Threshold solutions for cubic Schrödinger systems

Abstract

We consider the following Scrödinger system $$\{\begin{array}{l}i{\partial }_{t}u+\Delta u+(|u{|}^2+\beta |v{|}^2)u=0,\\ i{\partial }_{t}v+\Delta v+(|v{|}^2+\beta |u{|}^2)v=0,\end{array}$$ with initial data $(u_0,v_0)\in H^1({\mathbb{R}}^3)\times H^1({\mathbb{R}}^3)$ at the so-called \textit{mass-energy threshold}, i.e., such that %$\mathcal{M}\mathcal{E}(u_0,v_0)=1$. $M(u_0,v_0)E(u_0,v_0)=M(\varphi ,\psi )E(\varphi ,\psi )$, where $(\varphi ,\psi )$ is a ground state. For a suitable range of values of $\beta >0$, we show the existence of special solutions to this system, which converge to a standing wave solution in one time direction, and either blows up or scatters in the opposite direction. Moreover, we classify general solutions at the ground state, showing a rigidity result regarding the possible long-time behaviors that might occur. Our results do not rely on the uniqueness of the corresponding ground state: indeed, the main results hold even in the case where the Weinstein functional is known to have more than one optimizer.