Local and global well-posedness for a quadratic Schr\"odinger system on spheres and Zoll manifolds

Abstract

We consider the initial value problem (IVP) associated to a quadratic Schr\"odinger system \begin{equation*} \begin{cases} i \partial_{t} v \pm \Delta_{g} v - v = \epsilon_{1} u \bar{v}, & t \in \mathbb{R},\; x \in M, \\[2ex] i \sigma \partial_{t} u \pm \Delta_{g} u - \alpha u = \frac{\epsilon_{2}}{2} v^{2}, & \sigma > 0, \;\alpha \in \mathbb{R},\; \epsilon_{i} \in \mathbb{C}\, (i = 1, 2),\\[2ex] (v(0), u(0)) = (v_0, u_0), \end{cases} \end{equation*} posed on a $d$-dimensional sphere $ \mathbb{S}^{d}$ or a compact Zoll manifold $M$. Considering $\sigma=\frac{\theta}{\beta}$ with $\theta, \beta\in \{n^2:n\in\mathbb{Z}\}$ we derive a bilinear Strichartz type estimate and use it to prove the local well-posedness results for given data $(v_0, u_0)\in H^s(M)\times H^s(M)$ whenever $s>\frac{1}{4}$ in the case $M = \mathbb{S}^{2}$ or a Zoll manifold, and $s > \frac{d - 2}{2}$ in the case $M = \mathbb{S}^{d}$ ($d \geq 3$) induced with the canonical metric. Moreover, in dimensions $2$ and $3$, we use a Gagliardo-Nirenberg type inequality to prove that the local solution can be extended globally in time whenever $s \geq 1$.