OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 17, Iss. 10 — Oct. 1, 2000
  • pp: 1870–1879

Resolution of a stochastic weakly damped nonlinear Schrödinger equation by a multilevel numerical method

Guy Moebs and Roger Temam  »View Author Affiliations


JOSA A, Vol. 17, Issue 10, pp. 1870-1879 (2000)
http://dx.doi.org/10.1364/JOSAA.17.001870


View Full Text Article

Acrobat PDF (231 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We consider a stochastic nonlinear Schrödinger equation related to signal propagation in waveguides and optical fibers. We first describe the modeling of the problem and the desired objectives concerning the transmission. We then present a new multilevel numerical method for its solution, which is based on a separation between low and high frequencies. We show that this method gives results of the same quality with significantly shorter CPU time compared with those of the other numerical methods commonly presented in the literature.

© 2000 Optical Society of America

OCIS Codes
(000.3870) General : Mathematics
(000.4430) General : Numerical approximation and analysis
(070.4340) Fourier optics and signal processing : Nonlinear optical signal processing
(190.3100) Nonlinear optics : Instabilities and chaos
(190.4370) Nonlinear optics : Nonlinear optics, fibers

Citation
Guy Moebs and Roger Temam, "Resolution of a stochastic weakly damped nonlinear Schrödinger equation by a multilevel numerical method," J. Opt. Soc. Am. A 17, 1870-1879 (2000)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-10-1870


Sort:  Author  |  Year  |  Journal  |  Reset

References

  1. B. M. Herbst, J. Ll. Morris, and A. R. Mitchell, “Numerical experience with the nonlinear Schrödinger equation,” J. Comput. Phys. 60, 282–305 (1985).
  2. B. M. Herbst and J. A. C. Weideman, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).
  3. T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger, equations,” J. Comput. Phys. 55, 203–230 (1984).
  4. M. Lax, J. H. Batteh, and G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
  5. C. Foias, O. Manley, and R. Temam, “Sur l’interaction des petits and grands tourbillons dans les écoulements turbulents,” C. R. Acad. Sci. Ser. I: Math. 307, 497–500 (1987).
  6. C. Foias, O. Manley, and R. Temam, “On the interaction of small and large eddies in two-dimensional turbulent flows,” Math. Modell. Numer. Anal. 22, 93–114 (1988).
  7. C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, and E. S. Titi, “On the computation of inertial manifolds,” Phys. Lett. A 131, 433–436 (1988).
  8. T. Dubois, F. Jauberteau, and R. Temam, Dynamical Multilevel Methods and the Numerical Solution of Turbulence (Cambridge U. Press, Cambridge, UK, 1998).
  9. C. Foias, G. Sell, and R. Temam, “Variétés inertielles des équations différentielles dissipatives,” C. R. Acad. Sci. Ser. I: Math. 301, 139–142 (1985).
  10. C. Foias, G. Sell, and R. Temam, “Inertial manifolds for nonlinear evolutionary equations,” J. Diff. Eqns. 77, 309–353 (1988).
  11. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences Series, Vol. 68 of 2nd ed. (Springer-Verlag, New York, 1997).
  12. G. Moebs, “An efficient parallelization of a multilevel split-step Fourier method for a weakly damped nonlinear Schrödinger equation,” (available from the author).
  13. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  14. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, UK, 1995).
  15. R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).
  16. J.-M. Ghidaglia, “Finite dimension behavior for the weakly damped driven Schrödinger equation,” Ann. Inst. Henri Poincaré 5, 365–405 (1988).
  17. O. Goubet, “Regularity of the attractor for the weakly damped driven Schrödinger equation,” Applic. Anal. 60, 99–119 (1996).
  18. I. Moise, R. Rosa, and X. Wang, “Attractors for non-compact semigroups via energy equations,” Nonlinearity 11, 1369–1393 (1998).
  19. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics (Springer-Verlag, New York, 1988).
  20. D. Gottlieb and S. A. Orszag, “Numerical analysis of spectral methods: theory and applications,” Vol. 26 of CBMS–NSF Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, Pa., 1997).
  21. J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
  22. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited