Threshold solutions for cubic Schrödinger systems

Abstract

We consider the following Scrödinger system $$\begin{cases}\displaystyle i\partial_t u + Δu +(|u|^2+β|v|^2) u= 0, \\ \displaystyle i\partial_t v + Δv +(|v|^2+β|u|^2) v = 0,\end{cases}$$ 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 %$\mathcal{ME}(u_0,v_0) = 1$. $M(u_0,v_0)E(u_0,v_0) = M(ϕ,ψ)E(ϕ,ψ)$, where $(ϕ,ψ)$ is a ground state. For a suitable range of values of $β>0$, 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.