82
Views
17
CrossRef citations to date
0
Altmetric
Original Articles

Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein–Gordon lattices

, &
Pages 1313-1334 | Received 31 Aug 2009, Accepted 31 Jan 2010, Published online: 26 Jul 2010

References

  • Bambusi , D and Penati , T . 2010 . Continuous approximation of ground states in DNLS lattices . Nonlinearity , 23 : 143 – 157 .
  • Mawhin , J and Willem , M . 1989 . Critical Point Theory and Hamiltonian Systems , New York : Springer-Verlag .
  • Bak , SN . 2004 . The constrained minimization method in the problem of the oscillations of a chain of nonlinear oscillators . Mat. Fiz. Anal. Geom. , 11 : 263 – 273 .
  • Bak , SN and Pankov , AA . 2004 . On periodic oscillations of an infinite chain of linearly coupled nonlinear oscillators . Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki , 9 : 13 – 16 .
  • Duistermaat , JJ . 1984 . “ Bifurcation of periodic solutions near equilibrium points of Hamiltonian systems ” . In Bifurcation Theory and Applications (Montecatini, 1983) , Edited by: Salvadori , Luigi . 57 – 105 . Berlin : Springer .
  • Bambusi , D and Paleari , S . 2001 . Families of periodic solutions of resonant PDEs . J. Nonlinear Sci. , 11 : 69 – 87 .
  • Bambusi , D and Paleari , S . 2002 . Families of periodic orbits for some PDE's in higher dimensions . Commun. Pure Appl. Anal. , 1 : 269 – 279 .
  • Kalyakin , LA . 1989 . Long-wave asymptotics. Integrable equations as the asymptotic limit of nonlinear systems . Uspekhi Mat. Nauk , 44 : 5 – 34 . 247
  • Kirrmann , P , Schneider , G and Mielke , A . 1992 . The validity of modulation equations for extended systems with cubic nonlinearities . Proc. R. Soc. Edinb. Sect. A , 122 : 85 – 91 .
  • Schneider , G and Wayne , CE . 2000 . “ Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam Model ” . In International Conference on Differential Equations , Vols 1 and 2 (Berlin, 1999) 390 – 404 . River Edge, NJ : World Science Publishing .
  • Bambusi , D , Carati , A and Ponno , A . 2002 . The nonlinear Schrödinger equation as a resonant normal form . Discrete Contin. Dyn. Syst. Ser. B , 2 : 109 – 128 .
  • Bambusi , D and Ponno , A . 2006 . On metastability in FPU . Commun. Math. Phys. , 264 : 539 – 561 .
  • Bambusi , D , Carati , A and Penati , T . 2009 . Boundary effects on the dynamics of chains of coupled oscillators . Nonlinearity , 22 : 923 – 946 .
  • Friesecke , G and Pego , RL . 1999 . Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit . Nonlinearity , 12 : 1601 – 1627 .
  • Friesecke , G and Pego , RL . 1999/2002/2004 . Solitary waves on Fermi–Pasta–Ulam lattices: I. Qualitative properties, renormalization and continuum limit: II. Linear implies nonlinear stability: III. Howland-type Floquet theory: IV. Proof of stability at low energy . Nonlinearity , 12/15/17 : 1601 – 1627/1343-1359/207-227/229-251 .
  • Hoffman , A and Wayne , CE . 2008 . Counter-propagating two-soliton solutions in the Fermi–Pasta–Ulam lattice . Nonlinearity , 21 : 2911 – 2947 .
  • Mizumachi , T and Pego , RL . 2008 . Asymptotic stability of Toda lattice solitons . Nonlinearity , 21 : 2099 – 2111 .
  • Iooss , G . 2000 . Travelling waves in the Fermi–Pasta–Ulam lattice . Nonlinearity , 13 : 849 – 866 .
  • Iooss , G and Kirchgässner , K . 2000 . Travelling waves in a chain of coupled nonlinear oscillators . Commun. Math. Phys. , 211 : 439 – 464 .
  • Iooss , G and James , G . 2005 . Localized waves in nonlinear oscillator chains . Chaos , 15 : 015113 15
  • Iooss , G and Pelinovsky , DE . 2006 . Normal form for travelling kinks in discrete Klein–Gordon lattices . Phys. D , 216 : 327 – 345 .
  • James , G . 2003 . Centre manifold reduction for quasilinear discrete systems . J. Nonlinear Sci. , 13 : 27 – 63 .
  • James , G and Sire , Y . 2005 . Travelling breathers with exponentially small tails in a chain of nonlinear oscillators . Commun. Math. Phys. , 257 : 51 – 85 .
  • James , G and Sire , Y . 2008 . “ Center manifold theory in the context of infinite one-dimensional lattices ” . In The Fermi-Pasta-Ulam Problem , Edited by: Gallavotti , Giovanni . 208 – 238 . Berlin : Springer .
  • Sire , Y . 2005 . Travelling breathers in Klein–Gordon lattices as homoclinic orbits to p-tori . J. Dyn. Differ. Equ. , 17 : 779 – 823 .
  • MacKay , RS and Aubry , S . 1994 . Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators . Nonlinearity , 7 : 1623 – 1643 .
  • Bambusi , D . 1996 . Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators . Nonlinearity , 9 : 433 – 457 .
  • Sepulchre , JA and MacKay , RS . 1997 . Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators . Nonlinearity , 10 : 679 – 713 .
  • Aubry , S , Kopidakis , G and Kadelburg , V . 2001 . Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems . Discrete Contin. Dyn. Syst. Ser. B , 1 : 271 – 298 .
  • Weinstein , MI . 1999 . Excitation thresholds for nonlinear localized modes on lattices . Nonlinearity , 12 : 673 – 691 .
  • Berestycki , H and Lions , PL . 1983 . Nonlinear scalar field equations: I. Existence of a ground state . Arch. Ration. Mech. Anal. , 82 : 313 – 345 .
  • Coleman , S , Glaser , V and Martin , A . 1978 . Action minima among solutions to a class of Euclidean scalar field equations . Commun. Math. Phys. , 58 : 211 – 221 .
  • Grillakis , M , Shatah , J and Strauss , W . 1987 . Stability theory of solitary waves in the presence of symmetry: I . J. Funct. Anal. , 74 : 160 – 197 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.