Threshold solutions for cubic Schr\"odinger systems

TL;DR

This paper studies the behavior of solutions to a coupled cubic Schrödinger system at the mass-energy threshold, identifying special solutions and classifying long-term dynamics without assuming ground state uniqueness.

Contribution

It introduces new solutions at the threshold and provides a classification of solution behaviors, even when multiple ground states exist.

Findings

Existence of special solutions converging to standing waves and exhibiting blow-up or scattering.

Rigidity results classifying possible long-time behaviors at the ground state.

Results hold without assuming uniqueness of ground states.

Abstract

We consider the following Scr\"odinger system $$\begin{cases}\displaystyle i\partial_t u + \Delta u +(|u|^2+\beta |v|^2) u= 0, \\ \displaystyle i\partial_t v + \Delta v +(|v|^2+\beta |u|^2) v = 0,\end{cases}$$ 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 %$\mathcal{ME}(u_0,v_0) = 1$. $M(u_0,v_0)E(u_0,v_0) = M(\phi,\psi)E(\phi,\psi)$, where $(\phi,\psi)$ 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.