Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the Gross--Pitaevskii Equation

Tzou, J. C.; Kevrekidis, P. G.; Kolokolnikov, T.; Carretero-González, R.

doi:10.1137/15M1038931

Title: Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the Gross--Pitaevskii Equation

Full Record
Other Related Research

Abstract

For a dissipative variant of the two-dimensional Gross--Pitaevskii equation with a parabolic trap under rotation, we study a symmetry breaking process that leads to the formation of vortices. The first symmetry breaking leads to the formation of many small vortices distributed uniformly near the Thomas$$-$$Fermi radius. The instability occurs as a result of a linear instability of a vortex-free steady state as the rotation is increased above a critical threshold. We focus on the second subsequent symmetry breaking, which occurs in the weakly nonlinear regime. At slightly above threshold, we derive a one-dimensional amplitude equation that describes the slow evolution of the envelope of the initial instability. Here, we show that the mechanism responsible for initiating vortex formation is a modulational instability of the amplitude equation. We also illustrate the role of dissipation in the symmetry breaking process. All analyses are confirmed by detailed numerical computations

Authors:

Tzou, J. C. ^[1]; Kevrekidis, P. G. ^[2]; Kolokolnikov, T. ^[1]; Carretero-González, R. ^[3]

Dalhousie Univ., Halifax (Canada). Dept. of Mathematics and Statistics
Univ. of Massachusetts, Amherst, MA (United States). Dept. of Mathematics and Statistics
San Diego State Univ., San Diego, CA (United States). Nonlinear Dynamical System Group Computational Science Research Center, and Dept. of Mathematics and Statistics

Publication Date:: Tue May 10 00:00:00 EDT 2016

Research Org.:: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE; US Air Force Office of Scientific Research (AFOSR); National Science Foundation (NSF)

OSTI Identifier:: 1394965

Report Number(s):: LA-UR-15-25770
Journal ID: ISSN 1536-0040

Grant/Contract Number:: AC52-06NA25396; RGPIN-33798; RGPAS/461907; FA950-12-1-0332

Resource Type:: Accepted Manuscript

Journal Name:: SIAM Journal on Applied Dynamical Systems

Additional Journal Information:: Journal Volume: 15; Journal Issue: 2; Journal ID: ISSN 1536-0040

Publisher:: Society for Industrial and Applied Mathematics

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Atomic and Nuclear Physics; Mathematics; nonlinear Schrodinger equation; Bose-Einstein condensates; vortex nucleation; dissipative Gross-Pitaevskii equation

Citation Formats


                    Tzou, J. C., Kevrekidis, P. G., Kolokolnikov, T., and Carretero-González, R. Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the Gross--Pitaevskii Equation.  United States: N. p., 2016. 
Web.  doi:10.1137/15M1038931.

Copy to clipboard


                    Tzou, J. C., Kevrekidis, P. G., Kolokolnikov, T., & Carretero-González, R. Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the Gross--Pitaevskii Equation.  United States.  https://doi.org/10.1137/15M1038931

Copy to clipboard


                    Tzou, J. C., Kevrekidis, P. G., Kolokolnikov, T., and Carretero-González, R. Tue .  
"Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the Gross--Pitaevskii Equation".  United States.  https://doi.org/10.1137/15M1038931.  https://www.osti.gov/servlets/purl/1394965.

Copy to clipboard


                    
@article{osti_1394965,

  title        = {Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the Gross--Pitaevskii Equation},

  author       = {Tzou, J. C. and Kevrekidis, P. G. and Kolokolnikov, T. and Carretero-González, R.},

  abstractNote = {For a dissipative variant of the two-dimensional Gross--Pitaevskii equation with a parabolic trap under rotation, we study a symmetry breaking process that leads to the formation of vortices. The first symmetry breaking leads to the formation of many small vortices distributed uniformly near the Thomas$-$Fermi radius. The instability occurs as a result of a linear instability of a vortex-free steady state as the rotation is increased above a critical threshold. We focus on the second subsequent symmetry breaking, which occurs in the weakly nonlinear regime. At slightly above threshold, we derive a one-dimensional amplitude equation that describes the slow evolution of the envelope of the initial instability. Here, we show that the mechanism responsible for initiating vortex formation is a modulational instability of the amplitude equation. We also illustrate the role of dissipation in the symmetry breaking process. All analyses are confirmed by detailed numerical computations},

  doi          = {10.1137/15M1038931},

  journal      = {SIAM Journal on Applied Dynamical Systems},

  number       = 2,

  volume       = 15,

  place        = {United States},

  year         = {Tue May 10 00:00:00 EDT 2016},

  month        = {Tue May 10 00:00:00 EDT 2016}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1137/15M1038931

