Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions

TL;DR

This paper investigates the existence, uniqueness, and blow-up behavior of solutions to a coupled Gross-Pitaevskii system modeling two-component Bose-Einstein condensates with attractive interactions, revealing critical blow-up phenomena and rates.

Contribution

It provides new results on the existence and non-existence of solutions, and characterizes the blow-up behavior and rates at critical interaction strengths.

Findings

Solutions blow up at the same point as interaction strength approaches critical value

Existence and uniqueness depend on trapping potentials

Optimal blow-up rate is established

Abstract

The paper is concerned with a system of two coupled time-independent Gross-Pitaevskii equations in $\mathbb{R}^2$, which is used to model two-component Bose-Einstein condensates with both attractive intraspecies and attractive interspecies interactions. This system is essentially an eigenvalue problem of a stationary nonlinear Schr\"odinger system in $\mathbb{R}^2$, solutions of the problem are obtained by seeking minimizers of the associated variational functional with constrained mass (i.e. $L^2-$norm constaints). Under certain type of trapping potentials $V_i(x)$ ($i=1,2$), the existence, non-existence and uniqueness of this kind of solutions are studied. Moreover, by establishing some delicate energy estimates, we show that each component of the solutions blows up at the same point (i.e., one of the global minima of $V_i(x)$) when the total interaction strength of intraspecies and interspecies goes to a critical value. An optimal blowing up rate for the solutions of the system is also given.