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Generalized solitary and periodic wave solutions to a (2+1)-dimensional Zakharov-Kuznetsov equation. (English) Zbl 1203.35205
Summary: The Exp-function method is employed to the Zakharov-Kuznetsov equation as a (2+1)-dimensional model for nonlinear Rossby waves. The observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient. The obtained results may be useful for better understanding the properties of two-dimensional coherent structures such as atmospheric blocking events.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35Q51Soliton-like equations
35C08Soliton solutions of PDE
35A24Methods of ordinary differential equations for PDE
35B10Periodic solutions of PDE
References:
[1]Ablowitz, M. J.; Segur, H.: Solitons and inverse scattering transform, (1981)
[2]Liu, G. T.; Fan, T. Y.: New applications of developed Jacobi elliptic function expansion methods, Phys. lett. A 345, 161-166 (2005)
[3]Malfliet, W.; Hereman, W.: The tanh method I: Exact solutions of nonlinear evolution and wave equations, Phys. scr. 54, 563-568 (1996) · Zbl 0942.35034 · doi:10.1088/0031-8949/54/6/003
[4]Wazwaz, A. M.: Distinct variants of the KdV equation with compact and noncompact structures, Appl. math. Comput. 150, 365-377 (2004) · Zbl 1039.35110 · doi:10.1016/S0096-3003(03)00238-8
[5]Bluman, G. W.; Kumei, S.: Symmetries and differential equations, (1989)
[6]Abdou, M. A.: The extended F-expansion method and its application for a class of nonlinear evolution equations, Chaos solitons fract. 31, 95-104 (2007) · Zbl 1138.35385 · doi:10.1016/j.chaos.2005.09.030
[7]Hirota, R.: The direct method in soliton theory, (2004)
[8]Weiss, J.; Tabor, M.; Carnevale, G.: The Painlevé property for partial differential equations, J. math. Phys. 24, 522-526 (1983) · Zbl 0514.35083 · doi:10.1063/1.525721
[9]Wang, M. L.: Exact solutions for a compound KdV – Burgers equation, Phys. lett. A 213, 279-287 (1996) · Zbl 0972.35526 · doi:10.1016/0375-9601(96)00103-X
[10]Miura, R. M.: Bäcklund transformation, (1996)
[11]Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994)
[12]He, J. H.: A new approach to nonlinear partial differential equations, Commun. nonlinear sci. Numer. simul. 2, 230-235 (1997) · Zbl 0923.35046 · doi:10.1016/S1007-5704(97)90029-0
[13]Liao, S. J.: Homotopy analysis method-A kind of nonlinear analytical technique not depending on small parameters, Shanghai J. Mech. 18, 196-200 (1997)
[14]He, J. H.: An approximate solution technique depending on an artificial parameter: a special example, Commun. nonlinear sci. Numer. simul. 3, 92-97 (1998) · Zbl 0921.35009 · doi:10.1016/S1007-5704(98)90070-3
[15]Wang, M.; Li, X.; Zhang, J.: The (G ' /G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. lett. A 372, 417-423 (2008) · Zbl 1217.76023 · doi:10.1016/j.physleta.2007.07.051
