Discretizations, integrable systems and computation

Journal of Physics A: Mathematical and General

Volumn 34, Issue 48, 2001, Pages 10671-10693

  Ablowitz, M J ^a   Herbst, B M ^b   Schober, C M ^c  

^a UNIVERSITY OF COLORADO   (United States)

^b STELLENBOSCH UNIVERSITY   (South Africa)

^c OLD DOMINION UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035824495     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/34/48/330     Document Type: Article

References (32)

1. Ablowitz M J and Ladik J F 1976 Nonlinear differential-difference equations and Fourier analysis J. Math. Phys. 17 1011-8
2. Ablowitz M J and Segur H 1981 Solitons and the Inverse Scattering Transform (Philadelphia, PA: SIAM)
3. Ablowitz M J and Clarkson P A 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge: Cambridge University Press)
4. Ablowitz M J and Herbst B M 1989 On homoclinic boundaries in the nonlinear Schrödinger equation Proc. CRM Workshop on Hamiltonian Systems, Transformation Groups and Spectral Transform Methods ed J Harnad and J E Marsden (Montreal: CRM) pp 121-32
5. Ablowitz M J and Herbst B M 1990 On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation SIAM J. Appl. Math. 50 339-51
6. Ablowitz M J, Schober C M and Herbst B M 1993 Numerical chaos, roundoff errors and homoclinic manifolds Phys. Rev. Lett. 71 2683-6
7. Ablowitz M J and Schober C M 1994 Effective chaos in the nonlinear Schrödinger equation Contemp. Math. 172 253-68
8. Ablowitz M J, Herbst B M and Schober C M 1996 Computational chaos in the nonlinear Schrödinger equation without homoclinic crossings Physica A 228 212-35
9. Ablowitz M J, Herbst B M and Schober C M 1996 On the numerical solution of the sine-Gordon equation. I. Integrable discretizations and homoclinic manifolds J. Comput. Phys. 126 299-314
10. Ablowitz M J, Herbst B M and Schober C M 1997 On the numerical solution of the sine-Gordon equation. II. Performance of numerical schemes J. Comput. Phys. 131 354-67
11. Ablowitz M J, Hammack J, Henderson D and Schober C M 2001 Long-time dynamics of the modulational instability of deep water waves Physica D 152-3 416-33
12. Ablowitz M J and Schober C M 2001 Chaotic dynamics of modulational instability in optical fibres Opt. Lett. at press
13. Bobenko A and Pinkall U 1994 Discrete surfaces with constant negative curvature SFB 288 preprint no 127
14. McLaughlin D W, Bishop A R and Overman E A 1986 Coherence and chaos in the driven damped sine-Gordon equation: measurement of the soliton spectrum Physica D 19 1-41
15. McLaughlin D W, Bishop A R, Forest M G and Overman E A 1986 A quasi-periodic route to chaos in a near-integrable PDE Physica D 23 293-328
16. McLaughlin D W, Ercolani N and Forest M G 1986 Modulational instabilities of periodic sine-Gordon waves: a geometric analysis Lect. Appl. Math. 23 149-66
17. McLaughlin D W 1992 Whiskered Tori for the NLS Equation. Solitons in the 80's ed T Fokas and V Zakharov
18. Ercolani N, Forest M G and McLaughlin D W 1990 Geometry of the modulational instability part iii: homoclinic orbits for the periodic sine-Gordon equation Physica D 43 349-84
19. McLaughlin D W and Overman E A 1994 Whiskered tori for integrable PDE's: chaotic behaviour in near integrable PDE's Surv. Appl. Math. 1 83-200
20. McLaughlin D W and Y Li 1994 Morse and Melnikov functions for NLS PDE's Commun. Math. Phys. 162 175-214
21. Forest M G and Lee J E 1986 Geometry and modulation theory for the periodic nonlinear Scrhödinger equation Oscillation Theory, Computation and Methods of Compensated Compactness (IMA in Math and its Appl.) vol 2 ed C Dafermos, J L Ercksen, D Kinderlehrer and M Slemrod (Berlin: Springer) pp 35-70
22. McLaughlin D W, Cai D and Shatah J 1999 Spatial-temporal chaos and effective stochastic dynamics for a near integrable nonlinear system Phys. Lett. 253 280-6
23. McLaughlin D W and Cai D 2000 Chaotic and turbulent behaviour of unstable 1-D nonlinear dispersive waves J. Math. Phys. 41 4125-53
24. McLaughlin D W and Schober C M 1992 Chaotic and homoclinic behaviour for numerical discretizations of the nonlinear Schrödinger equation Physica D 57 447-65
25. Calini A, Ercolani N M, McLaughlin D W and Schober C M 1996 Mel'nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation Physica D 89 227-60
26. Herbst B M and Ablowitz M J 1989 Numerically induced chaos in the nonlinear Schrödinger equation Phys. Rev. Lett. 62 2065-8
27. Hirota R 1977 Nonlinear partial difference equations III; discrete sine-Gordon equation J. Phys. Soc. Japan 43 2079-86
28. Molina M I and Tsironis G P 1993 Dynamics of self-trapping in the discrete nonlinear Schrödinger equation Physica D 65 267-73
29. Sanz-Serna J M and Calvo M P 1994 Numerical Hamiltonian Problems (London: Chapman and Hall)
30. Islas A L, Karpeev D A and Schober C M 2001 Geometric integrators for the nonlinear Schrödinger equation J. Comput. Phys. 172 1-33
31. Schober C M 1999 Symplectic integrators for the Ablowitz-Ladik discrete nonlinear Schrödinger equation Phys. Lett. A 259 140-51
32. Channell P J and Scovel C 1990 Symplectic integration of Hamiltonian systems Nonlinearity 3 1-13