Cross-Invariant Sets of the Cubic Nonlinear Schrödinger System with Partial Confinement

Abstract

This paper studies the cubic nonlinear Schrödinger system with partial confinement:

\begin{cases}-i\varphi_t + (x_1^2 + x_2^2)\varphi = \Delta\varphi +\mu_1|\varphi|^2\varphi + \beta|\psi|^2\varphi, & (t,x) \in \mathbb{R}^+\times \mathbb{R}^3, \\ -i\psi_t + (x_1^2 + x_2^2)\psi =\Delta\psi + \mu_2|\psi|^2\psi + \beta|\varphi|^2\psi, & (t,x) \in\mathbb{R}^+ \times \mathbb{R}^3,\end{cases}

which models the Bose-Einstein condensates with multiple states or the propagation of mutually incoherent wave packets in nonlinear optics. The cross-invariant sets of the evolution flow are obtained by constructing the cross-constrained variational problem. Furthermore, the sharp condition for global existence and blowup of the solutions is derived.