Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation

Communications on Pure and Applied Mathematics

Volumn 49, Issue 11, 1996, Pages 1175-1255

  Li, Y ^a   McLaughlin, David W ^b   Shatah, Jalal ^b   Wiggins, S ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b NEW YORK UNIVERSITY   (United States)

^c CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030506927     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(199611)49:11<1175::AID-CPA2>3.0.CO;2-9     Document Type: Article

References (63)

1. Arnold, V. I., Instabilities of dynamical systems with many degrees of freedom, Soviet Math. Dokl. 5, 1964, pp. 581-585.
2. Ball, J. M., Saddle point analysis for an ordinary differential equation in a Banach space, and an application to dynamic buckling of a beam, pp. 93-160 in: Nonlinear Elasticity, R. W. Dickey, ed., 1973.
3. Bartuccelli, M. Constantin, P., Doering, C., Gibbon, J. D., and Gisselfält, M., On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D 44, 1990, pp 421-444.
4. Bates, P., and Jones. C., Invariant manifolds for semilinear partial differential equations, pp. 1-38 in: Dynam. Report. Serv. Dynam. Syst. Appl., Vol. 2, Wiley, 1989.
5. Bates, P., and Lu, K., private communication, 1995.
6. Bikbaev, R. F., and Kuksin, S. B., A periodic boundary value problem for the sine-Gordon Equation, its Hamiltonian perturbations, and KAM-deformations of finite-gap tori, Algebra i Analyz. 4, 1992, pp. 42-78.
7. Bimir, B., and Grauer, R., An explicit description of the global attractor of the damped and driven sine-Gordon equation, Comm. Math. Phys. 162, 1994, pp. 539-590.
8. Bishop, A. R., Forest, M. G., McLaughlin, D. W., and Overman II, E. A., A quasiperiodic route to chaos in a near-integrable PDE. Spatio-temporal coherence and chaos in physical systems (Los Alamos, N. M., 1986), Phys. D 23, 1986, pp. 293-328.
9. Cazenave, T., An Introduction to Nonlinear Schrödinger Equations, Vol. 22, IMUFRJ Rio De Janeiro, 1989.
10. Chow, S., Lin, X., and Lu, K., Smooth invariant foliations in infinite-dimensional spaces, J. Differential Equations 94, 1991, pp. 266-291.
11. Chow, S., and Lu, K., Invariant manifolds for flows in Banach spaces, J. Differential Equations 74, 1988, pp. 285-317.
12. Constantin, P., A construction of inertial manifolds, the connection between infinite-dimensional and finite-diemnsional dynamical systems, Contemp. Math. 99, 1989, pp. 27-62.
13. Constantin, P., Foias, C., Nicolaenko, B., and Temam, R., Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, New York, 1989.
14. Doering, C., Gibbon, J. D., Holm, D. D., and Nicolaenko, B., Low-dimensional behaviour in the complex Ginzburg-Landau equation, Nonlinearlity 1, 1988, pp. 279-309.
15. Ercolani, N. M., Forest, M. G., and McLaughlin, D. W., Geometry of the modulational instability I, II., unpublished manuscript, 1987.
16. Ercolani, N. M., Forest, M. G., and McLaughlin, D. W., Geometry of the modulational instability. III. Homoclinic orbits for the periodic sine-Gordon equation, Phys. D 43, 1990. pp. 349-84.
17. Ercolani, N. M., and McLaughlin, D. W., Toward a topological classification of integrable PDEs. The geometry of Hamiltonian systems (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ. No. 22, Springer-Verlag, New York, 1991.
18. Fenichel, N., Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21, 1971, pp. 193-226.
19. Fenichel, N., Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23, 1974, pp. 1109-1137.
20. Fenichel, N., Asymptotic stability with rate conditions. II, Indiana Univ. Math. J. 26, 1977, pp. 81-93.
21. Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31, 1979, pp. 53-98.
22. Fomenko, A. T., Topological classification of all integrable Hamiltonian differentail equations of general type with two degrees of freedom, The Geometry of Hamiltonian Systems, Math. Sci. Res. Inst. Publ., No. 22, Springer-Verlag, New York, 1991.
23. Grauer, R., and Birnir, B., The center manifold and bifurcations of damped and driven sine-Gordon breathers, Phys. D 56, 1992, pp. 165-184.
24. Grebert, B., and Guillot, J. C., Gaps of one-dimensional periodic AKNS systems. Forum Math., 1993, pp. 459-504.
25. Guckenheimer, J., and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, No. 42, Springer-Verlag, New York, 1990.
26. Hadamard, J., Sur literation et les solutions asymptotiques des equations differentielles, Bull. Soc. Math. France 29, 1901, pp. 224-228.
27. Hale, J., Ordinary Differential Equations, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y, 1980.
28. Haller, G., Orbits homoclinic to resonances: the Hamiltonian PDE case, in preparation, 1995.
29. Haller, G., and Wiggins, S., Orbits homoclinic to resonances: the Hamiltonian case, Phys. D 66, 1993, pp. 298-346.
