Two variational problems related to the nonlinear Schrödinger equation

Abstract: The nonlinear Schrödinger equation is a partial differential equation which appears as a model in several branches of physics, including Bose-Einstein condensation and hydrodynamics. In this thesis, we investigate a particular class of solutions, namely periodic standing waves. We derive explicit formulas for the solutions using elliptic functions and study their variational properties. The standing waves are critical points of the energy subject to the constraint of fixed mass. In particular, we characterize the constrained minimizers and investigate a phase transition which occurs for a critical value of a parameter in the equation. We also study a second class of solutions which are obtained by seeking critical points of the energy subject to the constraints of fixed mass and angular momentum. This leads to periodic standing waves in a moving reference frame or, equivalently, quasi periodic standing waves in a fixed reference frame. In this case, we are not able to completely identify the minimizers, and instead complement with numerical computations.