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New exact solutions to the Zakharov-Kuznetsov equation and its generalized form. (English) Zbl 1221.35328
Summary: The extended hyperbolic function method is used for analytic treatment of the $(2 + 1)$-dimensional Zakharov-Kuznetsov (ZK) equation and its generalized form. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the solitons solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.

35Q53KdV-like (Korteweg-de Vries) equations
35C08Soliton solutions of PDE
Full Text: DOI
[1] Zakharov, V. E.; Kuznetsov, E. A.: On three-dimensional solitons, Sov phys 39, 285-288 (1974)
[2] Schamel, H.: A modified KdV equation for ion acoustic waves due to resonant electrons, J plasma phys 9, No. 3, 377-387 (1973)
[3] Monrl, 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)
[4] Monrl, S.; Parkes, E. J.: Stability of solitary wave solutions to a modified Zakharov -- Kuznetsov equation, J plasma phys 64, No. 3, 411-426 (2000)
[5] 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
[6] Wazwaz, A. M.: The extended tanh method for the Zakharov -- Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms, Commun nonlinear sci numer simul 13, 1039-1047 (2008) · Zbl 1221.35373 · doi:10.1016/j.cnsns.2006.10.007
[7] Li, B.; Chen, Y.; Zhang, H. Q.: 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
[8] Wazwaz, A. M.: Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method, Chaos solitons fractals 12, No. 8, 1549-1556 (2001) · Zbl 1022.35051 · doi:10.1016/S0960-0779(00)00133-8
[9] 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
[10] Wazwaz, A. M.: A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Appl math comput 56, 269-276 (2001) · Zbl 0999.65109 · doi:10.1016/S0378-4754(01)00291-9
[11] Wazwaz, A. M.: Compactons and solitary patterns structures for variants of the KdV and the KP equations, Appl math comput 139, No. 1, 37-54 (2003) · Zbl 1029.35200 · doi:10.1016/S0096-3003(02)00120-0
[12] Feng, B.; Malomed, B.; Kawahara, T.: Cylindrical solitary pulses in a two-dimensional stabilized Kuramoto -- Sivashinsky system, Physica D, 127-138 (2003) · Zbl 1006.76015 · doi:10.1016/S0167-2789(02)00721-2
[13] 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
[14] Wazwaz, A. M.: Nonlinear dispersive special type of the Zakharov -- Kuznetsov equation $ZK(n,n)$ with compact and noncompact structures, Appl math comput 161, 577-590 (2005) · Zbl 1061.65105 · doi:10.1016/j.amc.2003.12.050
[15] Wazwaz, A. M.: Special types of the nonlinear dispersive Zakharov -- Kuznetsov equation with compactons, solitons and periodic solutions, Appl math comput 81, No. 9, 1107-1119 (2004) · Zbl 1059.35131 · doi:10.1080/00207160410001684253
[16] 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
[17] Wu, H. X.; Fan, T. Y.: Nonlinear dispersive $ZK(n, n)$ equations: new compacton solutions and solitary pattern solutions, Appl math comput 187, 1308-1314 (2007) · Zbl 05163811
[18] Shang, Y. D.: The extended hyperbolic function method and exact solutions of the long -- short wave resonance equations, Chaos solitons fractals 36, 762-771 (2008) · Zbl 1153.35374 · doi:10.1016/j.chaos.2006.07.007
[19] Shang, Y. D.: Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation, Appl math comput 187, 1286-1297 (2007) · Zbl 1112.76010 · doi:10.1016/j.amc.2006.09.038
[20] Fan, E. G.: Integrable systems and computer algebra, (2004)