Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Leyun Wu, Chilin Zhang

TL;DR

This work extends De Giorgi-type symmetry classification to a coupled two-component Bose-Einstein condensate model with Rabi coupling, establishing that, in \mathbb{R}^3, any positive solution with monotone behavior in the third coordinate is one-dimensional. The authors prove a Liouville-type result via a Schrödinger-type linearization and a negative-definite ratio energy, augmented by a growth-bound analysis of the Ginzburg-Landau energy and its limiting profiles. In the special α=2 regime, the system decouples to an Allen-Cahn-type equation, recovering 1D symmetry in low dimensions and highlighting high-dimensional obstructions. These results provide a rigorous baseline for understanding phase-transition interfaces in multi-component condensates and inform energy-method frameworks for coupled elliptic systems.

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 first paper of the series, we focus on the local problem of the form $Δu = u(u^2+v^2-1) + v(αuv - ω)$, $Δv = v(u^2+v^2-1) + u(αuv - ω)$, and prove that positive global solutions in $\mathbb{R}^3$ satisfying $\partial u/\partial x_3 > 0 > \partial v/\partial x_3$ must be one-dimensional.

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

TL;DR

This work extends De Giorgi-type symmetry classification to a coupled two-component Bose-Einstein condensate model with Rabi coupling, establishing that, in \mathbb{R}^3, any positive solution with monotone behavior in the third coordinate is one-dimensional. The authors prove a Liouville-type result via a Schrödinger-type linearization and a negative-definite ratio energy, augmented by a growth-bound analysis of the Ginzburg-Landau energy and its limiting profiles. In the special α=2 regime, the system decouples to an Allen-Cahn-type equation, recovering 1D symmetry in low dimensions and highlighting high-dimensional obstructions. These results provide a rigorous baseline for understanding phase-transition interfaces in multi-component condensates and inform energy-method frameworks for coupled elliptic systems.

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 first paper of the series, we focus on the local problem of the form , , and prove that positive global solutions in satisfying must be one-dimensional.

Paper Structure

This paper contains 13 sections, 11 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The system of Bose-Einstein condensates
  3. Classical theory in a special case Lg
  4. Gibbons' conjecture: similarities and differences
  5. Other comments
  6. Main results and key ideas
  7. The linearized equation and the Liouville theorem
  8. The Schrödinger equation
  9. The Caccioppoli inequality and the Liouville theorem
  10. Growth rate of the Ginzburg-Landau energy
  11. Evolution of the energy
  12. Limiting profile
  13. Proof of Lemma \ref{['lem. energy growth']} and Theorem \ref{['thm. main theorem']}

Key Result

Corollary 2.1

Let $\alpha=2$ and $0<\omega<1$. Then all global positive solutions to eq. main satisfy $u(x)+v(x)\equiv\sqrt{1+\omega}$. Moreover, let $0\leq a<b$ be constants satisfying $a^{2}+b^{2}=1$ and $ab=\frac{\omega}{2}$. Assume that $\frac{\partial u}{\partial x_{n}}>0>\frac{\partial v}{\partial x_{n}}$, then we have the following results:

Figures (1)

  • Figure 2.1: Steady states $(a, b), (b, a)$ and $\Bigl(\sqrt{\tfrac{1+\omega}{2+\alpha}},\,\sqrt{\tfrac{1+\omega}{2+\alpha}}\Bigr)$.

Theorems & Definitions (27)

  • Corollary 2.1
  • Remark 2.1
  • Theorem 2.1
  • proof : Proof of Corollary \ref{['cor. decouple corollary']}
  • Lemma 3.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.2
  • proof
  • ...and 17 more