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The extended tanh method for the Zakharov-Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms. (English) Zbl 1221.35373
Summary: The reliable extended tanh method, that combines tanh with coth, is used for analytic treatment of the Zakharov-Kuznetsov (ZK) equation, the modified ZK equation, and the generalized forms of these equations. New travelling wave solutions with solitons and periodic structures are determined. The power of the employed method is confirmed.
##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35Q51 Soliton-like equations
##### References:
 [1] Monro, S.; Parkes, E. J.: The derivation of a modified Zakharov – Kuznetsov equation and the stability of its solutions, J plasma phys 62, No. 3, 305-317 (1999) [2] Monro, S.; Parkes, E. J.: Stability of solitary-wave solutions to a modified Zakharov – Kuznetsov equation, J plasma phys 64, No. 3, 411-426 (2000) [3] Schamel, H.: A modified Korteweg – de-Vries equation for ion acoustic waves due to resonant electrons, J plasma phys 9, No. 3, 377-387 (1973) [4] Ablowitz, M. J.; Clarkson, P. A.: Solitons, nonlinear evolution equations and inverse scattering, (1991) · Zbl 0762.35001 [5] Wadati, M.: The exact solution of the modified kortweg – de Vries equation, J phys soc jpn 32, 1681-1687 (1972) [6] Li, B.; Chen, Y.; Zhang, H.: Exact travelling wave solutions for a generalized Zakharov – Kuznetsov equation, Appl math comput 146, 653-666 (2003) · Zbl 1037.35070 · doi:10.1016/S0096-3003(02)00610-0 [7] Kadomtsev, B. B.; Petviashvili, V. I.: Sov phys JETP, Sov phys JETP 39, 285-295 (1974) [8] Zakharov, V. E.; Kuznetsov, E. A.: On three-dimensional solitons, Sov phys 39, 285-288 (1974) [9] Shivamoggi, B. K.: The Painlevé analysis of the Zakharov – Kuznetsov equation, Phys scr 42, 641-642 (1990) · Zbl 1063.35550 · doi:10.1088/0031-8949/42/6/001 [10] Malfliet, W.: Solitary wave solutions of nonlinear wave equations, Am J phys 60, No. 7, 650-654 (1992) · Zbl 1219.35246 · doi:10.1119/1.17120 [11] Malfliet, W.: The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations, J comput appl math 164 – 165, 529-541 (2004) · Zbl 1038.65102 · doi:10.1016/S0377-0427(03)00645-9 [12] Wazwaz, A. M.: The tanh method for generalized forms of nonlinear heat conduction and Burgers – Fisher equations, Appl math comput 169, 321-338 (2005) · Zbl 1121.65359 · doi:10.1016/j.amc.2004.09.054 [13] Wazwaz, A. M.: Nonlinear dispersive special type of the Zakharov – Kuznetsov equation $ZK\left(n,n\right)$ with compact and noncompact structures, Appl math comput 161, 577-590 (2005) · Zbl 1061.65105 · doi:10.1016/j.amc.2003.12.050 [14] Wazwaz, A. M.: Special types of the nonlinear dispersive Zakharov – Kuznetsov equation with compactons, solitons and periodic solutions, Int J comput math 81, No. 9, 1107-1119 (2004) · Zbl 1059.35131 · doi:10.1080/00207160410001684253 [15] Wazwaz, A. M.: Exact solutions with solitons and periodic structures for the Zakharov – Kuznetsov (ZK) equation and its modified form, Commun nonlinear sci numer simul 10, 597-606 (2005) · Zbl 1070.35075 · doi:10.1016/j.cnsns.2004.03.001 [16] Wazwaz, A. M.: Partial differential equations: methods and applications, (2002) [17] Wazwaz, A. M.: A computational approach to soliton solutions of the Kadomtsev – petviashili equation, Appl math comput 123, No. 2, 205-217 (2001) · Zbl 1024.65098 · doi:10.1016/S0096-3003(00)00065-5 [18] Wazwaz, A. .M.: An analytic study of compactons structures in a class of nonlinear dispersive equations, Math comput simul 63, No. 1, 35-44 (2003) · Zbl 1021.35092 · doi:10.1016/S0378-4754(02)00255-0 [19] Wazwaz, A. M.: Existence and construction of compacton solutions, Chaos solitons fractals 19, No. 3, 463-470 (2004) · Zbl 1068.35124 · doi:10.1016/S0960-0779(03)00171-1 [20] Wazwaz, A. M.: Compactons in a class of nonlinear dispersive equations, Math comput modell 37, No. 3/4, 333-341 (2003) · Zbl 1044.35078 · doi:10.1016/S0895-7177(03)00010-4