundary of the condensate. For small c, we prove the existence of vortex free solutions. Our proof relies on the uniqueness of solutions at c=0, that we derive using a special decoupling of the energy and a Pohozaev identity. Another key tool is to estimate the momentum in terms of the energy.