Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (I): The De Giorgi conjecture for the local problem in ${\mathbb{R}}^3$

Abstract

In this series of papers, we investigate coupled systems arising in the study of two-component Bose--Einstein condensates, and we establish classification results for solutions of De Giorgi conjecture type. In the first paper of the series, we focus on the local problem of the form $\hat{I}”u=u(u^2+v^2-1)+v(\hat{I}\pm uv-\ddot{I}‰)$, $\hat{I}”v=v(u^2+v^2-1)+u(\hat{I}\pm uv-\ddot{I}‰)$, and prove that positive global solutions in ${\mathbb{R}}^3$ satisfying $\partial u/\partial x_3>0>\partial v/\partial x_3$ must be one-dimensional.