Explicit symplectic methods for the nonlinear schrodinger equation

Communications in Computational Physics

Volumn 6, Issue 3, 2009, Pages 639-654

  Guan, Hu ^a   Jiao, Yandong ^a   Liu, Ju ^b   Tang, Yifa ^a  

^a CHINESE ACADEMY OF SCIENCES   (China)

^b XI'AN JIAOTONG UNIVERSITY   (China)

Author keywords

Explicit symplectic method; L L N splitting; Nonlinear Schrodinger equation

Indexed keywords

EID: 67650941643     PISSN: 18152406     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (45)

1. M.J. Ablowtiz and J.E Ladik, Nonlinear differential-difference equations, J. Math. Phys., 16 (1975), 598-603.
2. M.J. Ablowtiz and J.E Ladik, Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys., 17 (1976), 1011-1018.
3. M.J. Ablowtiz, B. Prinari and A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, Cambridge, 2004.
4. S.K. Adhikari, Dissipation-managed soliton in a quasi-one-dimensional Bose-Einstein condensate, Laser Phys. Lett., 3(11) (2006), 553-557.
5. G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms, J. Stat. Phys., 74 (1994), 1117-1143.
6. T.B. Benjamin and J.E. Feir, The disintegration of wavetrains in deep water, part 1, J. Fluid Mech., 27 (1967), 417-430.
7. J.B. Chen, M.Z. Qin and Y.E Tang, Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation, Comput. Math. Appl., 43(8/9) (2002), 1095-1106.
8. M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a non-linear Schrödinger equation, J. Comput. Phys., 44(2) (1981), 277-288.
9. R.K. Dodd, J.C. Eibeck, J.D. Gibbon and H.C. Morris, Solitons and Nonlinear Wave Equation, Academic Press, 1982.
10. K. Feng, Collected Works of Feng Kang (II), National Defence Industry Press, Beijing, 1995.
11. Q.D. Feng and Y.E Tang, Calculation of formal energies of symplectic schemes for Hamiltonian systems with applications, (2001), preprint.
12. E. Forest and R.D. Ruth, Fourth-order symplectic integration, Physica D, 43 (1990), 105-117.
13. B.Y. Guo, The convergence of numerical method for nonlinear Schrodinger equation, J. Comput. Math., 4(2) (1986), 121-130.
14. E. Hairer, Backward analysis of numerical integrators and symplectic methods, Annals Numer. Math., 1 (1994), 107-132.
15. E. Hairer, Ch. Lubich and G. Wanner, Geometric Numerical Integration, Springer, 2002.
16. A. Hasegawa, Solitons in Optical Communications, Clarendon Press, Oxford, 1995.
17. B.M. Herbst, J.L. Morris and A.R. Mitchell, Numerical experience with the nonlinear Schrödinger equation, J. Comput. Phys., 60(2) (1985), 282-305.
18. B.M. Herbst, E Varadi and M.J. Ablowitz, Symplectic methods for the nonlinear Schrödinger equation, Math. Comput. Simul., 37 (1994), 353-369.
19. A.L. Islas, D.A. Karpeev and CM. Schober, Geometric integrators for the nonlinear Schrödinger equation, J. Comput. Phys., 173 (1) (2001), 116-148.
20. S. Jiménez, P. Pascual, C. Aguirre and L. Vazquez, A panoramic view of some perturbed nonlinear wave equations, Inter. J. Bifur. Chaos, 14(1) (2004), 1-40.
21. T. Kapitula and P. Kevrekidis, Stability of waves in discrete systems, Nonlinearity, 14 (2001), 533-566.
22. V.V. Konotop and V.E. Vekslerchik, Randomly modulated dark soliton, J. Phys. A: Math. Gen., 24 (1991), 767-785.
23. G.L. Lamb, Elements of Soliton Theory, John Wiley and Sons, New York, 1980.
24. X.S. Liu, Y.Y. Qi and P.Z. Ding, Periodic and chaotic breathers in the nonlinear Schrodinger equation, Chin. Phys. Lett., 21 (11) (2004), 2081-2084.
25. X.S. Liu, Y.Y. Qi, J.F. He and P.Z. Ding, Recent progress in symplectic algorithms for use in quantum systems, Commun. Comput. Phys., 2 (2007), 1-53.
26. B.A. Malomed, Variational methods in nonlinear fiber optics and related fields, Progress in Optics., 43 (2002), 69-191.
27. J.W. Miles, An envelope soliton problem, SIAM J. Appl. Math., 41(2) (1981), 227-230.
28. M. Onorato, A.R. Osborne, M. Serio and S. Bertone, Freak waves in random oceanic sea states, Phys. Rev. Lett., 86 (2001), 5831-5834.
29. V.M. Pérez-García and X.Y Liu, Numerical methods for the simulation of trapped nonlinear Schrodinger systems, Appl. Math. Comput., 144(2-3) (2003), 215-235.
30. R.I. McLachlan, Symplectic integration of Hamiltonian wave equation, Numer. Math., 66 (1994), 465-492.
31. V.M. Pérez-García, H. Michinel, J.I. Cirac, M. Lewenstein and P. Zoller, Dynamics of Bose- Einstein condensates: Variational solutions of the Gross-Pitaevskii equations, Phys. Rev. A, 56(2) (1997), 1424-1432.
32. J.I. Ramos, Linearly implicit methods for the nonlinear Schrödinger equation in nonhomogeneous media, Appl. Math. Comput., 133(1) (2002), 1-28.
33. J.M. Sanz-Serna and J.G. Verwer, Conservative and nonconservative schemes for the solution of the nonlinear Schrodinger equation, IMA J. Numer. Anal., 6(1) (1986), 25-42.
34. J.C. Scovel, Symplectic numerical integration of Hamiltonian systems, The Geometry of Hamiltonian Systems, MSRI Series 22, T. Ratiu (Ed.), Springer-Verlag, New York, 1991, pp. 463-496.
35. J.C. Scovel and Y.F. Tang, Symplectic methods for the nonlinear Schrödinger equation, (1996), preprint.
36. C. Sulem and PL. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, New York, 1999.
37. M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A, 146(6) (1990), 319-323.
38. Y.F. Tang, Formal energy of a symplectic scheme for Hamiltonian systems and its applications (I), Comput. Math. Appl., 27(7) (1994), 31-39.
39. Y.F. Tang, J.W. Cao, X.T. Liu and Y.C. Sun, Symplectic methods for the Ablowitz-Ladik discrete nonlinear Schrodinger equation, J. Phys. A: Math. Theor., 40(10) (2007), 2425-2437.
40. Y.F. Tang, V.M. Pérez-García and L. Vázquez, Symplectic methods for the Ablowitz-Ladik model, Appl. Math. Comput., 82 (1997), 17-38.
41. Y.F. Tang, L. Vazquez, F. Zhang and V.M. Pérez-Garcia, Symplectic methods for the nonlinear Schrödinger equation, Comput. Math. Appl., 32(5) (1996), 73-83.
42. H. Yoshida, Construction of higher-order symplectic integrators, Phys. Lett. A, 150(5-7) (1990), 262-268.
43. V.E. Zakharov, Collapse and self-focusing of Langmuir waves, Handbook of Plasma Physics, 2 (1984), 81-121.
44. V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, 34(1) (1972), 62-69.
45. F. Zhang, V.M. Pérez-García and L. Vázquez, Numerical simulation of nonlinear Schrödionger systems: A new conservative scheme, Appl. Math. Comput., 71(2-3) (1995), 165-177.