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Time-fractional study of electron acoustic solitary waves in plasma of cold electron and two isothermal ions

  • S. A. EL-WAKIL (a1), ESSAM M. ABULWAFA (a1), EMAD K. EL-SHEWY (a1) (a2) and ABEER A. MAHMOUD (a1)

Abstract

In this paper, a homogeneous system of unmagnetized collisionless plasma consisting of a cold electron fluid, low-temperature ion obeying Boltzmann-type distribution and high-temperature ion obeying non-thermal distribution is considered. The perturbation method with two different forms of stretching will be considered to drive the KdV and modified KdV (mKdV) equations. The Agrawal's method is applied to formulate the time-fractional KdV and mKdV equations. A variational iteration method is used to solve these equations. The results show that the fractional order of the differential equations can be used to modify the shape of the solitary pulse instead of adding higher order dissipation terms to the equations. This study may be useful to construct the compressive and rarefactive electrostatic potential pulses associated with the broadband electrostatic noise type emissions.

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Agrawal, O. P. 2002 Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368379.
Agrawal, O. P. 2006 Fractional variational calculus and the transversality conditions. J. Phys. A: Math. Gen. 39, 10375–10 384.
Agrawal, O. P. 2007 Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A: Math. Theor. 40, 62876303.
Ajlouni, A. M. S. and Al-Rabai'ah, H. A. 2010 Fractional-calculus diffusion equation. Nonlinear Biomed. Phy. 4 (1), 310.
Asbridge, R., Bame, S. J. and Strong, I. B. 1968 Outward flow of protons from the Earth's bow shock. J. Geophys. Res. 73 (12), 57775782.
Atanacković, T. M., Konjik, S., Oparnica, L. J. and Pilipović, S. 2010 Generalized Hamilton's principle with fractional derivatives. J. Phys. A: Math. Theor. 43, 255203 (12 pages).
Baleanu, D., Golmankhaneh, A. K., Nigmatullin, R. and Golmankhaneh, A. K. 2010 Fractional Newtonian mechanics. Cent. Eur. J. Phys. 8 (1), 120125.
Buti, B., Mohan, M. and Shukla, P. K. 1980 Exact electron-acoustic solitary waves J. Plasma Phys. 23 (2), 341347.
Cairns, R. A., Mamum, A. A., Bingham, R., Boström, R., Dendy, R. O., Nairn, C. M. C. and Shukla, P. K. 1995 Electrostatic solitary structures in nonthermal plasmas. Geophys. Res. Lett. 22 (20), 27092712.
Cattell, C. A., Dombeck, J., Wygant, J. R., Hudson, M. K., Mozer, F. S., Temerin, M. A., Peterson, W. K., Kletzing, C. A., Russell, C. T. and Pfaff, R. F. 1999 Comparisons of polar satellite observations of solitary wave velocities in the plasma sheet boundary and the high altitude cusp to those in the auroral zone. Geophys. Res. Lett. 26 (3), 425428.
Cattell, C. A., Mozer, F. S., Anderson, R. R., Hones, E. W. and Sharp, R. D. 1986 ISEE observations of the plasma sheet boundary, plasma sheet, and neutral sheet: 2. Waves. J. Geophys. Res. 91, 56815688.
Chandrashekar, C. M. 2010 Fractional recurrence in discrete-time quantum walk. Cent. Eur. J. Phys. 8 (6), 979988.
Chatterjee, P. and Roychoudhury, R. 1995 Arbitrary-amplitude electron acoustic solitary waves in a plasma. J. Plasma Phys. 53 (1), 2529.
Devanandhan, S., Singh, S. V. and Lakhina, G. S. 2011a Electron acoustic solitary waves with kappa-distributed electrons, Phys. Scr. 84, 025507 (6 pages).
Devanandhan, S., Singh, S. V., Lakhina, G. S. and Bharuthram, R. 2011b Electron acoustic solitons in the presence of an electron beam and superthermal electrons. Nonlinear Process. Geophys. 18, 627634.
Dubouloz, I., Pottellette, R., Malingre, M., Holmgren, G. and Lindqvist, P. A. 1991 Detailed analysis of broadband electrostatic noise in the dayside auroral zone. J. Geophys. Res. 96, 35653579.
El-Shewy, E. K. 2007 Linear and nonlinear properties of electron-acoustic solitary waves with non-thermal electrons. Chaos, Solitons Fractals 31 (4), 10201024.
El-Wakil, S. A., Abulwafa, E. M., Zahran, M. A. and Mahmoud, A. A. 2011a Time-fractional KdV equation: formulation and solution using variational methods. Nonlinear Dyn. 65 (1–2), 5563.
El-Wakil, S. A., Abulwafa, E. M., Zahran, M. A. and Mahmoud, A. A. 2011b Formulation of some fractional evolution equations used in mathematical physics. Nonlinear Sci. Lett. A 2 (1), 3746.
