Multisymplectic integration of N-coupled nonlinear Schrödinger equation with destabilized periodic wave solutions

Chaos, Solitons and Fractals

Volumn 41, Issue 2, 2009, Pages 735-751

  Aydin, Ayhan ^a  

^a ATILIM UNIVERSITY   (Turkey)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL TECHNIQUES;

CONSERVATION PROPERTIES; DINGER EQUATION; MULTI-SYMPLECTIC; NUMERICAL RESULTS; PERIODIC SOLUTION; PERIODIC WAVE SOLUTIONS; PREISSMAN SCHEME; STABILITY ANALYSIS;

NONLINEAR EQUATIONS;

EID: 67349109102     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.chaos.2008.03.011     Document Type: Article

References (53)

1. Agrawal G.P. Nonlinear fiber optics. 2nd ed. (1995), Academic Press, New York
2. Akhmediev A., and Ankiewicz A. Solitons: nonlinear pulses and beams (1997), Chapman & Hall, London
3. Scott A.C. Launching a Davydov soliton: I. Soliton analysis. Phys Scr 29 (1984) 279-283
4. Shakravarty S., Ablowitz M.J., Sauer J.R., and Jenkins R.B. Multisoliton interactions and wavelength-division multiplexing. Opt Lett 20 (1995) 136-138
5. Yeh C., and Bergman L. Enhanced pulse compression in a nonlinear fiber by a wavelength division multiplexed optical pulse. Phys Rev E 57 (1998) 2398-2404
6. Kivshar Y.S., and Agrawal G.P. Optical solitons: from fibers to photonic crystals (2003), Academic Press, San Diego
7. Dhar A.K., and Dhas K.P. Fourth order nonlinear evolution equation for two Stokes wave trains in deep water. Phys Fluids A 3 (1991) 3021-3026
8. El Naschie M.S. Deterministic quantum mechanics versus classical mechanical indeterminism. Int J Nonlinear Sci Numer Simul 8 1 (2007) 1-6
9. Chow K.W., and Lai D.W.C. Periodic solutions for systems of coupled nonlinear Schrödinger equations with three and four components. Phys Rev E 68 (2003) 017601
10. Tan B., and Boyd J.P. Coupled-mode envelope solitary waves in a pair of cubic Schrödinger equations with cross modulation: analytical solutions and collisions with applications to Rossby waves. Chaos, Solitons & Fractals 11 (2000) 1113-1129
11. Wazwaz A.-M. A study on linear and nonlinear Schrodinger equations by the variational iteration method. Chaos, Solitons & Fractals 37 (2008) 1136-1142
12. Mandal B., and Chowdhury A.R. Compression, splitting and switching of bright and dark solitons in nonlinear directional coupler. Chaos, Solitons & Fractals 27 (2006) 103-113
13. Herbst B.M., Varadi F., and Ablowitz M.J. Symplectic methods for the nonlinear Schrödinger equation. Math Comput Simul 37 (1994) 353-369
14. Chen J.B., and Qin M.Z. Symplectic and multisymplectic methods for the nonlinear Schrödinger equation. Comput Math Appl 43 (2002) 1095-1106
15. Islas A.L., and Schober C.M. On the preservation of phase structure under multi-symplectic discretization. J Comput Phys 197 (2004) 585-609
16. Sun J.Q., Gu X.Y., and Ma Z.Q. Numerical study of the soliton waves of the coupled nonlinear Schrödinger system. Physica D 196 (2004) 311-328
17. Sun J.Q., and Qin M.Z. Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system. Comput Phys Commun 155 (2003) 221-235
18. Tan B., and Boyd J.P. Stability and long time evolution of the periodic solution to the two coupled nonlinear Schrödinger equations. Chaos, Solitons & Fractals 12 (2001) 721-734
19. Tsang S.C., and Chow K.W. The evolution of periodic waves of the coupled nonlinear Schrödinger equations. Math Comput Simul 66 (2004) 551-564
20. Ablowitz M.J., and Segur H. Solitons and the inverse scattering transform (1981), SIAM, Philadelphia
21. Lamb G.L. Elements of soliton theory (1980), John Wiley & Sons, New York
22. Manakov S.V. On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov Phys JEPT 38 (1974) 248-253 [English Trans.]
23. Zakharov V.E., and Schulman E.I. To the integrability of the system of two coupled nonlinear Schrödinger equations. Physica D 4 (1982) 270-274
24. Xu G.Q., and Li Z.B. On the Painlevé integrability, periodic wave solutions and soliton solutions of generalized coupled higher-order nonlinear Schrödinger equations. Chaos, Solitons & Fractals 26 (2005) 1363-1375
