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Circle-like concentrated solutions for two-component Bose-Einstein condensates

Qidong Guo, Qiaoqiao Hua, Chongyang Tian

Abstract

We investigate the normalized solutions of the following two-component Bose-Einstein condensates (BEC) system \begin{equation}\left\{ \begin{split} -Δu + (λ+P(x))u &= αu^3 +βuv^2, && \text{in } \mathbb{R}^2,\\-Δv + (λ+Q(x))v &= γv^3 +βu^2 v, && \text{in } \mathbb{R}^2, \end{split} \right.\end{equation} with $L^2$-constraint $$\int_{\mathbb{R}^2}(u^2+v^2)\,dx = 1.$$ For any $α>0$, $γ> 0$ and $\ β\in (-\sqrt{αγ},0)\cup(0,\min \{α,γ\})\cup \left(\max \{α,γ\} , + \infty\right)$, we establish the existence of synchronized solutions concentrating on high-dimensional subsets of $\mathbb{R}^2$ by employing a finite-dimensional reduction method combined with some local Pohozaev identities. More precisely, we construct vector radial solutions that concentrate on circles when $ \frac{α+ γ- 2β}{αγ- β^2}$ tends to zero. Our results fill the blank in the system for high-dimensional concentrated normalized solutions.

Circle-like concentrated solutions for two-component Bose-Einstein condensates

Abstract

We investigate the normalized solutions of the following two-component Bose-Einstein condensates (BEC) system \begin{equation}\left\{ \begin{split} -Δu + (λ+P(x))u &= αu^3 +βuv^2, && \text{in } \mathbb{R}^2,\\-Δv + (λ+Q(x))v &= γv^3 +βu^2 v, && \text{in } \mathbb{R}^2, \end{split} \right.\end{equation} with -constraint For any , and , we establish the existence of synchronized solutions concentrating on high-dimensional subsets of by employing a finite-dimensional reduction method combined with some local Pohozaev identities. More precisely, we construct vector radial solutions that concentrate on circles when tends to zero. Our results fill the blank in the system for high-dimensional concentrated normalized solutions.
Paper Structure (4 sections, 19 theorems, 185 equations)

This paper contains 4 sections, 19 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Necessity
  3. Existence
  4. Normalized solutions

Key Result

Theorem 1.1

Let $\alpha>0$, $\gamma>0$, $\beta \in (-\sqrt{\alpha\gamma},0)\cup(0,\min \{\alpha,\gamma\})\cup \left(\max \{\alpha,\gamma\} , + \infty\right)$, $P(x)=P(|x|)$ and $Q(x)=Q(|x|)$ are $C^1$ functions with $P,Q,P^\prime,Q^\prime$ being bounded. Suppose $(u_\lambda,v_\lambda)$ is a concentrated radial and Then it must holds that Moreover, $r_\lambda$ satisfies

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 1.3: cf. Proposition 2.3 in SPZW
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 29 more