The Schrödinger equations have had a profound impact on a wide range of fields of modern science, including quantum mechanics, superfluidity, geometrical optics, BoseEinstein condensates, and the analysis of dispersive phenomena in the theory of PDE. The main purpose of this thesis is to explore two Schrödingertype equations appearing in the socalled Bohmian formulation of quantum mechanics and in the study of excitonpolariton condensates.
For the first topic, the linear Schrödinger equation is the starting point in the formulation of a phasespace model proposed in [1] for the Bohmian interpretation of quantum mechanics. We analyze this model, a nonlinear Vlasovtype equation, as a Hamiltonian system defined on an appropriate Poisson manifold built on Wasserstein spaces, the aim being to establish its existence theory. For this purpose, we employ results from the theory of PDE, optimal transportation, differential geometry and algebraic topology.
The second topic of the thesis is the study of a nonlinear Schrödinger equation, called the complex GrossPitaevskii equation, appearing in the context of BoseEinstein condensation of excitonpolaritons. This model can be roughly described as a drivendamped GrossPitaevskii equation which shares some similarities with the complex GinzburgLandau equation. The difficulties in the analysis of this equation stem from the fact that, unlike the complex GinzburgLandau equation, the complex GrossPitaevskii equation does not include a viscous dissipation term. Our approach to this equation will be in the framework of numerical computations, using two main tools: collocation methods and numerical continuation for the stationary solutions and a timesplitting spectral method for the dynamics. After performing a linear stability analysis on the computed stationary solutions, we are led to postulate the existence of radially symmetric stationary ground state solutions only for certain values of the parameters in the equation; these parameters represent the “strength” of the driving and damping terms. Moreover, numerical continuation allows us to show, for fixed parameters, the ground and some of the excited state solutions of this equation. Finally, for the values of the parameters that do not produce a stable radially symmetric solution, our dynamical computations show the emergence of rotating vortex lattices.
Date of Award  May 16 2018 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Sciences and Engineering


Supervisor  Peter Markowich (Supervisor) 

 PDE's
 Optimal transport
 Schrödinger
 hamiltonian flow
 existence
 BoseEinstein condensates