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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 $\mathbb{R}^{3}$

Leyun Wu, Chilin Zhang

TL;DR

This work extends De Giorgi–type rigidity to a nonlocal, two-component Gross–Pitaevskii system modeling Rabi-coupled Bose–Einstein condensates with fractional Laplacian interactions. The authors adapt extension techniques and energy-sliding methods to a coupled nonlinear system, deriving a Liouville-type theorem: any globally bounded, monotone positive solution is one-dimensional in low dimensions, with explicit asymptotics connecting homogeneous equilibria. The analysis hinges on a linearized Schrödinger framework, trace and energy estimates, and a limiting-profile decomposition that yields a complete classification of possible asymptotic states. These results deepen the understanding of phase transitions and interface structures in fractional multi-component condensates and connect nonlocal PDE rigidity to fractional Gross–Pitaevskii models.

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} (-Δ)^{s}u+u(u^{2}+v^{2}-1)+v(αuv-ω)=0, (-Δ)^{s}v+v(u^{2}+v^{2}-1)+u(αuv-ω)=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.

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 $\mathbb{R}^{3}$

TL;DR

This work extends De Giorgi–type rigidity to a nonlocal, two-component Gross–Pitaevskii system modeling Rabi-coupled Bose–Einstein condensates with fractional Laplacian interactions. The authors adapt extension techniques and energy-sliding methods to a coupled nonlinear system, deriving a Liouville-type theorem: any globally bounded, monotone positive solution is one-dimensional in low dimensions, with explicit asymptotics connecting homogeneous equilibria. The analysis hinges on a linearized Schrödinger framework, trace and energy estimates, and a limiting-profile decomposition that yields a complete classification of possible asymptotic states. These results deepen the understanding of phase transitions and interface structures in fractional multi-component condensates and connect nonlocal PDE rigidity to fractional Gross–Pitaevskii models.

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} (-Δ)^{s}u+u(u^{2}+v^{2}-1)+v(αuv-ω)=0, (-Δ)^{s}v+v(u^{2}+v^{2}-1)+u(αuv-ω)=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 , any positive entire solution in satisfying the monotonicity condition must be one-dimensional. Moreover, when , the same conclusion holds for monotone solutions in . 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.

Paper Structure

This paper contains 16 sections, 15 theorems, 175 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Allen-Cahn equation and minimal surfaces
  3. The fractional De Giorgi conjecture
  4. The Gross-Pitaevskii system and our contribution
  5. Notations
  6. Main results
  7. Preliminaries
  8. Basic estimates
  9. Range of the solution
  10. Linearization and the Schrödinger Operator
  11. The extension method
  12. The Schrödinger equation
  13. The Caccioppoli inequality
  14. Evolution of the Ginzburg-Landau energy
  15. The limiting profile
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(\alpha,\omega)$ be a pair of parameters satisfying $0<\omega<\frac{\alpha}{2}$. Let $(n,s)$ be a pair such that either $s\in[\frac{1}{2},1)$ with $n\geq3$, or $s\in(0,\frac{1}{2})$ with $n\geq2$. Assume that $(u,v)$ is a pair of positive and globally bounded functions in $C^{1,1}_{\rm loc}(\ma Then the pair $(u,v)$ must be one-dimensional. More precisely, there exists a unit vector $\vec{\nu

Figures (1)

  • Figure 1.1: Steady states $(a, b), (b, a)$ and $\Bigl(\sqrt{\tfrac{1+\omega}{2+\alpha}},\,\sqrt{\tfrac{1+\omega}{2+\alpha}}\Bigr)$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2: Trace inequality
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • ...and 23 more