American Institute of Mathematical Sciences

Advanced
May  2012, 32(5): 1775-1799. doi: 10.3934/dcds.2012.32.1775

Breather continuation from infinity in nonlinear oscillator chains

Guillaume James 1, and  Dmitry Pelinovsky 2,

1. 

Laboratoire Jean Kuntzmann, UMR CNRS 5224, BP 53, 38041 Grenoble Cedex 9, France
2. 

Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

Received  November 2010 Revised  September 2011 Published  January 2012

Existence of large-amplitude time-periodic breathers localized near a single site is proved for the discrete Klein--Gordon equation, in the case when the derivative of the on-site potential has a compact support. Breathers are obtained at small coupling between oscillators and under nonresonance conditions. Our method is different from the classical anti-continuum limit developed by MacKay and Aubry, and yields in general branches of breather solutions that cannot be captured with this approach. When the coupling constant goes to zero, the amplitude and period of oscillations at the excited site go to infinity. Our method is based on near-identity transformations, analysis of singular limits in nonlinear oscillator equations, and fixed-point arguments.
Keywords: nonlocal bifurcations, Discrete Klein--Gordon equation, discrete breathers, fixed-point arguments., asymptotic methods.
Mathematics Subject Classification: Primary: 37K50; Secondary: 37K60, 37N2.
Citation: Guillaume James, Dmitry Pelinovsky. Breather continuation from infinity in nonlinear oscillator chains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1775-1799. doi: 10.3934/dcds.2012.32.1775
References:
[1]

J. F. R. Archilla, J. Cuevas, B. Sánchez-Rey and A. Alvarez, Demonstration of the stability or instability of multibreathers at low coupling,, Physica D, 180 (2003), 235.   Google Scholar

[2]

S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization,, Lattice Dynamics (Paris, 103 (1997), 201.  doi: 10.1016/S0167-2789(96)00261-8.  Google Scholar

[3]

S. Aubry, G. Kopidakis and V. Kadelburg, Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems,, Discrete and Continuous Dynamical Systems B, 1 (2001), 271.   Google Scholar

[4]

D. Bambusi, Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators,, Nonlinearity, 9 (1996), 433.  doi: 10.1088/0951-7715/9/2/009.  Google Scholar

[5]

J. Fura and S. Rybicki, Periodic solutions of second order Hamiltonian systems bifurcating from infinity,, Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, 24 (2007), 471.   Google Scholar

[6]

G. James, Centre manifold reduction for quasilinear discrete systems,, J. Nonlinear Sci., 13 (2003), 27.  doi: 10.1007/s00332-002-0525-x.  Google Scholar

[7]

G. James, A. Levitt and C. Ferreira, Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing,, Applicable Analysis, 89 (2010), 1447.  doi: 10.1080/00036810903437788.  Google Scholar

[8]

G. James, B. Sánchez-Rey and J. Cuevas, Breathers in inhomogeneous nonlinear lattices: An analysis via center manifold reduction,, Rev. Math. Phys., 21 (2009), 1.  doi: 10.1142/S0129055X09003578.  Google Scholar

[9]

V. Koukouloyannis and P. Kevrekidis, On the stability of multibreathers in Klein-Gordon chains,, Nonlinearity, 22 (2009), 2269.  doi: 10.1088/0951-7715/22/9/011.  Google Scholar

[10]

R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators,, Nonlinearity, 7 (1994), 1623.  doi: 10.1088/0951-7715/7/6/006.  Google Scholar

[11]

R. S. MacKay and J.-A. Sepulchre, Stability of discrete breathers,, Localization in Nonlinear Lattices (Dresden, 119 (1998), 148.  doi: 10.1016/S0167-2789(98)00073-6.  Google Scholar

[12]

J. L. Marín and S. Aubry, Finite size effects on instabilities of discrete breathers,, Physica D, 119 (1998), 163.   Google Scholar

[13]

A. Mielke and C. Patz, Dispersive stability of infinite-dimensional Hamiltonian systems on lattices,, Applicable Analysis, 89 (2010), 1493.  doi: 10.1080/00036810903517605.  Google Scholar

[14]

A. Pankov, "Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices,", Imperial College Press, (2005).  doi: 10.1142/9781860947216.  Google Scholar

[15]

M. Peyrard, S. Cuesta-López and G. James, Modelling DNA at the mesoscale: A challenge for nonlinear science?,, Nonlinearity, 21 (2008).  doi: 10.1088/0951-7715/21/6/T02.  Google Scholar

[16]

M. Peyrard, S. Cuesta-López and G. James, Nonlinear analysis of the dynamics of DNA breathing,, J. Biol. Phys., 35 (2009), 73.  doi: 10.1007/s10867-009-9127-2.  Google Scholar

[17]

M. Peyrard, Nonlinear dynamics and statistical physics of DNA,, Nonlinearity, 17 (2004).  doi: 10.1088/0951-7715/17/2/R01.  Google Scholar

[18]

J.-A. Sepulchre and R. S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators,, Nonlinearity, 10 (1997), 679.  doi: 10.1088/0951-7715/10/3/006.  Google Scholar

[19]

D. Treschev, Travelling waves in FPU lattices,, Discrete and Continuous Dynamical Systems A, 11 (2004), 867.  doi: 10.3934/dcds.2004.11.867.  Google Scholar

[20]

G. Weber, Sharp DNA denaturation due to solvent interaction,, Europhys. Lett., 73 (2006).  doi: 10.1209/epl/i2005-10466-6.  Google Scholar

show all references

References:
[1]

J. F. R. Archilla, J. Cuevas, B. Sánchez-Rey and A. Alvarez, Demonstration of the stability or instability of multibreathers at low coupling,, Physica D, 180 (2003), 235.   Google Scholar

