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

TL;DR

This paper studies the existence and concentration behavior of normalized solutions for a two-component Bose-Einstein condensate system in two dimensions, revealing solutions that concentrate on geometric subsets like circles.

Contribution

It introduces a novel finite-dimensional reduction method combined with Pohozaev identities to establish high-dimensional concentration solutions in BEC systems.

Findings

Existence of synchronized solutions concentrating on high-dimensional subsets.

Construction of radial solutions that concentrate on circles under specific parameter limits.

Fills a gap in understanding high-dimensional normalized solutions for the system.

Abstract

We investigate the normalized solutions of the following two-component Bose-Einstein condensates (BEC) system \begin{equation}\left\{ \begin{split} -\Delta u + (\lambda+P(x))u &= \alpha u^3 +\beta uv^2, && \text{in } \mathbb{R}^2,\\-\Delta v + (\lambda+Q(x))v &= \gamma v^3 +\beta u^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 $\alpha>0$, $\gamma > 0$ and $\ \beta \in (-\sqrt{\alpha\gamma},0)\cup(0,\min \{\alpha,\gamma\})\cup \left(\max \{\alpha,\gamma\} , + \infty\right)$, 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{\alpha + \gamma - 2\beta}{\alpha\gamma - \beta^2}$ tends to zero. Our results fill the blank in the system for high-dimensional concentrated normalized solutions.