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Bäcklund transformation and soliton interactions for the Zakharov-Kuznetsov equation in plasmas. (English) Zbl 1248.35176
Summary: Symbolically investigated in this paper is the Zakharov-Kuznetsov equation which describes the propagation of the electrostatic excitations in a magnetized, rotating and collisionless three-component plasma. Bilinear form and Bäcklund transformation for the Zakharov-Kuznetsov equation are derived with the Hirota method and symbolic computation. \(N\)-soliton solutions in terms of the Wronskian determinant are constructed, and the verification is finished through the direct substitution into bilinear equations. Propagation characteristics and interaction behaviors of the solitons are discussed through a graphical analysis. During the propagation, the one-soliton width and amplitude are both unchanged and not related to the coefficients, while the soliton interactions are elastic.
35Q51 Soliton equations
35C08 Soliton solutions
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
68W30 Symbolic computation and algebraic computation
Full Text: DOI
[1] Meseguer, A.; Mellibovsky, F.; Panizzi, S.; Tyagi, M.; Sujith, R.I.; Tang, L.; Paidoussis, M.P.; Helfrich, K.R.; Sinha, S.C.; Redkar, S.; Butcher, E.A., Appl. numer. math., J. math. anal. appl., Physica D, J. sound vib., Dyn. atmos. oceans, Commun. nonlinear sci. numer. simul., 11, 510, (2006)
[2] Fisher, L.S.; Golovin, A.A.; Shang, Y.D.; Tsoy, E.N.; Akhmediev, N.; Tian, B.; Shan, W.R.; Zhang, C.Y.; Wei, G.M.; Gao, Y.T.; Gao, Y.T.; Tian, B.; Zhang, C.Y., J. colloid interface sci., Chaos solitons fractals, Opt. commun., Eur. phys. J. B, Acta mech., 182, 17, (2006), (Rapid Not.)
[3] Tian, B.; Gao, Y.T., Phys. plasmas, Phys. lett. A, 340, 449, (2005), 342 (2005) 228; 359 (2006) 241
[4] Das, G.; Sarma, J.; Yan, Z.Y.; Zhang, H.Q., Phys. plasmas, J. phys. A, Eur. phys. J. D, Phys. lett. A, 340, 243, (2005), 362 (2007) 283
[5] Barnett, M.P.; Capitani, J.F.; Von Zur Gathen, J.; Gerhard, J.; Gao, Y.T.; Tian, B., Int. J. quantum chem., Phys. plasmas, Phys. plasmas (lett.), 13, 120703, (2006)
[6] Chow, K.W.; Lai, W.C.; Shek, C.K.; Tso, K.; Zhu, Y.J.; Zhang, J.F.; Khater, A.H.; Malfliet, W.; Callebaut, D.K.; Kamel, E.S.; Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., Chaos solitons fractals, Commun. nonlinear sci. numer. simul., J. comput. appl. math., J. phys. soc. Japan, 50, 3813, (1981)
[7] Leble, S.B.; Ustinov, N.V.; Hu, X.B.; Wang, D.L.; Qian, X.M.; Radha, R.; Lakshmanan, M.; Faucher, M.; Winternitz, P.; Kaya, D.; Sayed, S.M., Inverse problems, Phys. lett. A, Phys. lett. A, Phys. rev. E, Phys. lett. A, 320, 192, (2003)
[8] Hammack, J.; McCallister, D.; Scheffner, N.; Segur, H., J. fluid mech., 285, 95, (1995)
[9] Nishinari, K.; Abe, K.; Satsuma, J., Phys. plasmas, 1, 2559, (1994)
[10] Geng, X.G.; Dai, H.H.; Geng, X.G.; Cao, C.W.; Cao, C.W.; Geng, X.G.; Wu, Y.T.; Cao, C.W.; Geng, X.G.; Bonopelchenko, B.; Sidorenko, J.; Strampp, W., Physica A, Phys. lett. A, J. phys. A, J. phys. A, Phys. lett. A, 157, 17, (1991)
[11] Matveev, V.B.; Salle, M.A.; Dubrousky, V.G.; Konopelchenko, B.G., Darboux transformation and solitons, J. phys. A, 27, 4619, (1994), Springer Berlin
[12] Ablowitz, M.J.; Clarkson, P.A.; Wadati, M.; Konno, K.; Ichikawa, Y.H., Solitons, nonlinear evolution equations and inverse scattering, J. phys. soc. Japan, 53, 2642, (1983), Cambridge Univ. Press New York
[13] Wadati, M.; Wadati, M.; Sanuki, H.; Konno, K.; Konno, K.; Wadati, M., J. phys. soc. Japan, Progr. theoret. phys., Progr. theoret. phys., 53, 1652, (1975)
[14] Tian, B.; Gao, Y.T.; Gao, Y.T.; Tian, B., Phys. plasmas, Phys. lett. A, Europhys. lett., 77, 15001, (2007)
[15] Caruello, F.; Tabor, M.; Weiss, J.; Tabor, M.; Carnevale, G., Physica D, J. math. phys., 24, 522, (1983)
[16] Hirota, R.; Hirota, R., The direct method in soliton theory, Phys. rev. lett., 27, 1192, (1971), Cambridge Univ. Press Cambridge · Zbl 1168.35423
[17] Satsuma, J.; Freeman, N.C.; Horrocks, G.; Wilkinson, P.; Deng, S.F.; Chen, D.Y.; Zhang, D.J., J. phys. soc. Japan, Phys. lett. A, J. phys. soc. Japan, 72, 2184, (2003)
[18] Nimmo, J.J.; Freeman, N.C.; Freeman, N.C.; Nimmo, J.J.; Nimmo, J.J.; Freeman, N.C.; Freeman, N.C., Phys. lett. A, Phys. lett. A, Phys. lett. A, IMA J. appl. math., 32, 125, (1984)
[19] Zakharov, V.E.; Kuznetsov, E.A., Sov. phys., 39, 285, (1974)
[20] Infeld, E.; Fryczs, P., J. plasma phys., 37, 97, (1987)
[21] Shivamoggi, B.K., Phys. scr., 42, 641, (1990)
[22] Infeld, E.; Shivamoggi, B.K., J. plasma phys., J. plasma phys., 41, 83, (1989)
[23] Kourakis, I.; Moslem, W.M.; Abdelsalam, U.M.; Sabry, R.; Shukla, P.K., Plasma fusion res., 4, 18, (2009)
[24] He, J.H., Chaos solitons fractals, 26, 695, (2005)
[25] Hu, H.H.; Ma, H.C.; Yu, Y.D.; Ge, D.J., Commun. theor. phys., Comput. math. appl., 58, 2523, (2009)
[26] Hirota, R.; Satsumam, J.; Hirota, R.; Hu, X.B.; Tang, X.Y., J. phys. soc. Japan, J. math. anal. appl., 288, 326, (2003)
[27] Yu, X.; Gao, Y.T.; Sun, Z.Y.; Liu, Y.; Sun, Z.Y.; Gao, Y.T.; Yu, X.; Liu, Y., Phys. rev. E, Nonlinear dynamics, Europhys. lett., 93, 40004, (2011)
[28] Sun, Z.Y.; Gao, Y.T.; Liu, Y.; Yu, X.; Wang, L.; Gao, Y.T.; Gai, X.L.; Sun, Z.Y.; Wang, L.; Gao, Y.T.; Gai, X.L., Phys. rev. E, Phys. scr., Z. naturforsch. A, 65, 818, (2010)
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