Feldman, W. C., Anderson, R. C., Bame, S. J., Gary, S. P., Gosling, J. T., McComas, D. J., Thomsen, M. F., Paschmann, G. and Hoppe, M. M. 1983 Electron velocity distributions near the earth's bow shock. J. Geophys. Res. 88, 96110.
Golmankhaneh, A. K., Golmankhaneh, A. K. and Baleanu, D. 2011 On nonlinear fractional Klein–Gordon equation, Signal Process. 91 (3), 446451.
Gurnett, D. A., Frank, L. A. and Lepping, R. P. 1976 Plasma waves in the distant magnetotail J. Geophys. Res. 81, 60596071.
He, J.-H. 1997a A new approach to nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 2 (4), 230235.
He, J.-H. 1997b Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbo-machinery aerodynamics. Int. J. Turbo Jet-Engines 14 (1), 2328.
He, J.-H. 1999 Variational-iteration – a kind of nonlinear analytical technique: some examples. Int. J. Nonlinear Mech. 34, 699708.
He, J.-H. 2004 Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons Fractals 19, 847851.
Herzallah, M. A. E., Muslih, S. I., Baleanu, D. and Rabei, E. M. 2011 Hamilton-Jacobi and fractional like action with time scaling. Nonlinear Dyn. 66 (4), 549555.
Ikezawa, S. and Nakamura, Y. 1981 Observation of electron plasma waves in plasma of two-temperature electrons. J. Phys. Soc. Japan 50 (3), 962967.
Inokuti, M., Sekine, H. and Mura, T. 1978 General use of the Lagrange Multiplier in non-linear mathematical physics. In: Variational Method in the Mechanics of Solids (ed. Nemat-Nasser, S.). Oxford: Pergamon Press, p. 156162.
Kakad, A. P., Singh, S. V., Reddy, R. V., Lakhina, G. S. and Tagare, S. G. 2009 Electron acoustic solitary waves in the Earth's magnetotail region. Adv. Space Res. 43 (12), 19451949.
Kilbas, A., Srivastava, H. M. and Trujillo, J. J. 2006 Theory and Applications of Fractional Differential Equations. New York: Elsevier Science Inc.
Lundin, R., Zakharov, A., Pellinen, R., Hultqvist, B., Borg, H., Dubinin, E., Barabash, S., Pissarenko, N., Koskinen, H. and Liede, I. 1989 First measurements of the ionospheric plasma escape from Mars. Nature 341, 609612.
Matsumoto, H., Kojima, H., Miyatake, T., Omura, Y., Okada, M., Nagano, I. and Tsutsui, M. 1994 Electrostatic solitary waves (ESW) in the magnetotail: BEN wave forms observed by GEOTAIL. Geophys. Res. Lett. 21 (25), 29152918.
Mozer, F. S., Ergun, R., Temerin, M., Cattell, C., Dombeck, J. and Wygant, J. 1997 New features of time domain electric-field structures in the auroral acceleration region. Phys. Rev. Lett. 79 (7), 12811284.
Muslih, S. I. and Agrawal, O. P. 2010 Riesz fractional derivatives and fractional dimensional space. Int. J. Theor. Phys. 49, 270275.
Podlubny, I. 1999 Fractional Differential Equations. San Diego: Academic Press.
Pottelette, R., Ergun, R. E., Treumann, R. A., Berthomier, M., Carlson, C. W., McFadden, J. P., and Roth, I. 1999 Modulated electron-acoustic waves in auroral density cavities: FAST observations. Geophys. Res. Lett. 26 (16), 26292632.
Rabei, E. M., Rawashdeh, I. M., Muslih, S. I. and Baleanu, D. 2011 Hamilton-Jacobi formulation for systems in terms of Riesz's fractional derivatives. Int. J. Theor. Phys. 50 (5), 15691576.
Singh, S. V. and Lakhina, G. S. 2001 Generation of electron-acoustic waves in the magnetosphere Planet. Space Sci. 49 (1), 107114.
Singh, S. V. and Lakhina, G. S. 2004 Electron acoustic solitary waves with non-thermal distribution of electrons. Nonlinear Process. Geophys. 11, 275279.
Singh, S. V., Lakhina, G. S., Bharuthram, R. and Pillay, S. R. 2011 Electrostatic solitary structures in presence of non-thermal electrons and a warm electron beam on the auroral field lines, Phys. Plasmas 18, 122306 (7 pages).
Singh, S. V., Reddy, R. V. and Lakhina, G. S. 2001 Broadband electrostatic noise due to nonlinear electron acoustic waves. Adv. Space Res. 28 (11), 16431648.
Tagare, S. G., Singh, S. V., Reddy, R. V. and Lakhina, G. S. 2004 Electron acoustic solitons in the Earth's magnetotail. Nonlinear Process. Geophys. 11, 215218.
Tejedor, V. and Metzler, R. 2010 Anomalous diffusion in correlated continuous time random walks. J. Phys. A: Math. Theor. 43, 082002 (6 pages).
Washimi, H. and Taniuti, T. 1966 Propagation of ion-acoustic solitary waves of small amplitude. Phys. Rev. Lett. 17 (19), 996998.
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