[16]&idot; Aslan, .; Öziş, T.: Analytic study on two nonlinear evolution equations by using the (G ' /G)-expansion method, Appl. math. Comput. 209, 425-429 (2009)
[17]&idot; Aslan, .; Öziş, T.: On the validity and reliability of the (G ' /G)-expansion method by using higher-order nonlinear equations, Appl. math. Comput. 211, 531-536 (2009)
[18]Öziş, T.; &idot; Aslan, .: Symbolic computation and construction of new exact traveling wave solutions to Fitzhugh – Nagumo and Klein – Gordon equations, Z. naturforsch. 64a, 15-20 (2009)
[19]Öziş, T.; &idot; Aslan, .: Symbolic computations and exact and explicit solutions of some nonlinear evolution equations in mathematical physics, Commun. theor. Phys. 51, 577-580 (2009)
[20]He, J. H.; Wu, X. H.: Exp-function method for nonlinear wave equations, Chaos solitons fract. 30, 700-708 (2006) · Zbl 1141.35448 · doi:10.1016/j.chaos.2006.03.020
[21]He, J. H.; Zhang, L. N.: Generalized solitary solution and compacton-like solution of the Jaulent – Miodek equations using the exp-function method, Phys. lett. A 372, 1044-1047 (2008) · Zbl 1217.35152 · doi:10.1016/j.physleta.2007.08.059
[22]Wu, X. H.; He, J. H.: Solitary solutions, periodic solutions and compacton-like solutions using the exp-function method, Comput. math. Appl. 54, 966-986 (2007) · Zbl 1143.35360 · doi:10.1016/j.camwa.2006.12.041
[23]He, J. H.; Abdou, M. A.: New periodic solutions for nonlinear evolution equations using exp-function method, Chaos solitons fract. 34, 1421-1429 (2007) · Zbl 1152.35441 · doi:10.1016/j.chaos.2006.05.072
[24]Öziş, T.; &idot; Aslan, .: Exact and explicit solutions to the (3+1)-dimensional Jimbo – Miwa equation via the exp-function method, Phys. lett. A 372, 7011-7015 (2008)
[25]Zakharov, V. E.; Kuznetsov, E. A.: On three-dimensional solitons, Soviet phys. 39, 285-288 (1974)
[26]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
[27]Munro, S.; Parkes, E. J.: The derivation of a modified Zakharov – Kuznetsov equation and the stability of its solutions, J. plasma phys. 62, 305-317 (1999)
[28]Das, J.; Bandyopadhyay, A.; Das, K. P.: Stability of an alternative solitary-wave solution of an ion-acoustic wave obtained from the mkdv – KdV – ZK equation in magnetized non-thermal plasma consisting of warm adiabatic ions, J. plasma phys. 72, 587-604 (2006)
[29]Biswas, A.; Zerrad, E.: 1-soliton solution of the Zakharov – Kuznetsov equation with dual-power law nonlinearity, Commun. nonlinear sci. Numer. simul. 14, 3574-3577 (2009) · Zbl 1221.35312 · doi:10.1016/j.cnsns.2008.10.004
[30]Iwasaki, H.; Toh, S.; Kawahara, T.: Cylindrical quasi-solitons of the Zakharov – Kuznetsov equation, Physica D 43, 293-303 (1990) · Zbl 0714.35076 · doi:10.1016/0167-2789(90)90138-F
[31]G.A. Gottwalld, The Zakharov – Kuznetsov equation as a two-dimensional model for nonlinear Rossby wave, arXiv: nlin, 0312009, 2003.
