Global Bifurcation In Four-Component Bose-Einstein Condensates In Space

TL;DR

This paper investigates the complex bifurcation structure of multi-component Bose-Einstein condensates in a spherical domain, revealing multiple global bifurcations and branches using advanced equivariant degree theory.

Contribution

It introduces a novel application of $G$-equivariant gradient degree to analyze high-dimensional bifurcations in coupled BEC systems, advancing bifurcation theory methods.

Findings

Multiple global bifurcations at critical values with high-dimensional kernels.

Existence of at least two global branches for $k=0$.

Existence of at least six global branches for $k=1$.

Abstract

We analyze a system of coupled Bose-Einstein condensates in the domain of a unitary ball in $\mathbb{R}^3$. The coupling is due to atom-to-atom interactions that occur between different gas components. The multi-component Bose-Einstein condensate is described by a system of Gross-Pitaevskii equations, which has an explicit trivial branch of constant solutions bifurcating from the zero-solution. Our main theorem establishes that this trivial branch undergoes multiple global bifurcations at any critical values with kernels of dimensions at least $3(2k+1)$, for $k \in \mathbb{N}^+$. Handling these high dimension kernels poses a challenge from the perspective of bifurcation theory. Our methodology, which relies on the $G$-equivariant gradient degree, effectively manages these complexities and establishes the existence of at least two global branch in the particular case of $k = 0$ and at least six branches in the case of $k = 1$.