On the Well-posedness and Stability of Cubic and Quintic Nonlinear Schrödinger Systems on \mathbb T³

2021/12/22 by Chen, Thomas, Urban, Amie
#82C10 #82D10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2112.12119

Abstract

In this paper, we study cubic and quintic nonlinear Schrödinger systems on 3-dimensional tori, with initial data in an adapted Hilbert space H^s_\underlineλ, 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 H¹_\underlineλ 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 H¹_\underlineλ 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 Schrödinger systems.

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