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

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 $\mathrm{\Delta }u=u(u^2+v^2-1)+v(\alpha uv-\omega )$, $\mathrm{\Delta }v=v(u^2+v^2-1)+u(\alpha uv-\omega )$, and prove that positive global solutions in ${\mathbb{R}}^3$ satisfying $\mathrm{\partial }u\mathrm{/}\mathrm{\partial }{x}_3>0>\mathrm{\partial }v\mathrm{/}\mathrm{\partial }{x}_3$ must be one-dimensional.