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

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.