Computational Solutions for the Korteweg–deVries Equation in Warm Plasma
El-Shewy Emad *, Abdelwahed Hesham G., Abd-El-Hamid Hamdi M.
Theoretical Physics Group
Faculty of Science, Mansoura University, Mansoura, Egypt
*e-mail: e_k_el_shewy@mans.edu.eg; emadshewy@yahoo.com
Received:
Received: 5 October 2009; revised: 2 May 2010; accepted: 6 May 2010; published online: 29 June 2010
DOI: 10.12921/cmst.2010.16.01.13-18
OAI: oai:lib.psnc.pl:710
Abstract:
The reductive perturbation method has been employed to derive the Korteweg–deVries (KdV) equation for small but finite amplitude electrostatic ion-acoustic waves. An algebraic method with computerized symbolic computation, which greatly exceeds the applicability of the existing tanh, extended tanh methods in obtaining a series of exact solutions of the KdV equation. Numerical studies have been made using plasma parameters to reveal different solutions, i.e. bell-shaped solitary pulses, rational pulses and solutions with singularity at a finite points called “blowup” solutions in addition to the propagation of explosive pulses. The result of the present investigation may be applicable to some plasma environments, such as ionosphere.
Key words:
explosive solutions, ion-acoustic waves, reductive perturbation, symbolic computations
References:
[1] G. Whitham, Linear and nonlinear waves. New York, Wiley (1974).
[2] R. Davidson, Methods in nonlinear plasma theory. New York, Academic Press (1972).
[3] S. K. El-Labany, J. Plasma Phys. 50, 495 (1993)..
[4] M. Sokała, J. Computational Methods in Science and Technology 14(2), 111 (2008).
[5] H. Washimi, T. Taniuti, J Phys Rev Lett. 17. 996 (1966).
[6] T. Taniuti, C. Wei, J Phys Soc Jpn 24, 491 (1968).
[7] S.A. Elwakil, E.K. El-Shewy, R. Sabry, Int. J. Nonlinear Sci. Numer. Simulation 5, 403 (2004).
[8] S.A. Elwakil, M.T. Attia, M.A. Zahran, E.K. El-Shewy, H.G. Abdelwahed, Z. Naturforsch. 61a, 316 (2006).
[9] M.J. Ablowitz, P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering. Cambridge, Cambridge University Press (1991).
[10] W. Malfliet, Solitary wave solutions of nonlinear wave equations. American Journal of Physics 60(7), 650-654 (1992).
[11] E. Fan, Phys. Lett. A 277, 212 (2000).
[12] S.A. Elwakil, S.K. El-Labany, M.A. Zahran, R. Sabry, Phys. Lett. A 299, 179 (2002).
[13] M.A. Abdou, Applied Mathematics and Computation 190, 988-996 (2007).
[14] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos, Solitons & Fractals 2007;34:1421.
[15] M.A. Abdou, Chaos Soliton Fractals 31, 95-104 (2007).
[16] M.A. Abdou, A. Elhanbaly, Communications in Nonlinear Science and Numerical Simulation 12, 1229-1241 (2007).
[17] E. Fan, Chaos, Solitons and Fractals 16, 819 (2003).
[18] W.M. Moslem, R. Sabry, U.M. Abdelsalam, I. Kourakis,
P.K. Shukla, New Journal of Physics 11, 033028 (2009).
[19] M. Adler, J. Moser, Commun. Math. Phys. 19, 1 (1978)
The reductive perturbation method has been employed to derive the Korteweg–deVries (KdV) equation for small but finite amplitude electrostatic ion-acoustic waves. An algebraic method with computerized symbolic computation, which greatly exceeds the applicability of the existing tanh, extended tanh methods in obtaining a series of exact solutions of the KdV equation. Numerical studies have been made using plasma parameters to reveal different solutions, i.e. bell-shaped solitary pulses, rational pulses and solutions with singularity at a finite points called “blowup” solutions in addition to the propagation of explosive pulses. The result of the present investigation may be applicable to some plasma environments, such as ionosphere.
Key words:
explosive solutions, ion-acoustic waves, reductive perturbation, symbolic computations
References:
[1] G. Whitham, Linear and nonlinear waves. New York, Wiley (1974).
[2] R. Davidson, Methods in nonlinear plasma theory. New York, Academic Press (1972).
[3] S. K. El-Labany, J. Plasma Phys. 50, 495 (1993)..
[4] M. Sokała, J. Computational Methods in Science and Technology 14(2), 111 (2008).
[5] H. Washimi, T. Taniuti, J Phys Rev Lett. 17. 996 (1966).
[6] T. Taniuti, C. Wei, J Phys Soc Jpn 24, 491 (1968).
[7] S.A. Elwakil, E.K. El-Shewy, R. Sabry, Int. J. Nonlinear Sci. Numer. Simulation 5, 403 (2004).
[8] S.A. Elwakil, M.T. Attia, M.A. Zahran, E.K. El-Shewy, H.G. Abdelwahed, Z. Naturforsch. 61a, 316 (2006).
[9] M.J. Ablowitz, P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering. Cambridge, Cambridge University Press (1991).
[10] W. Malfliet, Solitary wave solutions of nonlinear wave equations. American Journal of Physics 60(7), 650-654 (1992).
[11] E. Fan, Phys. Lett. A 277, 212 (2000).
[12] S.A. Elwakil, S.K. El-Labany, M.A. Zahran, R. Sabry, Phys. Lett. A 299, 179 (2002).
[13] M.A. Abdou, Applied Mathematics and Computation 190, 988-996 (2007).
[14] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos, Solitons & Fractals 2007;34:1421.
[15] M.A. Abdou, Chaos Soliton Fractals 31, 95-104 (2007).
[16] M.A. Abdou, A. Elhanbaly, Communications in Nonlinear Science and Numerical Simulation 12, 1229-1241 (2007).
[17] E. Fan, Chaos, Solitons and Fractals 16, 819 (2003).
[18] W.M. Moslem, R. Sabry, U.M. Abdelsalam, I. Kourakis,
P.K. Shukla, New Journal of Physics 11, 033028 (2009).
[19] M. Adler, J. Moser, Commun. Math. Phys. 19, 1 (1978)
