Search Menu
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information
Search Scholarly Publications

Title: When linear stability does not exclude nonlinear instability

Abstract

We describe a mechanism that results in the nonlinear instability of stationary states even in the case where the stationary states are linearly stable. In this study, this instability is due to the nonlinearity-induced coupling of the linearization’s internal modes of negative energy with the continuous spectrum. In a broad class of nonlinear Schrödinger equations considered, the presence of such internal modes guarantees the nonlinear instability of the stationary states in the evolution dynamics. To corroborate this idea, we explore three prototypical case examples: (a) an antisymmetric soliton in a double-well potential, (b) a twisted localized mode in a one-dimensional lattice with cubic nonlinearity, and (c) a discrete vortex in a two-dimensional saturable lattice. In all cases, we observe a weak nonlinear instability, despite the linear stability of the respective states.

Authors:
 [1];  [2];  [3]
+ Show Author Affiliations
  1. Univ. of Massachusetts, Amherst, MA (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. McMaster Univ., Hamilton, ON (Canada)
  3. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1233191
Alternate Identifier(s):
OSTI ID: 1193887
Report Number(s):
LA-UR-14-29116
Journal ID: ISSN 0031-9007; PRLTAO; TRN: US1600428
Grant/Contract Number:  
2014/133; 2839; FA9550-12-1-0332; AC52-06NA25396
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review Letters
Additional Journal Information:
Journal Volume: 114; Journal Issue: 21; Journal ID: ISSN 0031-9007
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; 97 MATHEMATICS AND COMPUTING

Citation Formats

Kevrekidis, P. G., Pelinovsky, D. E., and Saxena, A. When linear stability does not exclude nonlinear instability. United States: N. p., 2015. Web. doi:10.1103/PhysRevLett.114.214101.
Copy to clipboard
Kevrekidis, P. G., Pelinovsky, D. E., & Saxena, A. When linear stability does not exclude nonlinear instability. United States. https://doi.org/10.1103/PhysRevLett.114.214101
Copy to clipboard
Kevrekidis, P. G., Pelinovsky, D. E., and Saxena, A. Fri . "When linear stability does not exclude nonlinear instability". United States. https://doi.org/10.1103/PhysRevLett.114.214101. https://www.osti.gov/servlets/purl/1233191.
Copy to clipboard
@article{osti_1233191,
title = {When linear stability does not exclude nonlinear instability},
author = {Kevrekidis, P. G. and Pelinovsky, D. E. and Saxena, A.},
abstractNote = {We describe a mechanism that results in the nonlinear instability of stationary states even in the case where the stationary states are linearly stable. In this study, this instability is due to the nonlinearity-induced coupling of the linearization’s internal modes of negative energy with the continuous spectrum. In a broad class of nonlinear Schrödinger equations considered, the presence of such internal modes guarantees the nonlinear instability of the stationary states in the evolution dynamics. To corroborate this idea, we explore three prototypical case examples: (a) an antisymmetric soliton in a double-well potential, (b) a twisted localized mode in a one-dimensional lattice with cubic nonlinearity, and (c) a discrete vortex in a two-dimensional saturable lattice. In all cases, we observe a weak nonlinear instability, despite the linear stability of the respective states.},
doi = {10.1103/PhysRevLett.114.214101},
journal = {Physical Review Letters},
number = 21,
volume = 114,
place = {United States},
year = {Fri May 29 00:00:00 EDT 2015},
month = {Fri May 29 00:00:00 EDT 2015}
}
Copy to clipboard
Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1103/PhysRevLett.114.214101
Copyright Statement
Other availability
Search WorldCat Search WorldCat to find libraries that may hold this journal
Citation Metrics:
Cited by: 17 works
Citation information provided by
Web of Science
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

Works referenced in this record:

The disintegration of wave trains on deep water Part 1. Theory
journal, February 1967

Freak Waves in Random Oceanic Sea States
journal, June 2001

Emergent Nonlinear Phenomena in Bose-Einstein Condensates
book, January 2008

  • Kevrekidis, Panayotis G.; Frantzeskakis, Dimitri J.; Carretero-González, Ricardo
  • Atomic, Optical, and Plasma Physics
  • DOI: 10.1007/978-3-540-73591-5

Spectral and Dynamical Stability of Nonlinear Waves
book, January 2013

Nonlinear Waves in Integrable and Nonintegrable Systems
book, January 2010

Localization in Periodic Potentials
journal, January 2011

Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems
journal, August 2004

  • Kapitula, Todd; Kevrekidis, Panayotis G.; Sandstede, Björn
  • Physica D: Nonlinear Phenomena, Vol. 195, Issue 3-4
  • DOI: 10.1016/j.physd.2004.03.018

Internal Modes of Solitary Waves
journal, June 1998

  • Kivshar, Yuri S.; Pelinovsky, Dmitry E.; Cretegny, Thierry
  • Physical Review Letters, Vol. 80, Issue 23
  • DOI: 10.1103/PhysRevLett.80.5032

Internal modes of envelope solitons
journal, May 1998

  • Pelinovsky, Dmitry E.; Kivshar, Yuri S.; Afanasjev, Vsevolod V.
  • Physica D: Nonlinear Phenomena, Vol. 116, Issue 1-2
  • DOI: 10.1016/S0167-2789(98)80010-9

On instability of excited states of the nonlinear Schrödinger equation
journal, January 2009

Direct Observation of Tunneling and Nonlinear Self-Trapping in a Single Bosonic Josephson Junction
journal, June 2005

Dynamics of a Tunable Superfluid Junction
journal, January 2011

Classical Bifurcation at the Transition from Rabi to Josephson Dynamics
journal, November 2010

Spontaneous symmetry breaking in photonic lattices: Theory and experiment
journal, June 2005

Symmetry-Breaking Instability of Multimode Vector Solitons
journal, July 2002

Beam interactions in one-dimensional saturable waveguide arrays
journal, October 2006

Dipole solitons in optically induced two-dimensional photonic lattices
journal, January 2004

  • Yang, Jianke; Makasyuk, Igor; Bezryadina, Anna
  • Optics Letters, Vol. 29, Issue 14
  • DOI: 10.1364/OL.29.001662

Discrete solitons in optics
journal, July 2008

Stability of discrete solitons in nonlinear Schrödinger lattices
journal, December 2005

  • Pelinovsky, D. E.; Kevrekidis, P. G.; Frantzeskakis, D. J.
  • Physica D: Nonlinear Phenomena, Vol. 212, Issue 1-2
  • DOI: 10.1016/j.physd.2005.07.021

Existence and stability of quasiperiodic breathers in the discrete nonlinear Schrödinger equation
journal, September 1997

Breathers on a background: periodic and quasiperiodic solutions of extended discrete nonlinear wave systems
journal, February 2003

Observation of Vortex-Ring “Discrete” Solitons in 2D Photonic Lattices
journal, March 2004

Observation of Discrete Vortex Solitons in Optically Induced Photonic Lattices
journal, March 2004

  • Neshev, Dragomir N.; Alexander, Tristram J.; Ostrovskaya, Elena A.
  • Physical Review Letters, Vol. 92, Issue 12
  • DOI: 10.1103/PhysRevLett.92.123903

Stability of vortex solitons in a photorefractive optical lattice
journal, January 2004

Observation of Two-Dimensional Dynamic Localization of Light
journal, June 2010

    Sort by:
    [ × clear filter / sort ]

    Similar Records in DOE PAGES and OSTI.GOV collections: