volume | PIER Journals

Abstract

This paper is a theoretical study of solitons in multidimensions, also known as optical bullets, that is governed by the nonlinear Schrodinger's equation in 1 + 3 dimensions. The parameter dynamics of such multidimensional solitons has been obtained. The study is extended to obtain the adiabatic evolution of soliton parameters in presence of the perturbation terms. Furthermore, the parameter dynamics for the vector multidimensional solitons and including the presence of the perturbation terms has been obtained.

1. Abdullaev, F., S. Darmanyan, and P. Khabibullaev, Optical Solitons, Springer Verlag, New York, 1993.

2. Ablowitz, M. J. and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, USA, 1981.

3. Ablowitz, M. J., G. Biondini, and S. Blair, "Localized multidimensional optical pulses in non-resonant quadratic materials," Mathematics and Computers in Simulation, Vol. 56, No. 6, 511-519, 2001. Google Scholar

4. Afanasjev, V. V., P. L. Chu, and Y. S. Kivshar, "Breathing spatial solitons in non-Kerr media," Optics Letters. Google Scholar

5. Akhmediev, N. N., V. I. Korneev, and R. F. Nabiev, "Modulation instability of the ground state of the nonlinear wave equation: optical machine gun," Optics Letters, Vol. 15, 393-395, 1992. Google Scholar

6. Akhmediev, N. N. and J. M. Soto-Crespo, "Generation of a train of three-dimensional nonlinear Schrodinger equation," Physical Review A, Vol. 47, No. 2, 1358-1364, 1993. Google Scholar

7. Akhmediev, N. N. and A. Ankiewicz, Solitons Nonlinear Pulses and Beams, Chapman and Hall, UK, 1997.

8. Akhmediev, N. N., A. Ankiewicz, and R. Grimshaw, "Hamiltonian-versus-energy diagrams in soliton theory," Physical Review E, Vol. 59, No. 5, 6088-6096, 1999. Google Scholar

9. Akhmediev, N. N., "Spatial solitons in Kerr and Kerr-like media," Optical and Quantum Electronics, Vol. 30, 535-569, 1998. Google Scholar

10. Bang, O., J. J. Rasmussen, and P. L. Christiansen, "Subcritical localization in the discrete nonlinear Schr¨odinger equation with arbitrary power nonlinearity," Nonlinearity, Vol. 7, No. 1, 205-218, 1994. Google Scholar

11. Biswas, A., "Perturbation of solitons due to power law nonlinearity," Chaos, Solitons and Fractals, Vol. 12, No. 3, 579-588, 2001. Google Scholar

12. Biswas, A., "Solitons in multiple-core couplers," Journal of Nonlinear Optical Physics and Materials, Vol. 10, No. 3, 2001. Google Scholar

13. Biswas, A., "Solitons in nonlinear fiber arrays," Journal of Electromagnetic Waves and Applications, Vol. 15, No. 9, 1189-1196, 2001. Google Scholar

14. Biswas, A., "Perturbation of solitons with non-Kerr law nonlinearity," Chaos, Solitons and Fractals, Vol. 13, No. 4, 815-823, 2002. Google Scholar

15. Blagoeva, A. B., S. G. Dinev, A. A. Dreischuh, and A. Naidenov, "Light bullets formation in a bulk media," IEEE Journal of Quantum Electronics, Vol. 27, 2060-2062, 1991. Google Scholar

16. Busalev, V. S. and V. E. Grikurov, "Simulation of instability of bright solitons for NLS with saturating nonlinearity," Mathematics and Computers in Simulation, Vol. 56, No. 6, 539-546, 2001. Google Scholar

17. Desyatnikov, A., A. Maimistov, and B. Malomed, "Three dimensional spinning solitonsin dispersive media with cubicquintic nonlinearity," Physical Review E, Vol. 61, No. 3, 3107-3113, 2000. Google Scholar

18. Emundson, D. E. and R. H. Enns, "The particle-like nature of colliding light bullets," Physical Review A, Vol. 51, No. 3, 2491-2498, 1995. Google Scholar

19. Emundson, D. E. and R. H. Enns, "Bistable light bullets," Optics Letters, Vol. 17, 586-588, 1992. Google Scholar

20. Enns, R. H. and D. E. Emundson, "Guide to fabricating bistablesoliton-supporting media," Physical Review A, Vol. 47, No. 5, 4524-4527, 1993. Google Scholar

21. Enns, R. H. and S. S. Rangnekar, "Bistable spheroidal optical solitons," Physical Review A, Vol. 45, No. 5, 3354-3357, 1992. Google Scholar

22. Enns, R. H. and S. S. Rangnekar, "Variational approach to bistable solitary waves in d dimensions," Physical Review E, Vol. 48, No. 5, 3998-4007, 1993. Google Scholar

23. Faddeev, L. D. and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer Verlag, New York, 1987.

24. Fokas, A. S. and V. E. Zakharov, Important Developments in Soliton Theory, Springer Verlag, New York, 1993.

25. Forest, M. G., D. W. McLaughlin, D. J. Muraki, and O. C.Wright, "Nonfocusing instabilities in coupled, integrable nonlinear Schr¨odinger pdes," Journal of Nonlinear Science, Vol. 10, 291-331, 2000. Google Scholar

