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Exploring the behavior of solitons on a desktop personal computer

 

S. López-Aguayo, J.P. Ochoa-Ricoux and J.C. Gutiérrez-Vega

 

Photonics and Mathematical Optics Group, Optics Center, Tecnológico de Monterrey, Garza Sada 2501, Monterrey, 64849 México.

 

Recibido el 26 de enero de 2005;
Aceptado el 7 de abril de 2005.

 

Abstract

In recent years, there has been a growing interest in studying and applying nonlinear wave equations and their soliton solutions. In this paper we discuss at undergraduate level a simple finite-difference numerical method for solving nonlinear wave equations. This method is applied for studying the striking behavior of the optical solitons. The procedures presented can be reproduced by enthusiastic students and instructor with a minimum of programming experience. We provide a set of interesting problems that could be taken as starting point to numerically explore the solutions of nonlinear wave equations.

Keywords: Soliton; nonlinear differential equations; numerical methods.

 

Resumen

El interés en el estudio de las ecuaciones de onda no lineales y sus soluciones solitónicas se ha incrementado recientemente. En este artículo estudiamos un método numérico muy simple basado en diferencias finitas para solucionar ecuaciones de onda no lineales. Este método es aplicado en el estudio del comportamiento de los solitones ópticos. Los procedimientos expuestos en este trabajo pueden reproducirse por estudiantes e instructores con un mínimo de experiencia en programación. Adicionalmente, incluimos una lista de problemas que pueden servir como punto de inicio para explorar las interesantes soluciones de las ecuaciones no lineales.

Descriptores: Solitones; ecuaciones diferenciales no lineales; métodos numéricos.

 

PACS: 42.65.-k; 02.70.Bf

 

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Acknowledgments

This research was supported by the Consejo Nacional de Ciencia y Tecnología of Mexico grant 42808 and by the Tecnológico de Monterrey Research Chair in Optics grant CAT-007.

 

References

1. S. Russell, "Reports on waves," Reports of the 14th meeting of the British Association for the Advancement of Science, (London, 1844) p. 311.         [ Links ]

2. N.J. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15 (1965) 240.         [ Links ]

3. M. Remoissenet, Waves Called Solitons (Springer-Verlag, Heidelberg, 1999).         [ Links ]

4. G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).         [ Links ]

5. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon Press, Oxford, 1995).         [ Links ]

6. P.G. Drazin and R.S. Johnson, Solitons: An Introduction (Cambridge University Press, Cambridge, 1996).         [ Links ]

7. A. Hasegawa, Plasma Instabilities and Nonlinear Effects (Springer-Verlag, Berlin, 1983).         [ Links ]

8. P. Gray and C. Scott, Chemical Oscillations and Instabilities (Clarendon, Oxford, 1990).         [ Links ]

9. N.J. Giordano, Computational Physics (Prentice Hall, New Jersey, 1997).         [ Links ]

10. A. García, Numerical Methods for Physics (Prentice Hall, New Jersey, 2000).         [ Links ]

11. D. Kincaid and W. Cheney, Numerical Analysis (Brooks Cole, 1996).         [ Links ]

12. C. Pozrikidis, Numerical Computation in Science and Engineering (Oxford University Press, 1998).         [ Links ]

13. L. Lapidus and G.F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering (Wiley-Interscience, 1999).         [ Links ]

14. M. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation (IEE Press, London, 2000).         [ Links ]

15. V.E. Zhakarov and A.B. Shabat, Sov. Phys. JETP 34 (1972) 62.         [ Links ]

16. R. Hirota, J. Math. Phys. 14 (1973) 805.         [ Links ]

17. T.R. Taha and M.J. Ablowitz, J. Comp. Phys. 55 192 (1984).         [ Links ]

18. T.R. Taha and M.J. Ablowitz, J. Comp. Phys 55 (1984) 203.         [ Links ]

19. W. Malfliet, Am. J. Phys. 60 (1992) 650.         [ Links ]

20. D. Schrader, IEEE J. Quantum. Electron. 31 (1995) 2221.         [ Links ]

21. C. Alwyn, F.Y.F. Scott, Chu, and W.D. McLaughlin, Proc. IEEE 61 (1973) 1443.         [ Links ]

22. Y.A. Berezin and V.I. Karpman, Sov. Phys JETP 19 (1964) 1265.         [ Links ]

23. A. Barone, F. Esposito, C.J. Magee, and A.C. Scott, Riv. Nuovo Cimento 1 (1971) 227.         [ Links ]

24. D.J. Benney and A.C. Newell, J. Math. Phys. 46 (1967) 133.         [ Links ]

25. M.J. Ablowitz and P.A. Carkson, "Solitons, Nonlinear Evolution Equations and Inverse Scattering", (London Mathematical Society Lecture Note Series, Cambridge, 1991) p. 149.         [ Links ]

26. For the case β > 0, the NLSE also admits soliton solutions. The intensity shape of these solitons exhibits a dip in a uniform background, and consequently they are referred to as dark solitons, see Ref. 5.         [ Links ]