论文标题
立方Schrödinger系统的阈值解决方案
Threshold solutions for cubic Schrödinger systems
论文作者
论文摘要
我们考虑以下scödinger系统$$ \ begin {cases} \ displayStyle i \ partial_t u +Δu +（| u |^2 +β| v |^2） $（u_0，v_0）\在h^1（\ mathbb {r}^3）\ times h^1（\ mathbb {r}^3）$ in所谓的\ textit {mass-energy threshold}，即$ m（u_0，v_0）e（u_0，v_0）= m（ϕ，ψ）e（ϕ，ψ）$，其中$（ϕ，ψ）$是基态。对于合适的$β> 0 $的值范围，我们显示了该系统的特殊解决方案的存在，该解决方案沿一个时间方向收敛到驻波解决方案，然后朝相反的方向炸毁或散布。此外，我们对基础状态的通用解决方案进行了分类，显示出可能发生的长期行为的刚性结果。我们的结果并不依赖相应的基态的唯一性：实际上，即使在Weinstein功能具有多个优化器的情况下，主要结果也会成立。
We consider the following Scrödinger system $$\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.