26. Gagnon, L. and P. A. Belanger, "Adiabatic amplification of optical solitons," Physical Review A, Vol. 43, No. 11, 6187-6193, 1991. Google Scholar

27. Ghidaglia, J. M. and J. C. Saut, "Nonexistence of travelling wave solutions to nonelliptic nonlinear Schr¨odinger equations," Journal of Nonlinear Science, Vol. 6, No. 2, 139-145, 1996. Google Scholar

28. Hasegawa, A. and Y. Kodama, Solitons in Optical Communications, Oxford University Press, UK, 1995.

29. Hayata, K. and M. Koshiba, "Solution of self-trapped multidimensional optical beams by Galerkin’s method," Optics Letters, Vol. 17, 841-843, 1992. Google Scholar

30. Hayata, K. and M. Koshiba, "Bright-dark solitary-wave solutions of a multi-dimensional nonlinear Schrodinger’s equation," Physical Review E, Vol. 48, No. 3, 2312-2315, 1993. Google Scholar

31. Hayata, K. and M. Koshiba, "Algebraic solitary-wave solutions of a nonlinear Schr¨odinger’s equation," Physical Review E, Vol. 51, No. 2, 1499-1502, 1995. Google Scholar

32. Infield, E. and G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge University Press, 1990.

33. Jovanoski, Z. and R. A. Sammut, "Propagation of Gaussian beams in a nonlinear saturable medium," Physical Review E, Vol. 50, No. 5, 4087-4093, 1994. Google Scholar

34. Karpman, V. I. and A. G. Shagalov, "Stability of solitons described the nonlinear Schrodinger-type equations with higher order dispersion," Physica D, Vol. 144, No. 1-2, 194-210, 2000. Google Scholar

35. Kivshar, Y. S. and B. A. Malomed, "Dynamics of solitons in nearly integrable systems," Reviews in Modern Physics, Vol. 61, No. 4, 765-915, 1989. Google Scholar

36. Kivshar, Y. S., "Bright and dark spatial solitons in non-Kerr media," Optical and Quantum Electronics, Vol. 30, 535-569, 1998. Google Scholar

37. Kivshar, Y. S. and B. Luther-Davis, "Dark optical solitons: physics and applications," Physics Reports, Vol. 298, 81-197, 1998. Google Scholar

38. Kivshar, Y. S. and D. E. Pelinovsky, "Self-focusing and transverse instabilities of solitary waves," Physics Reports, Vol. 331, 117-195, 2000. Google Scholar

39. Manassah, J. T., P. L. Baldeck, and R. R. Alfano, "Self-focusing, self-phase modulation, and diffraction in bulk homogenous material," Optics Letters, Vol. 13, 1090-1092, 1988. Google Scholar

40. Mcleod, R., K. Wagner, and S. Blair, "(3+1)-dimensional optical soliton dragging logic," Physical Review A, Vol. 52, No. 4, 3254-3278, 1995. Google Scholar

41. Mihalache, D., D. Mazilu, L. C. Crasovan, B. A. Malomed, and F. Lederer, "Three-dimensional spinning solitons in the cubicquintic nonlinear medium," Physical Review E, Vol. 61, No. 6, 7142-7145, 2000. Google Scholar

42. Mihalache, D., M. Bertolotti, and C. Cibilia, "Nonlinear wave propagation in planar structures," Progress in Optics, Vol. XXVII, 228-309, 1989. Google Scholar

43. Pelinovsky, D. E., V. V. Afanasjev, and Y. S. Kivshar, "Nonlinear theory of oscillating, decaying and collapsing solitons in the generalized nonlinear Schr¨odinger’s equation," Physical Review E, Vol. 53, No. 2, 1940-1953, 1996. Google Scholar

44. Silberberg, Y., "Collapse of optical pulses," Optics Letters, Vol. 15, 1282-1284, 1990. Google Scholar

45. Sonar, S., J. Kumar, and P. K. Sen, "Suppression of soliton instability by higher order nonlinearity in long haul optical communication systems," Journal of Nonlinear Optical Physics and Materials, Vol. 8, No. 4, 497-502, 1999. Google Scholar

46. Skarka, V., V. I. Berezhiani, and R. Miklaszewski, "Generation of light spatiotemporal solitons from asymmetric pulses in saturating nonlinear media," Physical Review E, Vol. 59, No. 1, 1270-1273, 1999. Google Scholar

47. Snyder, A. W. and D. J. Mitchell, "Spatial solitons of the powerlaw nonlinearity," Optics Letters, Vol. 18, No. 2, 101-103, 1993. Google Scholar

48. Sulem, C. and P. L. Sulem, The Nonlinear Schrodinger’s Equation, Springer Verlag, New York, 1999.

49. Yang, J. and D. J. Kaup, "Stability and evolution of solitary waves in perturbed generalized nonlinear Schrodinger’s equation," SIAM Journal of Applied Mathematics, Vol. 60, No. 3, 967-989, 2000. Google Scholar

50. Zhou, C., X. T. He, and T. Cai, "Pattern structures on generalized nonlinear Schrodinger’s equations with various nonlinear terms," Physical Review E, Vol. 50, No. 5, 4136-4155, 1994. Google Scholar