- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Diverging probability-density functions for flat-top solitary waves

Avner Peleg1, Yeojin Chung2, Tomáš Dohnal3, and Quan M. Nguyen1

  • 1Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260, USA
  • 2Department of Mathematics, Southern Methodist University, Dallas, Texas 75275, USA
  • 3Institute for Applied and Numerical Mathematics 2, Universität Karlsruhe, Karlsruhe 76128, Germany

PDF

Phys. Rev. E 80, 026602 – Published 14 August, 2009

DOI: https://doi.org/10.1103/PhysRevE.80.026602

Abstract

We investigate the statistics of flat-top solitary wave parameters in the presence of weak multiplicative dissipative disorder. We consider first propagation of solitary waves of the cubic-quintic nonlinear Schrödinger equation (CQNLSE) in the presence of disorder in the cubic nonlinear gain. We show by a perturbative analytic calculation and by Monte Carlo simulations that the probability-density function (PDF) of the amplitude η exhibits loglognormal divergence near the maximum possible amplitude ηm, a behavior that is similar to the one observed earlier for disorder in the linear gain [A. Peleg et al., Phys. Rev. E 72, 027203 (2005)]. We relate the loglognormal divergence of the amplitude PDF to the superexponential approach of η to ηm in the corresponding deterministic model with linear/nonlinear gain. Furthermore, for solitary waves of the derivative CQNLSE with weak disorder in the linear gain both the amplitude and the group velocity β become random. We therefore study analytically and by Monte Carlo simulations the PDF of the parameter p, where p=η/(1−εsβ/2) and εs is the self-steepening coefficient. Our analytic calculations and numerical simulations show that the PDF of p is loglognormally divergent near the maximum p value.

View figure in article

Article Text

References (62)