25. Porsezian K., and Kalithasan B. Cnoidal and solitary wave solutions of the coupled higher order nonlinear Schrödinger equation in nonlinear optics. Chaos, Solitons & Fractals 31 (2007) 188-196
26. Kanna T., and Lakshmanan M. Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. Phys Rev Lett 86 (2001) 5043-5046
27. Kanna T., Tsoy E.N., and Akhmediev N. On the solution of multicomponent nonlinear Schrödinger equations. Phys Lett A 330 (2004) 224-229
28. Ankiewicz A., Krolikowski W., and Akhmediev N.N. Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions. Phys Rev E 59 (1999) 6079-6087
29. Akhmediev N., Krolikowski W., and Synder A.W. Partially coherent solitons of variable shape. Phys Rev Lett 81 (1998) 4632-4635
30. Nakkeeran K. Exact dark soliton solitions for a family of N coupled nonlinear Schrödinger equations in optical fiber media. Phys Rev E 64 (2001) 046611
31. Hairer E., Lubich C., and Wanner G. Geometric numerical integration: structure-preserving algorithms for ordinary differential equations vol. 31 (2002), Springer, New York
32. Bridges T.J., and Reich S. Numerical methods for Hamiltonian PDEs. J Phys A: Math Gen 39 (2006) 5287-5320
33. Kong L., Liu R., and Zheng X. A survey on symplectic and multi-symplectic algorithms. Appl Math Comput 186 (2007) 670-684
34. Leimkuhler B., and Reich S. Simulating Hamiltonian dynamics (2004), Cambridge University, Cambridge
35. Islas A.L., and Schober C.M. Backward error analysis for multisymplectic discretization of Hamiltonian PDEs. Math Comput Simul 69 (2005) 290-303
36. Moore B., and Reich S. Backward error analysis for multi-symplectic integration methods. Numer Math 95 (2003) 625-652
37. Ascher U., and McLachlan R. Multisymplectic box schemes and the Korteweg-de Vries equation. Appl Numer Math 48 (2004) 255-269
38. Zhao P.F., and Qin M.Z. Multisymplectic geometry and multisymplectic Preissman scheme for the KdV equation. J Phys A: Math Gen 33 (2000) 3613-3626
39. Aydi{dotless}n A, Karasözen B. Multi-symplectic integration of coupled nonlinear Schrödinger system with soliton solutions. Int J Comput Math [accepted for publication], doi:10.1080/00207160701713615.
40. Aydi{dotless}n A. Geometric integrators for coupled nonlinear Schrödinger equation. Ph.D. thesis, Department of Mathematics, Middle East Technical University; 2005. p. 34-5.
41. Kostov N.A., and Uzunov I.M. New kinds of periodical waves in birefringent optical fibers. Opt Commun 89 (1992) 389-392
42. Chow K.W., and Lai D.W.C. Periodic solutions of coupled nonlinear Schrödinger equations with five and six components. Phys Rev E 65 (2003) 026613
43. Aydi{dotless}n A., and Karasözen B. Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions. Comput Phys Commun 177 (2007) 566-583
44. Bridges T.J., and Derks G. The symplectic Evans matrix and the instability of solitary waves and fronts. Arch Rat Mech Anal 156 (2001) 1-87
45. Bridges T.J., and Reich S. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys Lett A 284 (2001) 184-193
46. Porsezian K. Bilinearization of coupled nonlinear Schrödinger type equations: integrability and solitons. J Nonlinear Math Phys 5 2 (1998) 126-131
47. Marsden J., Partick G.W., and Shkoller S. Multi-symplectic geometry, variational integrators, and nonlinear PDEs. Commun Math Phys 199 (1998) 351-395
48. Reich S. Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations. J Comput Phys 156 (1999) 1-27
49. Wang Y.S., and Qin M.Z. Multisymplectic Schemes for the nonlinear Klein-Gordon equation. Math Comput Model 36 (2002) 963-977
50. Chen J.B., and Qin M.Z. A multisymplectic variational integrator for the nonlinear Schrödinger equation. Numer Methods PDEs 18 (2002) 523-536
51. Newton P.K., and Keller J.B. Stability of plane wave solutions of nonlinear systems. Wave Motion 10 (1988) 183-191
52. Benjamin T.B., and Feir J.E. The disintegration of wave trains on deep water. Part I. Theory. J Fluid Mech 27 (1967) 417-430
53. Akhmediev N.N. Nonlinear physics - déjá vu in optics. Nature 413 (2001) 267-268