Stability of waves in discrete systems

Nonlinearity

Volumn 14, Issue 3, 2001, Pages 533-566

  Kapitula, Todd ^a   Kevrekidis, Panayotis ^b,c  

^a UNIVERSITY OF NEW MEXICO   (United States)

^b PRINCETON UNIVERSITY   (United States)

^c CENTER FOR NONLINEAR STUDIES   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035343981     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/14/3/306     Document Type: Article

References (48)

1. Ablowitz M and Herbst B 1993 Solitons and computation Important Developments in Soliton Theory ed A Fokas and V Zakharov (Berlin: Springer) pp 489-510
2. Ablowitz M and Ladik J 1975 Nonlinear differential-difference equations J. Math. Phys. 16 598-603
3. Ablowitz M and Ladik J 1976 Nonlinear differential-difference equations and Fourier analysis J. Math. Phys. 17 1011-8
4. Aceves A, de Angelis C, Peschel T, Muschall R, Lederer F, Trillo S and Wabnitz S 1996 Discrete self-trapping, soliton interactions and beam steering in nonlinear waveguide arrays Phys. Rev. E 53 1172-89
5. Aceves A, de Angelis C, Trillo S and Wabnitz S 1994 Storage and steering of self-trapped discrete solitons in nonlinear waveguide arrays Opt. Lett. 19 332-4
6. Alexander J, Gardner R and Jones C K R T 1990 A topological invariant arising in the stability of travelling waves J. Reine Angew. Math. 410 167-212
7. Alexeeva N, Barashenkov I and Pelinovsky D 1999 Dynamics of the parametrically driven NLS solitons beyond the onset of the oscillatory instability Nonlinearity 12 103-40
8. Balmforth N, Craster R and Kevrekidis P 2000 Being stable and discrete Physica D 135 212-32
9. Bountis T, Capel H, Kollmann M, Ross J, Bergamin J and van der Weele J 2000 Multibreathers and homoclinic orbits in 1-dimensional nonlinear lattices Phys. Lett. A 268 50-60
10. Darmanyan S, Kobyakov A and Lederer F 1998 Stability of strongly localized excitations in discrete media with cubic nonlinearity J. Exp. Theor. Phys. 86 682-6
11. Daumont I, Dauxois T and Peyrard M 1997 Modulational instability: first step towards energy localization in nonlinear lattices Nonlinearity 10 617-30
12. Eisenberg H, Silberberg Y, Morandotti R, Boyd A and Aitchison J 1998 Discrete spatial optical solitons in waveguide arrays Phys. Rev. Lett. 81 3383-6
13. Fiedler B and Scheurle J 1996 Discretization of Homoclinic Orbits, Rapid Forcing and 'Invisible' Chaos (Memoirs Am. Math. Soc. vol 570) (Providence, RI: American Mathematical Society)
14. Gaididei Y, Mingaleev S, Christiansen P and Rasmussen K 1997 Effects of nonlocal dispersive interactions on self-trapping excitations Phys. Rev. E 55 6141-50
15. Gardner R and Zumbrun K 1998 The gap lemma and geometric criteria for instability of viscous shock profiles Commun. Pure Appl. Math. 51 797-855
16. Grillakis M 1988 Linearized instability for nonlinear Schrödinger and Klein-Gordon equations Commun. Pure Appl. Math. 46 747-74
17. Grillakis M, Shatah J and Strauss W 1987 Stability theory of solitary waves in the presence of symmetry I J. Funct. Anal. 74 160-97
18. Grimshaw R 1995 Weakly nonlocal solitary waves in a singularly perturbed nonlinear Schrödinger equation Stud. Appl. Math. 94 257-70
19. Hakim V and Mallick K 1993 Exponentially small splitting of separatrices, matching in the complex plane and Borel summation Nonlinearity 6 57-70
20. Herbst B and Ablowitz M 1993 Numerical chaos, symplectic integrators and exponentially small splitting distances J. Comput. Phys. 105 122-32
21. Hirth J and Lothe J 1982 Theory of Dislocations (New York: Wiley)