  1. D. Pushkarov and S. Tanev, Opt. Commun. 124, 354 (1996).
  2. Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
  3. R. Grimshaw, in Environmental Stratified Flows, edited by R. Grimshaw (Kluwer Academic, Dordrecht, 2001), Chap. 1, pp. 1–27.
  4. L. Gagnon, J. Opt. Soc. Am. A 6, 1477 (1989).
  5. J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, Phys. Rev. E 55, 4783 (1997).
  6. R. Grimshaw, D. Pelinovsky, E. Pelinovsky, and A. Slunyaev, Chaos 12, 1070 (2002).
  7. C. Zhou, X. T. He, and S. Chen, Phys. Rev. A 46, 2277 (1992).
  8. M. Tribeche, S. Ghebache, K. Aoutou, and T. H. Zerguini, Phys. Plasmas 15, 033702 (2008).
  9. W. van Saarloos and P. C. Hohenberg, Phys. Rev. Lett. 64, 749 (1990); Physica D 56, 303 (1992).
  10. B. A. Malomed and A. A. Nepomnyashchy, Phys. Rev. A 42, 6009 (1990).
  11. Y. Kodama and A. Hasegawa, Opt. Lett. 8, 342 (1983).
  12. F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, IEEE Photon. Technol. Lett. 7, 101 (1995).
  13. K.-P. Ho, J. Lightwave Technol. 18, 915 (2000).
  14. A. Peleg, Opt. Lett. 29, 1980 (2004).
  15. Y. Chung and A. Peleg, Nonlinearity 18, 1555 (2005).
  16. Y. Chung and A. Peleg, Phys. Rev. A 77, 063835 (2008).
  17. A. Peleg, Phys. Lett. A 360, 533 (2007).
  18. A. Peleg, Phys. Lett. A 373, 2734 (2009) .
  19. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, CA, 2001).
  20. D. Mihalache, D. Mazilu, M. Bertolotti, and C. Sibilia, J. Opt. Soc. Am. B 5, 565 (1988).
  21. J. Herrmann, Opt. Commun. 87, 161 (1992).
  22. Y. Kartashov, V. A. Vysloukh, A. E. Egorov, and A. S. Zelenina, J. Opt. Soc. Am. B 21, 982 (2004).
  23. S. J. Shwetanshumala, A. Biswas, and S. Konar, J. Electromagn. Waves Appl. 20, 901 (2006).
  24. L. Hong, R. Beech, F. Osman, X. T. He, S. Y. Lou, and H. Hora, J. Plasma Phys. 70, 415 (2004).
  25. E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, Phys. Rev. Lett. 85, 1146 (2000).
  26. B. Tanatar, Europhys. Lett. 51, 261 (2000).
  27. X. Y. Tang and P. K. Shukla, Phys. Rev. A 76, 013612 (2007).
  28. R. Carretero-González, D. J. Frantzeskakis, and P. G. Kevrekidis, Nonlinearity 21, R139 (2008).
  29. Y. S. Kivshar and B. A. Malomed, J. Phys. A 19, L967 (1986).
  30. J. Soneson and A. Peleg, Physica D 195, 123 (2004).
  31. V. Hakim, P. Jakobsen, and Y. Pomeau, Europhys. Lett. 11, 19 (1990).
  32. R. J. Deissler and H. R. Brand, Phys. Rev. Lett. 72, 478 (1994).
  33. I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99 (2002).
  34. P. Coullet and L. Kramer, Chaos 14, 244 (2004).
  35. J. D. Moores, Opt. Commun. 96, 65 (1993).
  36. F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, Opt. Commun. 114, 447 (1995).
  37. N. N. Akhmediev and A. Ankiewicz, in Dissipative Solitons, edited by N. N. Akhmediev and A. Ankiewicz (Springer, Berlin, 2005), Chap. 1, pp. 1–18.
  38. J. N. Kutz, SIAM Rev. 48, 629 (2006).
  39. N. Tzoar and M. Jain, Phys. Rev. A 23, 1266 (1981).
  40. D. Anderson and M. Lisak, Phys. Rev. A 27, 1393 (1983).
  41. G. Yang and Y. R. Shen, Opt. Lett. 9, 510 (1984).
  42. K. Mio, T. Ogino, K. Minami, and S. Takeda, J. Phys. Soc. Jpn. 41, 265 (1976).
  43. E. Mjölhus, J. Plasma Phys. 16, 321 (1976).
  44. D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 798 (1978).
  45. J. Moses, B. A. Malomed, and F. W. Wise, Phys. Rev. A 76, 021802(R) (2007).
  46. T. Kakutani and N. Yamasaki, J. Phys. Soc. Jpn. 45, 674 (1978).
  47. C. Koop and G. Butler, J. Fluid Mech. 112, 225 (1981).
  48. M. J. Ablowitz and H. Segur, Solitons and The Inverse Scattering Transform (SIAM, Philadelphia, 1981).
  49. Ch.-Y. Lee and R. C. Beardsley, J. Geophys. Res. 79, 453 (1974).
  50. R. Grimshaw, E. Pelinovsky, and O. Poloukhina, Nonlinear Process. Geophys. 9, 221 (2002).
  51. T. P. Stanton and L. A. Ostrovsky, Geophys. Res. Lett. 25, 2695 (1998).
  52. D. R. G. Jeans and T. J. Sherwin, Cont. Shelf Res. 21, 1855 (2001).
  53. P. Holloway, E. Pelinovsky, and T. Talipova, in Environmental Stratified Flows, edited by R. Grimshaw (Kluwer Academic, Dordrecht, 2001), Chap. 2, pp. 29–60.
  54. A. Peleg, T. Dohnal, and Y. Chung, Phys. Rev. E 72, 027203 (2005).
  55. N. Akhmediev, A. Ankiewicz, and R. Grimshaw, Phys. Rev. E 59, 6088 (1999).
  56. Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, and M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998); T. Kapitula and B. Sandstede, J. Opt. Soc. Am. B 15, 2757 (1998); J. Yang and D. J. Kaup, SIAM J. Appl. Math. 60, 967 (2000).
  57. R. L. Stratonovich, Introduction to the Theory of Random Noise (Gordon and Breach, New York, 1963).
  58. C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 2004).
  59. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 2007).
  60. H. Yoshida, Phys. Lett. A 150, 262 (1990).
  61. E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, Physica D 87, 201 (1995).
  62. A. C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985).

Outline

Information

Phys. Rev. E 80, 026602– Published 14 August, 2009
  • Received 2 April 2009

© 2009 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation