A KAM theorem for Hamiltonian networks with long ranged couplings

Nonlinearity

Volumn 20, Issue 6, 2007, Pages 1313-1342

  Geng, Jiansheng ^a   Yi, Yingfei ^b  

^a NANJING UNIVERSITY   (China)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34248588202     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/20/6/001     Document Type: Article

References (37)

1. Aubry S 1995 Anti-integrability in dynamical and variational problems Physica D 86 284-96
2. Aubry S 1997 Breathers in nonlinear lattices: existence, linear stability and quantization Physica D 103 201-50
3. 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. B 1 271-98
4. Bambusi D 1996 Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators Nonlinearity 9 433-57
5. Bambusi D and Vella D 2002 Quasi periodic breathers in Hamiltonian lattices with symmetries Discrete Cont. Dyn. Syst. B 2 389-99
6. Bourgain J 1998 Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations Ann. Math. 148 363-439
7. Bourgain J 1995 Construction of periodic solutions of nonlinear wave equations in higher dimension Geom. Funct. Anal. 5 629-39
8. Bourgain J 1994 Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE Int. Math. Res. Not. 8 475-97
9. Bourgain J 2005 Green's Function Estimates for Lattice Schrödinger Operators and Applications (Annals of Mathematics Studies vol 158) (Princeton, NJ: Princeton University Press)
10. Bourgain J 1999 Nonlinear Schrödinger Equations (Park City Series vol 5) (Providence, RI: American Mathematical Society)
11. Braun O M and Kivshar Y S 1998 Nonlinear dynamics of the Frenkel-Kontorova model Phys. Rep. 306 1-108
12. Carpio A and Bonilla L L 2003 Oscillatory wave fronts in chains of coupled nonlinear oscillators Phys. Rev. E 67 056621
13. Chierchia L and You J 2000 KAM tori for 1D nonlinear wave equations with periodic boundary conditions Commun. Math. Phys. 211 498-525
14. Craig W and Wayne C E 1993 Newton's method and periodic solutions of nonlinear wave equations Commun. Pure. Appl. Math. 46 1409-98
15. Dauxois T, Peyrard M and Willis C R 1992 Localised breather-like solution in a discrete Klein-Gordon model and application to DNA Physica D 57 267-82
16. Fröhlich J, Spencer T and Wayne C E 1986 Localization in disordered nonlinear dynamical systems J. Stat. Phys. 42 247-74
17. Geng J and You J 2005 A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions J. Diff. Eqns 209 1-56
18. Geng J and You J 2006 A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces Commun. Math. Phys. 262 343-72
19. Geng J and Yi Y 2007 Quasi-periodic solutions in a nonlinear Schrödinger equation J. Diff. Eqns 233 512-42
20. James G 2003 Centre manifold reduction for quasilinear discrete systems J. Nonlin. Sci. 13 27-63
21. Johansson M and Aubry S 1997 Existence and stability of quasiperiodic breathers in the discrete nonlinear Schrödinger equation Nonlinearity 10 1151-78
22. Kalosakas G et al 2004 Sequence-specific thermal fluctuations identify start sites for DNA transcription Europhys. Lett. 68 127-33
23. Kresse O and Truskinovsky L 2003 Mobility of lattice defects: discrete and continuum approaches J. Mech. Phys. Solids 51 1305-32
24. Kuksin S B 1987 Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum Funct. Anal. Appl. 21 192-205
25. Kuksin S B 1993 Nearly Integrable Infinite Dimensional Hamiltonian Systems (Lecture Notes in Mathematics vol 1556) (Berlin: Springer)
26. Kuksin S B and Pöschel J 1996 Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schrödinger equation Ann. Math. 143 149-79
27. Li Y and Yi Y 2004 Persistence of lower dimensional tori of general types in Hamiltonian systems Trans. Am. Math. Soc. 357 1565-600
28. Mackay R S and Aubry S 1994 Proof of existence of breathers of time-reversible or Hamiltonian networks of weakly coupled oscillators Nonlinearity 7 1623-43
29. Pöschel J 1996 Quasi-periodic solutions for a nonlinear wave equation Comment. Math. Helvetici 71 269-96
30. Pöschel J 1996 A KAM theorem for some nonlinear partial differential equations Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 119-48
31. Pöschel J 1990 Small divisors with spatial structure in infinite dimensional Hamiltonian systems Commun. Math. Phys. 127 351-93
32. Prohofsky E W, Lu K C, Van Zandt L L and Putnam B F 1979 Breathing models and induced resonant melting of the double helix Phys. Lett. A 70 492-4
33. Wayne C E 1990 Periodic and quasi-periodic solutions for nonlinear wave equations via KAM theory Commun. Math. Phys. 127 479-528
34. Xu J and You J 2001 Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition J. Math. Pures Appl. 80 1045-67
35. You J 1999 Perturbation of lower dimensional tori for Hamiltonian systems J. Diff. Eqns 152 1-29
36. Yuan X 2002 Construction of quasi-periodic breathers via KAM technique Commun. Math. Phys. 226 61-100
37. Zhang C T 1987 Soliton excitations in deoxyribonucleic acid (DNA) double helices Phys. Rev. A 35 886-91