22. Johansson M and Aubry S 1997 Existence and stability of quasiperiodic breathers in the discrete nonlinear Schrödinger equation Nonlinearity 10 1151-78
23. Johansson M and Aubry S 2000 Growth and decay of discrete nonlinear Schrödinger breathers interacting with internal modes or standing-wave phonons Phys. Rev. E 61 5864-79
24. Johansson M and Kivshar Y 1999 Discreteness-induced oscillatory instabilities of dark solitons Phys. Rev. Lett. 82 85-8
25. Jones C K R T 1988 Instability of standing waves for non-linear Schrödinger-type equations Ergod. Theory Dynam. Syst. 8 119-38
26. Kapitula T 1998 Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation Physica D 116 95-120
27. Kapitula T 1999 The Evans function and generalized Melnikov integrals SIAM J. Math. Anal. 30 273-97
28. Kapitula T, Kevrekidis P and Jones C K R T 2001 Soliton internal mode bifurcations: pure power law? Phys. Rev. E 63 036602
29. Kapitula T and Rubin J 2000 Existence and stability of standing hole solutions to complex Ginzburg-Landau equations Nonlinearity 13 77-112
30. Kapitula T and Sandstede B 1998 Stability of bright solitary wave solutions to perturbed nonlinear Schrödinger equations Physica D 124 58-103
31. Kevrekidis P and Jones C K R T 2000 Bifurcation of internal solitary wave modes from the essential spectrum Phys. Rev. E 61 3114-21
32. Kevrekidis P, Jones C K R T and Kapitula T 2000 Exponentially small splitting of heteroclinic orbits: from the rapidly forced pendulum to discrete solitons Phys. Lett. A 269 120-9
33. Kevrekidis P, Bishop A and Rasmussen K 2001 Twisted localized modes Phys. Rev. E 63 036603
34. Kevrekidis P and Weinstein M 2000 Dynamics of lattice kinks Physica D 142 113-152
35. Kivshar Y, Królikowski W and Chubykalo O 1994 Dark solitons in discrete lattices Phys. Rev. E 50 5020-32
36. Kivshar Y, Pelinovsky D, Cretegny T and Peyrard M 1998 Internal modes of solitary waves Phys. Rev. Lett. 80 5032-5
37. Laedke E, Spatschek K and Turitsyn S 1994 Stability of discrete solitons and quasicollapse to intrisically localized modes Phys. Rev. Lett. 73 1055-9
38. Levy H and Lessman F 1992 Finite Difference Equations (New York: Dover)
39. MacKay R and Aubry S 1994 Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators Nonlinearity 7 1623-43
40. McGurn A 1999 Intrinsic localized modes in nonlinear photonic crystal waveguids: Dispersive modes Phys. Lett. A 260 314-21
41. Morandotti R, Peschel U, Aitchison J, Eisenberg H and Silberberg Y 1999 Dynamics of discrete solitons in optical waveguide arrays Phys. Rev. Lett. 83 2726-9
42. Pego R and Weinstein M 1992 Eigenvalues and instabilities of solitary waves Phil. Trans. R. Soc. A 340 47-94
43. Pelinovsky E and Shavratsky S 1980 Breaking of stationary waves in nonlinear dispersive media Phvsica D 1 317-28
44. Peyrard M and Bishop A 1989 Statistical mechanics of a nonlinear model for DNA denaturation Phys. Rev. Lett. 62 2755-8
45. Speight J and Ward R 1994 Kink dynamics in a novel discrete sine-Gordon equation Nonlinearity 7 475-84
46. Ustinov A, Doderer T, Vernik I, Pedersen N, Huebner R and Oboznov V 1993 Experiments with solitons in annular Josephson junctions Physica D 68 41-4
47. van der Zant H, Orlando T, Watanabe S and Strogatz S 1995 Kink propagation in a highly discrete system: observation of phase locking to linear waves Phys. Rev. Lett. 74 174-7
48. Weinstein M 1999 Excitation thresholds for nonlinear localized modes on lattices Nonlinearity 12 673-91