# Semi-classical analysis with new Galilean transformations for a Gross--Pitaevskii system with non-zero conditions at infinity

## TL;DR

This paper develops a semi-classical analysis for a coupled Gross--Pitaevskii system with non-zero conditions at infinity, introducing a new Galilean transformation to study superfluid behavior in rotating Bose--Einstein condensates.

## Contribution

It introduces a novel Galilean type transformation and applies a modulated energy functional approach to analyze superfluid dynamics in a semi-classical regime with non-zero boundary conditions.

## Key findings

- Controlled propagation of mass densities and momenta via a compressible Euler equation with Coriolis force.
- Precise description of the effect of rotating fields on superfluid far from obstacles.
- Established a new analytical framework for superfluid systems with non-zero conditions at infinity.

## Abstract

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 (BECs). Among such phenomena, the interaction of dark--bright solitons is one of the most important issues. In this work we investigate the semi-classical limit for a coupled system of Gross--Pitaevskii (GP) 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 \cite{ll08,lz05} to control the propagation of mass densities and linear momenta of the solution via a compressible Euler equation with Coriolis force in a semi-classical regime. Moreover, the effect of the rotating field on the superfluid in the region far away from the obstacle is precisely described.