[32]He, J. H.: Application of homotopy perturbation method to nonlinear wave equations, Chaos soliton fract. 26, 695-700 (2005) · Zbl 1072.35502 · doi:10.1016/j.chaos.2005.03.006
[33]Fu, Z.; Liu, S.; Liu, S.: Multiple structures of 2-D nonlinear Rossby wave, Chaos soliton fract. 24, 383-390 (2005) · Zbl 1067.35071 · doi:10.1016/j.chaos.2004.09.043
[34]Zhou, X. W.; Wen, Y. X.; He, J. H.: Exp-function method to solve the nonlinear dispersive K(m,n) equations, Int. J. Nonlinear sci. Numer. simul. 3, 301-306 (2008)
[35]Wu, X. H.; He, J. H.: EXP-function method and its application to nonlinear equations, Chaos soliton fract. 38, 903-910 (2008) · Zbl 1153.35384 · doi:10.1016/j.chaos.2007.01.024
[36]Abdou, M. A.; Abulwafa, E. M.: Application of the exp-function method to the Riccati equation and new exact solutions with three arbitrary functions of quantum Zakharov equations, Z. naturforsch. 63a, 646-652 (2008)
[37]Marinakis, V.: The exp-function method find n-soliton solutions, Z. naturforsch. 63a, 653-656 (2008)
[38]Dai, C. Q.; Wang, Y. Y.: Exact travelling wave solutions of Toda lattice equations obtained via the exp-function method, Z. naturforsch. 63a, 657-662 (2008)
[39]Zhang, S.; Wang, W.; Tong, J. L.: The exp-function method for the Riccati equation and exact solutions of dispersive long wave equations, Z. naturforsch. 63a, 663-670 (2008)
[40]Liu, S.; Fu, Z.; Liu, S.; Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. lett. A 289, 69-74 (2001) · Zbl 0972.35062 · doi:10.1016/S0375-9601(01)00580-1
[41]Ebaid, A.: Exact solitary wave solutions for some nonlinear evolution equations via exp-function method, Phys. lett. A 365, 213-219 (2007) · Zbl 1203.35213 · doi:10.1016/j.physleta.2007.01.009
[42]El-Wakil, S. A.; Madkour, M. A.; Abdou, M. A.: Application of exp-function method for nonlinear evolution equations with variable coefficients, Phys. lett. A 369, 62-69 (2007) · Zbl 1209.81097 · doi:10.1016/j.physleta.2007.04.075
[43]Zhang, S.: Application of exp-function method to a KdV equation with variable coefficients, Phys. lett. A 365, 448-453 (2007) · Zbl 1203.35255 · doi:10.1016/j.physleta.2007.02.004
[44]Zhang, S.: Exp-function method exactly solving a KdV equation with forcing term, Appl. math. Comput. 197, 128-134 (2008) · Zbl 1135.65388 · doi:10.1016/j.amc.2007.07.041
[45]Zhang, S.: Exp-function method for constructing explicit and exact solutions of a lattice equation, Appl. math. Comput. 199, 242-249 (2008) · Zbl 1142.65102 · doi:10.1016/j.amc.2007.09.051
[46]Zhu, S. D.: Discrete (2+1)-dimensional Toda lattice equation via exp-function method, Phys. lett. A 372, 654-657 (2008) · Zbl 1217.37064 · doi:10.1016/j.physleta.2007.07.085
[47]Dai, C.; Cen, X.; Wu, S.: Exact travelling wave solutions of the discrete sine – Gordon equation obtained via the exp-function method, Nonlinear anal. 70, 58-63 (2009) · Zbl 1183.34101 · doi:10.1016/j.na.2007.11.034
[48]C. Dai, X. Cen, S. Wu, The application of He’s exp-function method to a nonlinear differential-difference equation, Chaos Solitons Fract., doi:10.1016/j.chaos.2008.02.021.
[49]Zhang, S.: Application of exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer – Kaup – kupershmidt equations, Phys. lett. A 372, 1873-1880 (2008) · Zbl 1220.37071 · doi:10.1016/j.physleta.2007.10.086
[50]Dai, C.: New exact solutions of the (3+1)-dimensional Burgers system, Phys. lett. A 373, 181-187 (2009) · Zbl 1227.35231 · doi:10.1016/j.physleta.2008.11.018
[51]C. Dai, Y. Wang, New variable separation solutions of the (2+1)-dimensional asymmetric Nizhnik – Novikov – Veselov system, Nonlinear Anal., doi:10.1016/j.na.2008.12.052.
[52]Dai, C.; Chen, J.: Exact solutions of (2+1)-dimensional stochastic Broer – Kaup equation, Phys. lett. A 373, 1218-1225 (2009) · Zbl 1228.76023 · doi:10.1016/j.physleta.2009.02.018
[53]J. Zhu, New explicit exact solutions of the mKdV equation using the variational iteration method combined with Exp-function method, Chaos Solitons Fract., doi:10.1016/j.chaos.2007.08.051.