Localized waves in nonlinear oscillator chains

Chaos

Volumn 15, Issue 1, 2005, Pages

  Looss, Gérard ^a   James, Guillaume ^b  

^a INSTITUT UNIVERSITAIRE DE FRANCE   (France)

^b INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17744372552     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1836151     Document Type: Review

References (66)

1. G. Friesecke and R. L. Pego, "Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit," Nonlinearity 12, 1601-1627 (1999).
2. G. Friesecke and J. A. Wards, "Existence theorem for solitary waves on lattices," Commun. Math. Phys. 161, 391-418 (1994).
3. G. Iooss, "Traveling waves in the Fermi-Pasta-Ulam lattice," Nonlinearity 13, 849-866 (2000).
4. G. Iooss and K. Kirchgässner, "Traveling waves in a chain of coupled nonlinear oscillators," Commun. Math. Phys. 211, 439-464 (2000).
5. M. Remoissenet, "Low-amplitude breather and envelope solitons in quasione-dimensional physical models," Phys. Rev. B 33, 2386-2392 (1986).
6. A. Tsurui, "Wave modulations in anharmonic lattices," Prog. Theor. Phys. 48, 1196-1203 (1972).
7. G. James and Y. Sire, "Traveling breathers with exponentially small tails in a chain of nonlinear oscillators," Commun. Math. Phys. (to be published).
8. Y. Sire and G. James, "Traveling breathers in Klein-Gordon chains," C. R. Acad. Sci., Ser. I: Math. 338, 661-666 (2004).
9. E. Fermi, J. Pasta, and S. Ulam, "Studies of nonlinear problems," Technical Report LA-1940, Los Alamos National Laboratory, 1955.
10. T. A. Kontorova and Ya. J. Frenkel, "On the theory of plastic deformation and twinning," Zh. Eksp. Teor. Fiz. 8, 1340-1348 (1938).
11. A. J. Sievers and S. Takeno, "Intrinsic localized modes in anharmonic crystals," Phys. Rev. Lett. 61, 970-973 (1988).
12. T. Dauxois, M. Peyrard, and A. R. Bishop, "Dynamics and thermodynamics of a nonlinear model for DNA denaturation," Phys. Rev. E 47, 684-695 (1993).
13. S. R. Bickham, S. A. Kiselev, and A. J. Sievers, "Stationary and moving intrinsic localized modes in one-dimensional monoatomic lattices with cubic and quartic anharmonicity," Phys. Rev. B 47, 14206-14211 (1993).
14. S. Flach and C. R. Willis, "Movability of localized excitations in nonlinear discrete systems: a separatrix problem," Phys. Rev. Lett. 72, 1777-1781 (1994).
15. K. W. Sandusky, J. B. Page, and K. E. Schmidt, "Stability and motion of intrinsic localized modes in nonlinear periodic lattices," Phys. Rev. B 46, 6161-6168 (1992).
16. S. Takeno and K. Hori, "A propagating self-localized mode in a onedimensional lattice with quartic anharmonicity," J. Phys. Soc. Jpn. 59, 3037-3040 (1990).
17. S. Aubry and T. Crétégny, "Mobility and reactivity of discrete breathers," Physica D 119, 34-46 (1998).
18. T. Dauxois, M. Peyrard, and C. R. Willis, "Discreteness effects on the formation and propagation of breathers in nonlinear Klein-Gordon equations," Phys. Rev. E 48, 4768-4778 (1993).
19. J. C. Eilbeck, "Numerical simulations of the dynamics of polypeptide chains and proteins," Computer Analysis for Life Science-Progress and Challenges in Biological and Synthetic Polymer Research, edited by C. Kawabata and A. R. Bishop (Ohmsha, Tokyo, 1986), pp. 12-21.
20. S. Flach and K. Kladko, "Moving discrete breathers," Physica D 127, 61-72 (1999).
21. S. Flach and C. R. Willis, "Discrete breathers," Phys. Rep. 295, 181-264 (1998).
22. S. Aubry, G. Kopidakis, and V. Kadelburg, "Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems," Discrete Contin. Dyn. Syst. 1, 271-298 (2001).
