Nonlinear instabilities of multi-site breathers in Klein-Gordon lattices

Cuevas-Maraver, Jesus; Kevrekidis, Panayotis G.; Pelinovsky, Dmitry E.

doi:10.1111/sapm.12107

Nonlinear instabilities of multi-site breathers in Klein-Gordon lattices

Journal Article · Mon Aug 01 00:00:00 EDT 2016 · Studies in Applied Mathematics

DOI:https://doi.org/10.1111/sapm.12107· OSTI ID:1246360

Cuevas-Maraver, Jesus ^[1]; Kevrekidis, Panayotis G. ^[2]; Pelinovsky, Dmitry E. ^[3]

Univ. de Sevilla, Sevilla (Spain)
Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Univ. of Massachusetts, Amherst, MA (United States)
McMaster Univ., Hamilton, ON (Canada); Nizhny Novgorod State Technical Univ., Novgorod (Russia)

Here, we explore the possibility of multi-site breather states in a nonlinear Klein–Gordon lattice to become nonlinearly unstable, even if they are found to be spectrally stable. The mechanism for this nonlinear instability is through the resonance with the wave continuum of a multiple of an internal mode eigenfrequency in the linearization of excited breather states. For the nonlinear instability, the internal mode must have its Krein signature opposite to that of the wave continuum. This mechanism is not only theoretically proposed, but also numerically corroborated through two concrete examples of the Klein–Gordon lattice with a soft (Morse) and a hard (Φ⁴) potential. Compared to the case of the nonlinear Schrödinger lattice, the Krein signature of the internal mode relative to that of the wave continuum may change depending on the period of the multi-site breather state. For the periods for which the Krein signatures of the internal mode and the wave continuum coincide, multi-site breather states are observed to be nonlinearly stable.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC52-06NA25396

OSTI ID:: 1246360

Report Number(s):: LA-UR-15-22633

Journal Information:: Studies in Applied Mathematics, Journal Name: Studies in Applied Mathematics Vol. 6; ISSN 0022-2526

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

References (19)

Demonstration of the stability or instability of multibreathers at low coupling Archilla, J. F. R.; Cuevas, J.; Sánchez-Rey, B. Physica D: Nonlinear Phenomena, Vol. 180, Issue 3-4 https://doi.org/10.1016/S0167-2789(03)00064-2	journal	June 2003
Breathers in nonlinear lattices: Existence, linear stability and quantization Aubry, Serge Physica D: Nonlinear Phenomena, Vol. 103, Issue 1-4 https://doi.org/10.1016/S0167-2789(96)00261-8	journal	April 1997
Internal modes of envelope solitons Pelinovsky, Dmitry E.; Kivshar, Yuri S.; Afanasjev, Vsevolod V. Physica D: Nonlinear Phenomena, Vol. 116, Issue 1-2 https://doi.org/10.1016/S0167-2789(98)80010-9	journal	May 1998
Discrete breathers Flach, S.; Willis, C. R. Physics Reports, Vol. 295, Issue 5 https://doi.org/10.1016/S0370-1573(97)00068-9	journal	March 1998
Discrete breathers: classical and quantum MacKay, R. S. Physica A: Statistical Mechanics and its Applications, Vol. 288, Issue 1-4 https://doi.org/10.1016/S0378-4371(00)00421-0	journal	December 2000
Stability of discrete solitons in nonlinear Schrödinger lattices Pelinovsky, D. E.; Kevrekidis, P. G.; Frantzeskakis, D. J. Physica D: Nonlinear Phenomena, Vol. 212, Issue 1-2 https://doi.org/10.1016/j.physd.2005.07.021	journal	December 2005
On instability of excited states of the nonlinear Schrödinger equation Cuccagna, Scipio Physica D: Nonlinear Phenomena, Vol. 238, Issue 1 https://doi.org/10.1016/j.physd.2008.08.010	journal	January 2009
Escape dynamics in the discrete repulsive model Achilleos, V.; Álvarez, A.; Cuevas, J. Physica D: Nonlinear Phenomena, Vol. 244, Issue 1 https://doi.org/10.1016/j.physd.2012.10.008	journal	February 2013
Discrete solitons in optics Lederer, Falk; Stegeman, George I.; Christodoulides, Demetri N. Physics Reports, Vol. 463, Issue 1-3 https://doi.org/10.1016/j.physrep.2008.04.004	journal	July 2008
Discrete breathers — Advances in theory and applications Flach, Sergej; Gorbach, Andrey V. Physics Reports, Vol. 467, Issue 1-3 https://doi.org/10.1016/j.physrep.2008.05.002	journal	October 2008
Nonlinear dynamics and statistical physics of DNA Peyrard, Michel Nonlinearity, Vol. 17, Issue 2 https://doi.org/10.1088/0951-7715/17/2/R01	journal	January 2004
Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators MacKay, R. S.; Aubry, S. Nonlinearity, Vol. 7, Issue 6 https://doi.org/10.1088/0951-7715/7/6/006	journal	November 1994
Damped-driven granular chains: An ideal playground for dark breathers and multibreathers Chong, C.; Li, F.; Yang, J. Physical Review E, Vol. 89, Issue 3 https://doi.org/10.1103/PhysRevE.89.032924	journal	March 2014
Discrete Breathers in One-Dimensional Diatomic Granular Crystals Boechler, N.; Theocharis, G.; Job, S. Physical Review Letters, Vol. 104, Issue 24 https://doi.org/10.1103/PhysRevLett.104.244302	journal	June 2010
Intrinsic Localized Modes in Anharmonic Crystals Sievers, A. J.; Takeno, S. Physical Review Letters, Vol. 61, Issue 8 https://doi.org/10.1103/PhysRevLett.61.970	journal	August 1988
Internal Modes of Solitary Waves Kivshar, Yuri S.; Pelinovsky, Dmitry E.; Cretegny, Thierry Physical Review Letters, Vol. 80, Issue 23 https://doi.org/10.1103/PhysRevLett.80.5032	journal	June 1998
Colloquium : Nonlinear energy localization and its manipulation in micromechanical oscillator arrays Sato, M.; Hubbard, B. E.; Sievers, A. J. Reviews of Modern Physics, Vol. 78, Issue 1 https://doi.org/10.1103/RevModPhys.78.137	journal	January 2006
Dynamics of Bose-Einstein condensates in optical lattices Morsch, Oliver; Oberthaler, Markus Reviews of Modern Physics, Vol. 78, Issue 1 https://doi.org/10.1103/RevModPhys.78.179	journal	February 2006
The Development of Variable-Step Symplectic Integrators, with Application to the Two-Body Problem Calvo, M. P.; Sanz-Serna, J. M. SIAM Journal on Scientific Computing, Vol. 14, Issue 4 https://doi.org/10.1137/0914057	journal	July 1993

Similar Records

Interaction between sine-Gordon breathers

Journal Article · Wed Aug 01 00:00:00 EDT 2001 · Physical Review E · OSTI ID:40277020

Evidence for breather excitations in the sine-Gordon chain

Journal Article · Sun Apr 08 23:00:00 EST 1979 · Phys. Rev. Lett.; (United States) · OSTI ID:6176494

Biphonons in the Klein-Gordon lattice

Journal Article · Mon Feb 28 23:00:00 EST 2005 · Physical Review. B, Condensed Matter and Materials Physics · OSTI ID:20666239

Related Subjects

97 MATHEMATICS AND COMPUTING
Mathematics

Nonlinear instabilities of multi-site breathers in Klein-Gordon lattices

Citation Formats

References (19)

Similar Records

Related Subjects