Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (II): The De Giorgi conjecture for the nonlocal problem in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$

Abstract

In this series of papers, we investigate coupled systems arising in the study of two-component Bose-Einstein condensates, and we establish classification results for solutions of De Giorgi conjecture type. In the present (second) paper of the series, we focus on the nonlocal problem of the form \begin{equation*} \left\{\begin{aligned} (-\Delta)^{s}u+u(u^{2}+v^{2}-1)+v(\alpha uv-\omega)=0, (-\Delta)^{s}v+v(u^{2}+v^{2}-1)+u(\alpha uv-\omega)=0, \end{aligned} \right. \end{equation*} which models the stationary states of Rabi-coupled condensates with inter- and intra-species interactions. We prove that for $\frac{1}{2}\le s<1$, any positive entire solution $(u,v)$ in $\mathbb{R}^3$ satisfying the monotonicity condition $\partial_{x_3}u>0>\partial_{x_3}v$ must be one-dimensional. Moreover, when $0<s<\frac{1}{2}$, the same conclusion holds for monotone solutions in $\mathbb{R}^2$. Our work generalizes classical De Giorgi-type theorems to a new class of nonlocal coupled systems and, to the best of our knowledge, presents the first Liouville-type classification of monotone solutions for Rabi-coupled fractional Bose-Einstein condensates, with particular emphasis on fractional Gross-Pitaevskii models.