30. Holmes, P., Space- and time-periodic perturbations of the sine-Gordon equation, pp. 164-191 in: Dynamical Systems and Turbulence. Warwick 1980, Lecture Notes in Math. 898, Springer, Berlin-New York, 1981.
31. Holmes, P., and Marsden, J., A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Arch. Rational Mech. Anal. 76, 1981, pp. 135-165.
32. Holmes, P., and Marsden, J., Horeshoes in perturbations of Hamiltonian systems with two degrees of freedom, Comm. Math. Phys. 82, 1982, pp. 523-544.
33. Holmes, P., and Marsden, J., Melnikov's method and Arnol'd diffusion for perturbations of integrable Hamiltonian svstems, J. Math. Phys. 23, 1982, pp. 669-675.
34. Jones, C., Geometric singular perturbation theory, pp. 44-118 in: Dynamical Systems (Montecatini Terme. 1994), Springer, Berlin, 1995.
35. Jones, C., and Kopell, N., Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations, 1994, pp. 64-88.
36. Kaper, C. J. T., and Kopell, N., Tracking invariant manifolds up to exponentially small errors, SIAM J. Math. Anal. 27, 1996. pp. 558-577.
37. Kevrekidis, Y., McLaughlin, D. W., and Winograd, M., Movie.
38. Kovacic, G., Orbits homoclinic to resonances: chaos in a model of the forced and damped sine-Gordon equation, PhD thesis, California Institute of Technology, 1989.
39. Kovacic, G., Dissipative dynamics of orbits homoclinic to a resonance band, Phys. Lett. A 167, 1992, pp. 143-150.
40. Kovacic, G., Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems, J. Dynam. Differential Equations 5, 1993, pp. 559-597.
41. Kovacic, G., Singular perturbation theory for homoclinic orbits in a class of near-integrable dissipative systems, SIAM J. Math. Anal., 1995, pp. 1611-1643.
42. Kovacic, G., and Wiggins, S., Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation, Phys. D 57, 1992, pp. 185-225.
43. Li, Y., Chaotic behavior in PDE's, PhD thesis, Princeton University, 1993.
44. Li, Y., and McLaughlin, D. W., Morse and Mel'nikov functions for NLS PDEs, Comm. Math Phys. 162, 1994, pp. 175-214.
45. Li, Y., and McLaughlin, D. W., Homoclinic orbits and chaos in discretizedperturbed NLS systems I. Homoclinic orbits, J. Nonlinear Sci., to appear, 1997.
46. Li, Y., McLaughlin, D. W., and Wiggins, S., Invariant Manifolds and Their Fibrations for Perturbed NLS PDEs Graph Transform Approach, Springer-Verlag, New York, in preparation, 1995.
47. Li, Y., and Wiggins, S., Homoclinic orbits and chaos in discretized perturbed NLS systems II. Symbol dynamics, J. Nonlinear Sci., to appear, 1997.
48. Magnus, W., and Winkler, W., Hill's Equation, Interscience Publishers John Wiley & Sons, New York, 1966.
49. McLaughlin, D., Whiskered tori for NLS equations, pp. 537-558 in: Important Developments in Soliton Theory, Springer, Berlin, 1993.
50. McLaughlin, D., Whiskered tori and chaotic behavior in nonlinear waves, Proceedings the International Congress Zurich, 1995.
51. McLaughlin, D., and E. A. Overman IL, Whiskered tori for integrable pde's: chaotic behavior in near integrable pde's, pp. 83-203 in: Surveys in Applied Mathematics, Vol. 1, Plenum, New York, 1995.
52. McLaughlin, D., Overman, E. A., Wiggins, S., and Xiong, C., Homoclinic orbits in a four dimensional model of a perturbed NLS equation, pp. 190-287 in: Dynamics Reported: Expositions in Dynamical Systems (New Series), vol. 5, C. K. K. T. Jones, U. Kirchgraber, and H.-O. Walther, eds., Springer-Verlag, Berlin-New York, 1996.
53. Moser, J., Stable and Random Motion in Dynamical Systems, Princeton University Press, Princeton, N.J., 1973.
54. Perron, O., Die stabilitatsfrage bei differentialgleichungssysteme, Math. Z. 32, 1930, pp. 703-728.
55. Poschel, J., and Trubowitz, E., Inverse Spectral Theory, Academic Press Inc., Boston, 1987.
56. Sakamoto, K., Invariant manifolds in singular perturbation problems for ordinary differential equations, Proc. Roy. Soc. Edinburgh Sect. A 116, 1990, pp. 45-78.
57. Sattinger, D. H., and Zurkowski, V. D., Gauge theory of Backlund transformations, Phys. D 26, 1987, pp. 225-250.
58. Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73, 1967, pp. 747-817.
59. Strauss, W., Nonlinear invariant wave equations, pp. 197-249 in: Invariant Wave Equations (Proc "Ettore Majorana" Internat. School of Math. Phys., Erice, 1977), G. Velo and A. S. Wightman, eds., Springer-Verlag, Berlin, 1978.
60. Teman, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
61. Wiggins, S., Global Bifurcations and Chaos. Analytical Methods, Springer-Verlag, New York, 1988.
62. Wiggins, S., Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Springer-Verlag, New York, 1994.
63. Zakharov, V. E., and Shabat, A. B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34, 1, 1972, pp. 62-69.