On the Well-posedness and Stability of Cubic and Quintic Nonlinear Schr\"odinger Systems on ${\mathbb T}^3$

TL;DR

This paper establishes well-posedness and stability results for cubic and quintic nonlinear Schrödinger systems on 3D tori, including global solutions and stability of stationary states, applicable to both rational and irrational tori.

Contribution

It provides new results on local and global well-posedness, and stability of stationary states for nonlinear Schrödinger systems on general 3D tori, extending previous work to broader geometric settings.

Findings

Local well-posedness for cubic and quintic systems on 3D tori.

Global well-posedness for defocusing cubic and small data quintic systems.

Existence and stability of stationary states using the energy-Casimir method.

Abstract

In this paper, we study cubic and quintic nonlinear Schr\"odinger systems on 3-dimensional tori, with initial data in an adapted Hilbert space $H^s_{\underline{\lambda}},$ and all of our results hold on rational and irrational rectangular, flat tori. In the cubic and quintic case, we prove local well-posedness for both focusing and defocusing systems. We show that local solutions of the defocusing cubic system with initial data in $H^1_{\underline{\lambda}}$ can be extended for all time. Additionally, we prove that global well-posedness holds in the quintic system, focusing or defocusing, for initial data with sufficiently small $H^1_{\underline{\lambda}}$ norm. Finally, we use the energy-Casimir method to prove the existence and uniqueness, and nonlinear stability of a class of stationary states of the defocusing cubic and quintic nonlinear Schr\"odinger systems.