Mathematical Theory and Numerical Methods for Gross-Pitaevskii Equations and Applications

Abstract

Gross-Pitaevskii equation (GPE), first derived in early 1960s, is a widely used model in different subjects, such as quantum mechanics, condensed matter physics, nonlinear optics etc. Since 1995, GPE has regained considerable research interests due to the experimental success of Bose-Einstein condensates (BEC), which can be well described by GPE at ultra-cold temperature.

The purpose of this thesis is to carry out mathematical and numerical studies for GPE. We focus on the ground states and the dynamics of GPE. The ground state is defined as the minimizer of the energy functional associated with the corresponding GPE, under the constraint of total mass ($L^2$ norm) being normalized to 1. For the dynamics, the task is to solve the Cauchy problem for GPE.

This thesis mainly contains three parts. The first part is to investigate the dipolar GPE modeling degenerate dipolar quantum gas. For ground states, we prove the existence and uniqueness as well as non-existence. For dynamics, we discuss the well-posedness, possible finite time blow-up and dimension reduction. Convergence for this dimension reduction has been established in certain regimes. Efficient and accurate numerical methods are proposed to compute the ground states and the dynamics. Numerical results show the efficiency and accuracy of the numerical methods.

The second part is devoted to the coupled GPEs modeling a two component BEC. We show the existence and uniqueness as well as non-existence and limiting behavior of the ground states in different parameter regimes. Efficient and accurate numerical methods are designed to compute the ground states. Examples are shown to confirm the analytical analysis.

The third part is to understand the convergence of the finite difference discretizations for GPE. We prove the optimal convergence rates for the conservative Crank-Nicolson finite difference discretizations (CNFD) and the semi-implicit finite difference discretizations (SIFD) for rotational GPE, in two and three dimensions. We also consider the nonlinear Schr\"{o}dinger equation perturbed by the wave operator, where the small perturbation causes high oscillation of the solution in time. This high oscillation brings significant difficulties in proving uniform convergence rates for CNFD and SIFD, independent of the perturbation. We overcome the difficulties and obtain uniform error bounds for both CNFD and SIFD, in one, two and three dimensions. Numerical results confirm our theoretical analysis.