Transient behavior of collapsing ring solutions in the critical nonlinear Schrödinger equation

Jordan Allen-Flowers, Karl B Glasner

Research output: Contribution to journalArticle

Abstract

The critical nonlinear Schrödinger equation (NLS) possesses nearly self-similar ring profile solutions. We address the question of whether this profile is maintained all the way until the point of singularity. A perturbative analysis of the rescaled PDE and the resulting self-similar profile uncover slow dynamics that eventually drive the ring structure to the classical peak-shaped collapse instead. A numerical scheme capable of resolving self-similar behavior to high resolutions confirms our analysis. We also consider ring-type blowup arising either from azimuthally polarized solutions of a coupled NLS system or as vortex solutions of the usual NLS. In this case, the ring profile is maintained up to the time of singularity.

Original languageEnglish (US)
Pages (from-to)53-61
Number of pages9
JournalPhysica D: Nonlinear Phenomena
Volume284
DOIs
StatePublished - Sep 15 2014

Fingerprint

nonlinear equations
rings
profiles
pulse detonation engines
ring structures
vortices
high resolution

Keywords

  • Blowup
  • Nonlinear Schrödinger equation
  • Self-similarity

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Statistical and Nonlinear Physics

Cite this

Transient behavior of collapsing ring solutions in the critical nonlinear Schrödinger equation. / Allen-Flowers, Jordan; Glasner, Karl B.

In: Physica D: Nonlinear Phenomena, Vol. 284, 15.09.2014, p. 53-61.

Research output: Contribution to journalArticle

@article{ee3398b8c87c499797c505844ebbbb3d,
title = "Transient behavior of collapsing ring solutions in the critical nonlinear Schr{\"o}dinger equation",
abstract = "The critical nonlinear Schr{\"o}dinger equation (NLS) possesses nearly self-similar ring profile solutions. We address the question of whether this profile is maintained all the way until the point of singularity. A perturbative analysis of the rescaled PDE and the resulting self-similar profile uncover slow dynamics that eventually drive the ring structure to the classical peak-shaped collapse instead. A numerical scheme capable of resolving self-similar behavior to high resolutions confirms our analysis. We also consider ring-type blowup arising either from azimuthally polarized solutions of a coupled NLS system or as vortex solutions of the usual NLS. In this case, the ring profile is maintained up to the time of singularity.",
keywords = "Blowup, Nonlinear Schr{\"o}dinger equation, Self-similarity",
author = "Jordan Allen-Flowers and Glasner, {Karl B}",
year = "2014",
month = "9",
day = "15",
doi = "10.1016/j.physd.2014.06.009",
language = "English (US)",
volume = "284",
pages = "53--61",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",

}

TY - JOUR

T1 - Transient behavior of collapsing ring solutions in the critical nonlinear Schrödinger equation

AU - Allen-Flowers, Jordan

AU - Glasner, Karl B

PY - 2014/9/15

Y1 - 2014/9/15

N2 - The critical nonlinear Schrödinger equation (NLS) possesses nearly self-similar ring profile solutions. We address the question of whether this profile is maintained all the way until the point of singularity. A perturbative analysis of the rescaled PDE and the resulting self-similar profile uncover slow dynamics that eventually drive the ring structure to the classical peak-shaped collapse instead. A numerical scheme capable of resolving self-similar behavior to high resolutions confirms our analysis. We also consider ring-type blowup arising either from azimuthally polarized solutions of a coupled NLS system or as vortex solutions of the usual NLS. In this case, the ring profile is maintained up to the time of singularity.

AB - The critical nonlinear Schrödinger equation (NLS) possesses nearly self-similar ring profile solutions. We address the question of whether this profile is maintained all the way until the point of singularity. A perturbative analysis of the rescaled PDE and the resulting self-similar profile uncover slow dynamics that eventually drive the ring structure to the classical peak-shaped collapse instead. A numerical scheme capable of resolving self-similar behavior to high resolutions confirms our analysis. We also consider ring-type blowup arising either from azimuthally polarized solutions of a coupled NLS system or as vortex solutions of the usual NLS. In this case, the ring profile is maintained up to the time of singularity.

KW - Blowup

KW - Nonlinear Schrödinger equation

KW - Self-similarity

UR - http://www.scopus.com/inward/record.url?scp=84905178886&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84905178886&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2014.06.009

DO - 10.1016/j.physd.2014.06.009

M3 - Article

AN - SCOPUS:84905178886

VL - 284

SP - 53

EP - 61

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -