Interface layer of a two-component Bose-Einstein condensate

TL;DR

This paper analyzes the interface behavior of a two-component Bose-Einstein condensate under strong segregation, constructing and proving properties of heteroclinic solutions that describe the wave functions near the interface.

Contribution

It develops an asymptotic analysis of coupled ODEs for Bose-Einstein condensates, constructing unique heteroclinic solutions with spectral gap properties.

Findings

Constructed an approximate heteroclinic solution.

Proved linear nondegeneracy and spectral gap of solutions.

Established uniqueness of the energy minimizer.

Abstract

This paper deals with the study of the behaviour of the wave functions of a two-component Bose-Einstein condensate near the interface, in the case of strong segregation. This yields a system of two coupled ODE's for which we want to have estimates on the asymptotic behaviour, as the strength of the coupling tends to infinity. As in phase separation models, the leading order profile is a hyperbolic tangent. We construct an approximate solution and use the properties of the associated linearized operator to perturb it into a genuine solution for which we have an asymptotic expansion. We prove that the constructed heteroclinic solutions are linearly nondegenerate, in the natural sense, and that there is a spectral gap, independent of the large interaction parameter, between the zero eigenvalue (due to translations) at the bottom of the spectrum and the rest of the spectrum. Moreover, we prove a uniqueness result which implies that, in fact, the constructed heteroclinic is the unique minimizer (modulo translations) of the associated energy, for which we provide an expansion.