23. G. James, "Center manifold reduction for quasilinear discrete systems," J. Nonlinear Sci. 13, 27-63 (2003).
24. R. S. MacKay and S. Aubry, "Proof of existence of breathers for timereversible or Hamiltonian networks of weakly coupled oscillators," Non-linearity 7, 1623-1643 (1994).
25. A. J. Sievers and S. Takeno, "Intrinsic localized modes in anharmonic crystals," Phys. Rev. Lett. 61, 970-973 (1988).
26. M. Kastner and J-A. Sepulchre, "Effective Hamiltonian for traveling discrete breathers in the FPU chain," Discrete Com. Dyn. Sys. B (to be published).
27. Z7 J-A. Sepulchre, "Energy barriers in coupled oscillators: from discrete kinks to discrete breathers," Proceedings of the Conference on Localization and Energy Transfer in Nonlinear Systems, June 17-21, 2002, San Lorenzo de El Escorial, Madrid, Spain, edited by L. Vazquez, R. S. MacKay, and M-P. Zorzano (World Scientific, Singapore, 2003), pp. 102-129.
28. R. S. MacKay and J-A. Sepulchre, "Effective Hamiltonian for travelling discrete breathers," J. Phys. A 35, 3985-4002 (2002).
29. Y. Sire and G. James, "Numerical computation of travelling breathers in Klein-Gordon chains," preprint 04-24 lab.MIP, UPS Toulouse.
30. A. A. Aigner, A. R. Champneys, and V. M. Rothos, "A new barrier to the existence of moving kinks in Frenkel-Kontorova lattices," Physica D 186, 148-170 (2003).
31. D. B. Duncan, J. C. Eilbeck, H. Feddersen, and J. A. D. Wattis, "Solitons on lattices," Physica D 68, 1-11 (1993).
32. A. V. Savin, Y. Zolotaryuk, and J. C. Eilbeck, "Moving kinks and nanopterons in the nonlinear Klein-Gordon lattice," Physica D 138, 267-281 (2000).
33. M. J. Ablowitz and Z. Musslimani, "Discrete spatial solitons in a diffraction-managed nonlinear waveguide array: A unified approach," Physica D 184, 276-303 (2003).
34. J. Szeftel, G. Huang, and V. Konotop, "On the existence of moving breathers in one-dimensional anharmonic lattices," Physica D 181, 215-221 (2003).
35. M. J. Ablowitz, Z. Musslimani, and G. Biondini, "Methods for discrete solitons in nonlinear lattices," Phys. Rev. E 65, 026602 (2002).
36. H. Feddersen, "Solitary wave solutions to the discrete nonlinear Shrödinger equation," in Nonlinear Coherent Structures in Physics and Biology, Lecture Notes in Physics, edited by M. Remoissenet and M. Peyrard (Springer-Verlag, Berlin, 1991), Vol. 393, pp. 159-167.
37. M. J. Ablowitz and J. F. Ladik, "Nonlinear differential-difference equations and Fourier analysis," J. Math. Phys. 17, 1011-1018 (1976).
38. S. Flach, Y. Zolotaryuk, and K. Kladko, "Moving kinks and pulses: an inverse method," Phys. Rev. E 59, 6105-6115 (1999).
39. A. Berger, R. S. MacKay, and V. M. Rothos, "A criterion for nonpersistence of travelling breathers for perturbations of the Ablowitz-Ladik lattice," Discrete Contin. Dyn. Syst., Ser. B 4, 911-920 (2004).
40. N. Flytzanis, St. Pnevmatikos, and M. Remoissenet, "Kink, breather, and asymmetric envelope or dark solitons in nonlinear chains. I monoatomic chain," J. Phys. C 18, 4603-4629 (1985).
41. J. Giannoulis and A. Mielke, "The nonlinear Schrodinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities," Non-linearity 17, 551-565 (2004).
42. J. Giannoulis and A. Mielke, "Macroscopic dynamics of an oscillator chain described by the nonlinear Schrödinger equation, preprint" (Institut für Analysis, Dynamik und Modellierung, University of Stuttgart, 2004).
43. K. Kirchgässner, "Wave solutions of reversible systems and applications," J. Diff. Eqns. 45, 113-127 (1982).
