Abstract
The nonlinear Schrödinger type equations are of tremendous interest in both theory and applications. Various regimes of pulse propagation in optical fibers are modeled by some form of the nonlinear Schrödinger equation.
In this paper we introduce parallel split-step Fourier methods for the numerical simulations of the coupled nonlinear Schrödinger equation that describes the propagation of two orthogonally polarized pulses in a monomode birefringent fibers. These methods are implemented on the Origin 2000 multiprocessor computer. Our numerical experiments have shown that these methods give accurate results and considerable speedup.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Cheney and D. Kincaid, Numerical Mathematics and Computing. Brooks/Cole-Thomson, Belmont, CA, USA, 2004.
J. W. Cooley and J. W. Tukey, An algorithm for the machine computation of complex Fourier series. Mathematics of Computation, 19:297–301, 1965.
M. Frigo and S. G. Johnson, The Fastest Fourier Transform in the West. Technical report, MIT-LCS-TR-728, MIT Laboratory for Computer Science, 1997.
M. Frigo, A fast Fourier transform compiler, preprint, 1999.
R. H. Hardin and F. D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Review Chronicle, 15:423, 1973.
M. S. Ismail and T. R. Taha, Numerical simulation of coupled nonlinear Schrödinger equation. Special Issue of the Journal Mathematics and Computers in Simulation on “Optical Solitons”, 56(6):547–562,2001
C. R. Menyuk, Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes, Journal of the Optical Society of America, B, 5(2):392–402, 1988.
Q. Park and H. J. Shin, Painlevć analysis of the coupled nonlinear Schrödinger equation for polarized optical waves in an isotropic medium, Physical Review E, 59(2):2373–2379, 1999.
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems. Chapman & Hall, London, 1994.
L. Sirovich, Introduction to Applied Mathematics. Springer Verlag, New York, 1988.
M. Snir, S. Otto, S. H. Lederman, D. Walker, and J. Dongarra. MPI: The Complete Reference. MIT Press, London, 1996.
T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical nonlinear Schrödinger equation, Journal of Computational Physics, 55(2):203–230, 1984.
M. Wadati, T. Izuka, and M. Hisakado, A coupled nonlinear Schrödinger equation, Journal of the Physical Society of Japan, 61(7):2241–2245, 1992.
M. Wald, I. M. Uzunov, F. Lederer, and S. Wabnitz, Optimization of soliton transmissions in dispersion-managed fiber links, Optics Communications, 145:48–52, 1998.
J. A. C. Weideman and B. M. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM Journal on Numerical Analysis, 23(3):485–507, 1986.
S. Zoldi, V. Ruban, A. Zenchuk, and S. Burtsev, Parallel implementation of the split-step Fourier method for solving nonlinear Schrödinger systems. SIAM News, 32(1):8–9, 1999.
G. M. Muslu and H. A. Erbay, A split-step Fourier method for the complexmodified Korteweg-de Vries equation. Computers & Mathematics with Applications, 45:503–514, 2003.
R. McLachlan, Symplectic integration of Hamiltonian wave equations. Numerical Mathematics, 66:465–492, 1994.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Taha, T.R., Xu, X. Parallel Split-Step Fourier Methods for the Coupled Nonlinear Schrödinger Type Equations. J Supercomput 32, 5–23 (2005). https://doi.org/10.1007/s11227-005-0183-5
Issue Date:
DOI: https://doi.org/10.1007/s11227-005-0183-5