Nonlinear Schrödinger systems on the three-dimensional torus

In this dissertation, we study cubic and quintic nonlinear Schrödinger systems on 3-dimensional tori, with initial data in an adapted Hilbert space H [superscript s over lambda underscore], 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 [superscript 1 over lambda underscore] 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 [superscript 1 over lambda underscore] 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.