On the Well-Posedness and Stability of Cubic and Quintic Nonlinear Schrödinger Systems on 𝕋^3

In this paper, we study cubic and quintic nonlinear Schrödinger systems on three-dimensional tori, with initial data in an adapted Hilbert space H^s_λ, 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_λ 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_λ 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.