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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 and Continuous Dynamical Systems, 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), 235-255.

[2]

S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, Lattice Dynamics (Paris, 1995), Physica D, 103 (1997), 201-250. doi: 10.1016/S0167-2789(96)00261-8.

[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), 271-298.

[4]

D. Bambusi, Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 9 (1996), 433-457. doi: 10.1088/0951-7715/9/2/009.

[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), 471-490.

[6]

G. James, Centre manifold reduction for quasilinear discrete systems, J. Nonlinear Sci., 13 (2003), 27-63. doi: 10.1007/s00332-002-0525-x.

[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), 1447-1465. doi: 10.1080/00036810903437788.

[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), 1-59. doi: 10.1142/S0129055X09003578.

[9]

V. Koukouloyannis and P. Kevrekidis, On the stability of multibreathers in Klein-Gordon chains, Nonlinearity, 22 (2009), 2269-2285. doi: 10.1088/0951-7715/22/9/011.

[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), 1623-1643. doi: 10.1088/0951-7715/7/6/006.

[11]

R. S. MacKay and J.-A. Sepulchre, Stability of discrete breathers, Localization in Nonlinear Lattices (Dresden, 1997), Physica D, 119 (1998), 148-162. doi: 10.1016/S0167-2789(98)00073-6.

[12]

J. L. Marín and S. Aubry, Finite size effects on instabilities of discrete breathers, Physica D, 119 (1998), 163-174.

[13]

A. Mielke and C. Patz, Dispersive stability of infinite-dimensional Hamiltonian systems on lattices, Applicable Analysis, 89 (2010), 1493-1512. doi: 10.1080/00036810903517605.

[14]

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

[15]

M. Peyrard, S. Cuesta-López and G. James, Modelling DNA at the mesoscale: A challenge for nonlinear science?, Nonlinearity, 21 (2008), T91-T100. doi: 10.1088/0951-7715/21/6/T02.

[16]

M. Peyrard, S. Cuesta-López and G. James, Nonlinear analysis of the dynamics of DNA breathing, J. Biol. Phys., 35 (2009), 73-89. doi: 10.1007/s10867-009-9127-2.

[17]

M. Peyrard, Nonlinear dynamics and statistical physics of DNA, Nonlinearity, 17 (2004), R1-R40. doi: 10.1088/0951-7715/17/2/R01.

[18]

J.-A. Sepulchre and R. S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity, 10 (1997), 679-713. doi: 10.1088/0951-7715/10/3/006.

[19]

D. Treschev, Travelling waves in FPU lattices, Discrete and Continuous Dynamical Systems A, 11 (2004), 867-880. doi: 10.3934/dcds.2004.11.867.

[20]

G. Weber, Sharp DNA denaturation due to solvent interaction, Europhys. Lett., 73 (2006), 806. doi: 10.1209/epl/i2005-10466-6.

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), 235-255.

[2]

S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, Lattice Dynamics (Paris, 1995), Physica D, 103 (1997), 201-250. doi: 10.1016/S0167-2789(96)00261-8.

[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), 271-298.

[4]

D. Bambusi, Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 9 (1996), 433-457. doi: 10.1088/0951-7715/9/2/009.

[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), 471-490.

[6]

G. James, Centre manifold reduction for quasilinear discrete systems, J. Nonlinear Sci., 13 (2003), 27-63. doi: 10.1007/s00332-002-0525-x.

[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), 1447-1465. doi: 10.1080/00036810903437788.

[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), 1-59. doi: 10.1142/S0129055X09003578.

[9]

V. Koukouloyannis and P. Kevrekidis, On the stability of multibreathers in Klein-Gordon chains, Nonlinearity, 22 (2009), 2269-2285. doi: 10.1088/0951-7715/22/9/011.

[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), 1623-1643. doi: 10.1088/0951-7715/7/6/006.

[11]

R. S. MacKay and J.-A. Sepulchre, Stability of discrete breathers, Localization in Nonlinear Lattices (Dresden, 1997), Physica D, 119 (1998), 148-162. doi: 10.1016/S0167-2789(98)00073-6.

[12]

J. L. Marín and S. Aubry, Finite size effects on instabilities of discrete breathers, Physica D, 119 (1998), 163-174.

[13]

A. Mielke and C. Patz, Dispersive stability of infinite-dimensional Hamiltonian systems on lattices, Applicable Analysis, 89 (2010), 1493-1512. doi: 10.1080/00036810903517605.

[14]

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

[15]

M. Peyrard, S. Cuesta-López and G. James, Modelling DNA at the mesoscale: A challenge for nonlinear science?, Nonlinearity, 21 (2008), T91-T100. doi: 10.1088/0951-7715/21/6/T02.

[16]

M. Peyrard, S. Cuesta-López and G. James, Nonlinear analysis of the dynamics of DNA breathing, J. Biol. Phys., 35 (2009), 73-89. doi: 10.1007/s10867-009-9127-2.

[17]

M. Peyrard, Nonlinear dynamics and statistical physics of DNA, Nonlinearity, 17 (2004), R1-R40. doi: 10.1088/0951-7715/17/2/R01.

[18]

J.-A. Sepulchre and R. S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity, 10 (1997), 679-713. doi: 10.1088/0951-7715/10/3/006.

[19]

D. Treschev, Travelling waves in FPU lattices, Discrete and Continuous Dynamical Systems A, 11 (2004), 867-880. doi: 10.3934/dcds.2004.11.867.

[20]

G. Weber, Sharp DNA denaturation due to solvent interaction, Europhys. Lett., 73 (2006), 806. doi: 10.1209/epl/i2005-10466-6.

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