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 1 work

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Similar Records in DOE PAGES and OSTI.GOV collections:

Vortex nucleation in a dissipative variant of the nonlinear Schrödinger equation under rotation

Journal Article Carretero-González, R. ; Kevrekidis, P. G. ; Kolokolnikov, T. - Physica. D, Nonlinear Phenomena

In this work, we motivate and explore the dynamics of a dissipative variant of the nonlinear Schrödinger equation under the impact of external rotation. As in the well established Hamiltonian case, the rotation gives rise to the formation of vortices. We show, however, that the most unstable mode leading to this instability scales with an appropriate power of the chemical potential μ of the system, increasing proportionally to μ^2/3. The precise form of the relevant formula, obtained through our asymptotic analysis, provides the most unstable mode as a function of the atomic density and the trap strength. We show howmore »« less
Cited by 7
https://doi.org/10.1016/j.physd.2015.11.009

Full Text Available
Vortex lattices generated by the Kelvin-Helmholtz instability in the Gross-Pitaevskii equation

Journal Article Ohta, A ; Kashiwa, R ; Sakaguchi, H - Physical Review. A

Vortex streets are formed from sheared initial conditions in classical fluids even without viscosity, which is called the Kelvin-Helmholtz instability. We demonstrate that similar vortex streets are generated from sheared initial conditions by the direct numerical simulation of the Gross-Pitaevskii (GP) equation which describes the dynamics of the Bose-Einstein condensates. Furthermore, we show the vortex-lattice formation from sheared initial conditions analogous to the rigid-body rotation in the GP equation under a rotating harmonic potential. The vortex-lattice formation by the dynamical instability in the system without energy dissipation differs from the vortex-lattice formation process by the imaginary time evolution of themore »« less
https://doi.org/10.1103/PHYSREVA.82.055602
Energy cascade with small-scale thermalization, counterflow metastability, and anomalous velocity of vortex rings in Fourier-truncated Gross-Pitaevskii equation

Journal Article Krstulovic, Giorgio ; Brachet, Marc - Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics (Print)

The statistical equilibria of the (conservative) dynamics of the Gross-Pitaevskii equation (GPE) with a finite range of spatial Fourier modes are characterized using a new algorithm, based on a stochastically forced Ginzburg-Landau equation (SGLE), that directly generates grand-canonical distributions. The SGLE-generated distributions are validated against finite-temperature GPE-thermalized states and exact (low-temperature) results obtained by steepest descent on the (grand-canonical) partition function. A standard finite-temperature second-order {lambda} transition is exhibited. A mechanism of GPE thermalization through a direct cascade of energy is found using initial conditions with mass and energy distributed at large scales. A long transient with partial thermalization atmore »« less
https://doi.org/10.1103/PHYSREVE.83.066311
Matter-wave vortices in cigar-shaped and toroidal waveguides

Journal Article Salasnich, L ; Toigo, F ; Malomed, B A - Physical Review. A

We study vortical states in a Bose-Einstein condensate (BEC) filling a cigar-shaped trap. An effective one-dimensional (1D) nonpolynomial Schroedinger equation (NPSE) is derived in this setting, for the models with both repulsive and attractive interatomic interactions. Analytical formulas for the density profiles are obtained from the NPSE in the case of self-repulsion within the Thomas-Fermi approximation, and in the case of the self-attraction as exact solutions (bright solitons). A crucially important ingredient of the analysis is the comparison of these predictions with direct numerical solutions for the vortex states in the underlying 3D Gross-Pitaevskii equation. The comparison demonstrates that themore »« less
https://doi.org/10.1103/PHYSREVA.76.063614
Exact solutions of the Gross-Pitaevskii equation for stable vortex modes in two-dimensional Bose-Einstein condensates

Journal Article Lei, Wu ; Jiefang, Zhang ; Lu, Li ; ... - Physical Review. A

We construct exact solutions of the Gross-Pitaevskii equation for solitary vortices, and approximate ones for fundamental solitons, in two-dimensional models of Bose-Einstein condensates with a spatially modulated nonlinearity of either sign and a harmonic trapping potential. The number of vortex-soliton (VS) modes is determined by the discrete energy spectrum of a related linear Schroedinger equation. The VS families in the system with the attractive and repulsive nonlinearity are mutually complementary. Stable VSs with vorticity S{>=}2 and those corresponding to higher-order radial states are reported, in the case of the attraction and repulsion, respectively.
https://doi.org/10.1103/PHYSREVA.81.061805

Similar Records