Chaotic and turbulent behavior of unstable one-dimensional nonlinear dispersive waves

Journal of Mathematical Physics

Volumn 41, Issue 6, 2000, Pages 4125-4153

  Cai, David ^a   McLaughlin, David W ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034399126     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.533337     Document Type: Article

References (42)

1. M. Cross and P. Hohenberg, "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112 (1993).
2. D. W. McLaughlin and E. A. Overman, "Whiskered tori for integrable pdes and chaotic behavior in near integrable pdes," Surveys Appl. Math. 1, 83-203 (1995).
3. D. Cai, D. W. McLaughlin, and K. T. R. McLaughlin, "The nonlinear Schrödinger equation as both a pde and a dynamical system," Handbook in Dynamical Systems (to appear).
4. D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak, "Spectral bifurcations in dispersive wave turbulence," Proc. Natl. Acad. Sci. USA 96, 14 216-14 221 (1999).
5. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, "Method for solving the KdV equation," Phys. Rev. Lett. 19, 1095-1097 (1967).
6. P. D. Lax, "Integrals of nonlinear equations of evolution and solitary waves," Commun. Pure Appl. Math. 21, 467-490 (1968).
7. V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).
8. T. B. Benjamin and J. F. Feir, "The disintegration of wave trains on deep water," J. Fluid Mech. 27, 417-430 (1967).
9. V. E. Zakharov, Ph.D. thesis, 1966.
10. N. Ercolani, M. Forest, and D. McLaughlin, "Geometry of the modulational instability, III: Homoclinic orbits for the periodic sine Gordon equation," Physica D 43, 348-384 (1990).
11. Y. Li and D. W. McLaughlin, "Morse and Melnikov functions for NLS pdes," Commun. Math. Phys. 162, 175-214 (1994).
12. V. I. Arnold, "Instabilities of systems with several degrees of freedom," Sov. Math. Dokl. 5, 581-585 (1964).
13. N. M. Ercolani and D. W. McLaughlin, "Toward a topological classification of integrable PDE's," The Geometry of Hamiltonian Systems (Springer-Verlag, New York, 1991).
14. A. R. Bishop, M. G. Forest, D. W. McLaughlin, and E. A. Overman, II, "A quasiperiodic route to chaos in a near-integrable pde," Physica D 23, 293-328 (1986).
15. J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer-Verlag, New York, 1983).
16. S. Wiggins, Global Bifurcations and Chaos: Analytical Methods (Springer-Verlag, New York, 1988).
17. Y. Li, D. W. McLaughlin, J. Shatah, and S. Wiggins, "Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation," Commun. Pure Appl. Math. 49, 1175-1255 (1996)
18. D. W. McLaughlin and J. Shatah, "Melnikov analysis for PDE's," in Dynamical Systems and Probabilistic Methods in Partial Differential Equations, Berkeley, CA, 1994 (American Mathematical Society, Providence, RI, 1996). pp. 51-100.
19. S. Smale, "Differential dynamical systems," Bull. Am. Math. Soc. 73, 747-817 (1967).
20. Y. Li, "Smale horseshoes and symbol dynamics in perturbed NLS equations," J. Nonlinear Sci. 9, 363-415 (1998).
21. C. Zeng, "Homoclinic orbits for perturbed NLS equation," to appear in Comm. Pure Appl. Math.
22. G. Haller, "Homoclinic jumping in the perturbed nonlinear Schrödinger equation," Commun. Pure Appl. Math. 52, 1-47 (1999).
23. G. Haller, Chaos Near Resonance (Springer-Verlag, New York, 1999).
24. G. Cruz Pacheco, C. D. Levermore, and B. Luce, "Melnikov methods for Pde's with applications to perturbed nonlinear Schrödinger equation," Physica D, to appear, 2000.
25. M. G. Forest, C. G. Goedde, and A. Sinha, "Chaotic transport and integrable instabilities in a near-integrable, many-particle, Hamiltonian lattice," Physica D 67, 347-386 (1993).
26. M. J. Ablowitz and B. M. Herbst, "Numerically induced chaos in the nonlinear Schrödinger equation," Phys. Rev. Lett. 62, 2065-2068 (1989).
27. M. J. Ablowitz, B. M. Herbst, and C. M. Schober, "The nonlinear Schrödinger equation: asymmetric perturbations, traveling waves and chaotic structures," Math. Comput. Simul. 43, 3-12 (1997).
28. A. Calini, N. Ercolani, D. W. McLaughlin, and C. M. Schober, "Melnikov analysis of numerically induced chaos in the nonlinear Schrödinger equation," Physica D 80, 227-260 (1996).
29. M. J. Ablowitz, J. Hammack, D. Henderson, and C. Schober, in preparation.
30. R. E. Blahut, Principles and Practice of Information Theory (Addison-Wesley, New York, 1988).
31. D. Cai, D. McLaughlin, and J. Shatah, "Spatiotemporal chaos and effective stochastic dynamics for a near integrable nonlinear system," Phys. Lett. A 253, 280-286 (1999).
32. D. Cai, D. McLaughlin, and J. Shatah, "Spatial temporal chaos for perturbed NLS equations," in preparation.
33. S. Zaleski, "A stochastic model for the large scale dynamics of some fluctuating interfaces," Physica D 34, 427-438 (1989).
34. V. Yakhot, "Large scale properties of unstable systems governed by the Kuramoto Sivashinski equation," Phys. Rev. A 24, 642-644 (1981).
35. K. Sneppen, J. Krug, M. H. Jensen, C. Jayaprakash, and T. Bohr, "Dynamic scaling and crossover analysis for the Kuramoto-Sivashinsky equation," Phys. Rev. A 46, R7351-7354 (1992).
36. G. Goren, J. Eckmann, and I. Procaccia, "Scenario for the onset of space-time chaos," Phys. Rev. E 57, 4106-4134 (1998).
37. A. J. Majda, D. W. McLaughlin, and E. G. Tabak, "A one dimensional model for dispersive wave turbulence," J. Nonlinear Sci. 7, 9-44 (1997).
38. V. E. Zakharov, "Kolmogorov spectra in weak turbulence problems," in Basic Plasma Physics II (North-Holland, New York, 1984), pp. 1-36.
39. R. Jordan, B. Turkington, and C. Zirbel, "A mean field statistical theory for the NLS equation," Physica D 137, 353-378 (2000).
40. R. Jordan and C. Josserand, "Self organization in nonlinear wave turbulence," Phys. Rev. E 61, 1527-1539 (2000).
41. A. C. Newell and V. E. Zakharov, "Optical turbulence," in Turbulence: A Tentative Dictionary, edited by P. Tabeling and O. Cardoso (Plenum, New York, 1995), pp. 59-66.
42. A. Majda, I. Timofeyev, and E. Vanden Eijnden, "Models for stochastic climate prediction," Proc. Natl. Acad. Sci. USA 96, 14 687-14 691 (1999).