Semiclassical analysis with new Galilean transformations for a Gross-Pitaevskii system with nonzero conditions at infinity

摘要

Recently, a rich variety of the micro-phenomena of the superfluid passing an obstacle has been observed in the binary mixture of rotating Bose-Einstein condensates. Among such phenomena, the interaction of dark-bright solitons is one of the most important issues. In this work, we investigate the semiclassical limit for a coupled system of Gross-Pitaevskii equations with rotating fields and trap potentials in a two-dimensional exterior domain, where the superfluid is non-vanishing at infinity. We establish a new Galilean-type transformation and follow the argument of the modulated energy functional (a Lyapunov-type functional) in Lee and Lin (J Math Phys 49:043517, 2008) and Lin and Zhang (Arch Ration Mech Anal 179:79-107, 2005) to control the propagation of mass densities and linear momenta of the solution via a compressible Euler equation with Coriolis force in a semiclassical regime. Moreover, the effect of the rotating field on the superfluid in the region far away from the obstacle is precisely described.