Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose-Einstein condensation

We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: $$\{\begin{array}{ll}-\mathrm{\Delta }u=-u{v}^2& \text{in }{\mathbb{R}}^N\\ -\mathrm{\Delta }v=-u^{2}v& \text{in }{\mathbb{R}}^N,\end{array}$$ for every dimension $N\ge 2$. In particular, we prove a Gibbons-type conjecture proposed by H. Berestycki, T. C. Lin, J. Wei and C. Zhao.