Ground States of Two-Component Attractive Bose-Einstein Condensates II: Semi-trivial Limit Behavior

Abstract

As a continuation of [14], we study new pattern formations of ground states $(u_1,u_2)$ for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in $R^2$, where the intraspecies interaction $(-a,-b)$ and the interspecies interaction $-\beta$ are both attractive, $i.e,$ $a$, $b$ and $\beta$ are all positive. If $0<b<a^{\ast }:=\Vert w{\Vert }_2^2$ and $0<\beta <a^{\ast }$ are fixed, where $w$ is the unique positive solution of $\Delta w-w+w^3=0$ in $R^2$, the semi-trivial behavior of $(u_1,u_2)$ as $a\nearrow a^{\ast }$ is proved in the sense that $u_1$ concentrates at a unique point and while $u_2\equiv 0$ in $R^2$. However, if $0<b<a^{\ast }$ and $a^{\ast }\le \beta <{\beta }^{\ast }=a^{\ast }+\sqrt{(a^{\ast }-a)(a^{\ast }-b)}$, the refined spike profile and the uniqueness of $(u_1,u_2)$ as $a\nearrow a^{\ast }$ are analyzed, where $(u_1,u_2)$ must be unique, $u_1$ concentrates at a unique point, and meanwhile $u_2$ can either blow up or vanish, depending on how $\beta$ approaches to $a^{\ast }$.

45 pages, two figures; Accepted by Trans. Amer. Math. Soc

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