44. A. Mielke, "Reduction of quasilinear elliptic equations in cylindrical domains with applications," Math. Methods Appl. Sci. 10, 51-66 (1988).
45. A. Vanderbauwhede and G. Iooss, "Center manifold theory in infinite dimensions," Dynamics Reported 1 New Series, edited by C. Jones, U. Kirchgaber, and H. Walter (Springer, New York, 1992), pp. 125-163.
46. E. Lombardi, "Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems," Lecture Notes in Mathematics (Springer-Verlag, Berlin, 2000), Vol. 1741.
47. G. Iooss and E. Lombardi, "Polynomial normal forms with exponentially small remainder for analytic vector fields," Preprint Institut Non Linéaire de Nice, 2004.
48. Y. Sire, "Travelling breathers in Klein-Gordon lattices as homoclinic orbits to p-tori," preprint 04-19 lab.MIP, UPS Toulouse.
49. G. Friesecke and K. Matthies, "Atomic-scale localization of high-energy solitary waves on lattices," Physica D 171, 211-220 (2002).
50. D. Smets and M. Willem, "Solitary waves with prescribed speed on infinite lattices," J. Funct. Anal. 149, 266-275 (1997).
51. L. A. Kalyakin, "Long wave asymptotics, Imegrable equations as asymptotic limits of nonlinear systems," Russ. Math. Surveys 44 (1), 3-42 (1989).
52. G. Schneider and C. E. Wayne, "Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model," in B. Fiedler, K. Gröger and J. Sprekels, editors, International Conference on Differential Equations Appl., 5 (1), 69-82 (1998).
53. E. Lombardi, "Phenomena beyond all orders and bifurcations of reversible homoclinic connections near higher resonances," Peyresq Lectures on Nonlinear Phenomena, edited by R. Kaiser and J. Montaldi (World Scientific, Singapore, 2000), pp. 161-200.
54. G. Friesecke and R. L. Pego, "Solitary waves on FPU lattices: II. Linear implies nonlinear stability," Nonlinearity 15, 1343-1359 (2002).
55. G. Friesecke and R. L. Pego, "Solitary waves on FPU lattices: III. Howland-type Floquet theory," Nonlinearity 17, 207-227 (2004).
56. G. Friesecke and R. L. Pego, "Solitary waves on FPU lattices: IV. Proof of stability at low energy," Nonlinearity 17, 229-251 (2004).
57. G. James, "Existence of breathers on FPU lattices," C. R. Acad. Sci., Ser. I: Math. 332, 581-586 (2001).
58. G. James and P. Noble, "Breathers on diatomic Fermi-Pasta-Ulam lattices," Physica D 196, 124-171 (2004).
59. R. Livi, M. Spicci, and R. S. MacKay, "Breathers on a diatomic FPU chain," Nonlinearity 10, 1421-1434 (1997).
60. J. Mallet-Paret, "The global structure of traveling waves in spatially discrete systems," J. Dyn. Differ. Equ. 11, 49-127 (1999).
61. J. Mallet-Paret, "The Fredholm alternative for functional differential equations of mixed type," J. Dyn. Differ. Equ. 11, 1-47 (1999).
62. J. Mallet-Paret and S. Verduyn Lunel, "Exponential dichotomies and Wiener-Hopf factorizations for mixed type functional differential equations," preprint (2001).
63. J. Guckenheimer and P. Holmes, "Nonlinear oscillations, dynamical systems, and bifurcation of vector fields," Appl. Math. Sci. (Springer, Berlin, 1983), Vol. 42.
64. C. Elphick, E. Tirapegui, M. E. Brachet, P. Coullet, and G. Iooss, "A simple global characterization for normal forms of singular vector fields," Physica D 29, 95-127 (1987).
65. G. Iooss and M. Adelmeyer, "Topics in bifurcation theory and applications," Adv. Ser. Nonlinear Dynamics (World Scientific, Singapore, 1998), Vol. 3.
66. G. Iooss and M-C Pérouème, "Perturbed homoclinic solutions in reversible 1:1 resonance vertor fields," J. Diff. Eqns. 102, 62-88 (1993).