Global attractor for a Ginzburg–Landau type model of rotating Bose–Einstein condensates

Alexey Cheskidov,Daniel Marahrens,Christof Sparber

doi:10.4310/dpde.2017.v14.n1.a2

Abstract

We study the long time behavior of solutions to a nonlinear partial differential equation arising in the description of trapped rotating Bose-Einstein condensates. The equation can be seen as a hybrid between the well-known nonlinear Schr\odinger/Gross-Pitaevskii equation and the Ginzburg-Landau equation. We prove existence and uniqueness of global in-time solutions in the physical energy space and establish the existence of a global attractor within the associated dynamics. We also obtain basic structural properties of the attractor and an estimate on its Hausdorff and fractal dimensions.

Highlights

From a mathematical point of view, rotating BoseEinstein condensates (BECs) can be described within the realm of a mean-field model, the so-called Gross-Pitaevskii equation [29]
Equation (1.7) can be seen as a hybrid between the Gross-Pitaevskii/Nonlinear Schrodinger equation and the complex Ginzburg-Landau equation. Both kind of models have been extensively studied in the mathematical literature: For local and global well-posedness results on nonlinear Schrodinger equation (NLS), with or without quadratic potentials V, we refer to [11, 8, 9]
In order to set up a wellposedness result for the nonlinear equation (1.6), we need to study the regularizing properties of the linear semigroup associated to HΩ, i.e

Summary

Introduction

Equation (1.7) can be seen as a hybrid between the Gross-Pitaevskii/Nonlinear Schrodinger equation and the complex Ginzburg-Landau equation Both kind of models have been extensively studied in the mathematical literature: For local and global well-posedness results on NLS, with or without quadratic potentials V , we refer to [11, 8, 9]. In order to set up a wellposedness result for the nonlinear equation (1.6), we need to study the regularizing properties of the linear semigroup associated to HΩ, i.e. SΩ(t) := exp −e−iθtHΩ , t ∈ R+, As usual we identify SΩ(t) with its associated integral kernel via SΩ(t)f (x) = SΩ(t, x, y)f (y) dy, f ∈ L2(Rd).

Local well-posedness

Global existence and asymptotic vanishing of solutions

Bounds on the mass and energy

The global attractor and its properties