[2]

S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization,, Lattice Dynamics (Paris, 103 (1997), 201.  doi: 10.1016/S0167-2789(96)00261-8.  Google Scholar

[3]

S. Aubry, G. Kopidakis and V. Kadelburg, Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems,, Discrete and Continuous Dynamical Systems B, 1 (2001), 271.   Google Scholar

[4]

D. Bambusi, Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators,, Nonlinearity, 9 (1996), 433.  doi: 10.1088/0951-7715/9/2/009.  Google Scholar

[5]

J. Fura and S. Rybicki, Periodic solutions of second order Hamiltonian systems bifurcating from infinity,, Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, 24 (2007), 471.   Google Scholar

[6]

G. James, Centre manifold reduction for quasilinear discrete systems,, J. Nonlinear Sci., 13 (2003), 27.  doi: 10.1007/s00332-002-0525-x.  Google Scholar

[7]

G. James, A. Levitt and C. Ferreira, Continuation of discrete breathers from infinity in a nonlinear model for DNA breathing,, Applicable Analysis, 89 (2010), 1447.  doi: 10.1080/00036810903437788.  Google Scholar

[8]

G. James, B. Sánchez-Rey and J. Cuevas, Breathers in inhomogeneous nonlinear lattices: An analysis via center manifold reduction,, Rev. Math. Phys., 21 (2009), 1.  doi: 10.1142/S0129055X09003578.  Google Scholar

[9]

V. Koukouloyannis and P. Kevrekidis, On the stability of multibreathers in Klein-Gordon chains,, Nonlinearity, 22 (2009), 2269.  doi: 10.1088/0951-7715/22/9/011.  Google Scholar

[10]

R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators,, Nonlinearity, 7 (1994), 1623.  doi: 10.1088/0951-7715/7/6/006.  Google Scholar

[11]

R. S. MacKay and J.-A. Sepulchre, Stability of discrete breathers,, Localization in Nonlinear Lattices (Dresden, 119 (1998), 148.  doi: 10.1016/S0167-2789(98)00073-6.  Google Scholar

[12]

J. L. Marín and S. Aubry, Finite size effects on instabilities of discrete breathers,, Physica D, 119 (1998), 163.   Google Scholar

[13]

A. Mielke and C. Patz, Dispersive stability of infinite-dimensional Hamiltonian systems on lattices,, Applicable Analysis, 89 (2010), 1493.  doi: 10.1080/00036810903517605.  Google Scholar

[14]

A. Pankov, "Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices,", Imperial College Press, (2005).  doi: 10.1142/9781860947216.  Google Scholar

[15]

M. Peyrard, S. Cuesta-López and G. James, Modelling DNA at the mesoscale: A challenge for nonlinear science?,, Nonlinearity, 21 (2008).  doi: 10.1088/0951-7715/21/6/T02.  Google Scholar

[16]

M. Peyrard, S. Cuesta-López and G. James, Nonlinear analysis of the dynamics of DNA breathing,, J. Biol. Phys., 35 (2009), 73.  doi: 10.1007/s10867-009-9127-2.  Google Scholar

[17]

M. Peyrard, Nonlinear dynamics and statistical physics of DNA,, Nonlinearity, 17 (2004).  doi: 10.1088/0951-7715/17/2/R01.  Google Scholar

[18]

J.-A. Sepulchre and R. S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators,, Nonlinearity, 10 (1997), 679.  doi: 10.1088/0951-7715/10/3/006.  Google Scholar

[19]

D. Treschev, Travelling waves in FPU lattices,, Discrete and Continuous Dynamical Systems A, 11 (2004), 867.  doi: 10.3934/dcds.2004.11.867.  Google Scholar

[20]

G. Weber, Sharp DNA denaturation due to solvent interaction,, Europhys. Lett., 73 (2006).  doi: 10.1209/epl/i2005-10466-6.  Google Scholar

[1]

Panayotis Panayotaros. Continuation and bifurcations of breathers in a finite discrete NLS equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1227-1245. doi: 10.3934/dcdss.2011.4.1227
[2]

Jean-Pierre Eckmann, C. Eugene Wayne. Breathers as metastable states for the discrete NLS equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6091-6103. doi: 10.3934/dcds.2018136
[3]

Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271
[4]

Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273
[5]

Vassilis Rothos. Subharmonic bifurcations of localized solutions of a discrete NLS equation. Conference Publications, 2005, 2005 (Special) : 756-767. doi: 10.3934/proc.2005.2005.756
[6]

Michael Kastner, Jacques-Alexandre Sepulchre. Effective Hamiltonian for traveling discrete breathers in the FPU chain. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 719-734. doi: 10.3934/dcdsb.2005.5.719
[7]

Jesús Cuevas, Bernardo Sánchez-Rey, J. C. Eilbeck, Francis Michael Russell. Interaction of moving discrete breathers with interstitial defects. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1057-1067. doi: 10.3934/dcdss.2011.4.1057
[8]

Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077
[9]

Andrew Comech. Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2711-2755. doi: 10.3934/dcds.2013.33.2711
[10]

Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525
[11]

Caidi Zhao, Gang Xue, Grzegorz Łukaszewicz. Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4021-4044. doi: 10.3934/dcdsb.2018122
[12]

S. Aubry, G. Kopidakis, V. Kadelburg. Variational proof for hard Discrete breathers in some classes of Hamiltonian dynamical systems. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 271-298. doi: 10.3934/dcdsb.2001.1.271
[13]

Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147
[14]

Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971
[15]

Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55
[16]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[17]

Hunseok Kang. Asymptotic behavior of a discrete turing model. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 265-284. doi: 10.3934/dcds.2010.27.265
[18]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903
[19]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679
[20]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences