The ground state of two coupled Gross-Pitaevskii equations in the Thomas-Fermi limit

TL;DR

This paper proves the existence and uniqueness of solutions for two coupled Gross-Pitaevskii equations in the Thomas-Fermi limit and provides an asymptotic expansion of these solutions as the semi-classical parameter approaches zero.

Contribution

It introduces a rigorous proof of solution existence, uniqueness, and detailed asymptotic expansion in the context of coupled Gross-Pitaevskii equations in the Thomas-Fermi limit.

Findings

Existence and uniqueness of positive solutions established.

Asymptotic expansion of solutions in powers of  as   0.

Detailed characterization of solutions in the semi-classical limit.

Abstract

We prove existence and uniqueness of a positive solution to a system of two coupled Gross-Pitaevskii equations. We give a full asymptotic expansion of this solution into powers of the semi classical parameter $\varepsilon$ in the Thomas--Fermi limit $\varepsilon\to 0$.