Exact solutions with solitons and periodic structures for the Zakharov-Kuznetsov (ZK) equation and its modified form. (English) Zbl 1070.35075

Wazwaz, Abdul-Majid

Commun. Nonlinear Sci. Numer. Simul. 10, No. 6, 597-606 (2005).

Summary: We study the Zakharov-Kuznetsov equation and two of its modified forms. We employ the sine-cosine algorithm to back up our analysis. Exact solutions with solitons and periodic structures are obtained.

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35Q51	Soliton equations

Keywords:

Solitons; Periodic solutions; Zakharov-Kuznetsov equation; KP equation; KdV equation; Sine-cosine method

Software:

MACSYMA

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, 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, 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, 3, 377-387 (1973)
[4]	Ablowitz, M. J.; Clarkson, P. A., Solitons, nonlinear evolution equations and inverse scattering (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001
[5]	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
[6]	Wadati, M., Introduction to solitons, Pramana: J. Phys, 57, 5-6, 841-847 (2001)
[7]	Wadati, M., The exact solution of the modified Kortweg-de Vries equation, J. Phys. Soc. Jpn, 32, 1681-1687 (1972)
[8]	Wadati, M., The modified Kortweg-de Vries equation, J. Phys. Soc. Jpn, 34, 1289-1296 (1973) · Zbl 1334.35299
[9]	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
[10]	Kivshar, Y. S.; Pelinovsky, D. E., Self-focusing and transverse instabilities of solitary waves, Phys. Rep, 331, 117-195 (2000)
[11]	Kadomtsev, B. B.; Petviashvili, V. I., Sov. Phys. JETP, 39, 285-295 (1974)
[12]	Zakharov, V. E.; Kuznetsov, E. A., On three-dimensional solitons, Sov. Phys, 39, 285-288 (1974)
[13]	Shivamoggi, B. K., The Painlevé analysis of the Zakharov-Kuznetsov equation, Phys. Scripta, 42, 641-642 (1990) · Zbl 1063.35550
[14]	Hereman, W.; Takaoka, M., Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA, J. Phys. A, 23, 4805-4822 (1990) · Zbl 0719.35085
[15]	Hereman, W.; Korpel, A.; Banerjee, P. P., A general physical approach to solitary wave construction from linear solutions, Wave Motion, 7, 283-289 (1985)
[16]	Infeld, E.; Rowlands, G., Nonlinear waves, solitons and chaos (1990), Cambridge University Press · Zbl 0726.76018
[17]	Yan, Z., Modified nonlinearly dispersive \(mK (m,n,k)\) equations: I. New compacton solutions and solitary patterns solutions, Comput. Phys. Commun, 152, 25-33 (2003) · Zbl 1196.35192
[18]	Yan, Z.; Bluman, G., New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations, Comput. Phys. Commun, 149, 11-18 (2002) · Zbl 1196.68338
[19]	Wazwaz, A. M., Partial differential equations: methods and applications (2002), Balkema Publishers: Balkema Publishers The Netherlands
[20]	Wazwaz, A. M., New solitary-wave special solutions with compact support for the nonlinear dispersive \(K(m,n)\) equations, Chaos Solitons Fract, 13, 2, 321-330 (2002) · Zbl 1028.35131
[21]	Wazwaz, A. M., Exact specific solutions with solitary patterns for the nonlinear dispersive \(K(m,n)\) equations, Chaos Solitons Fract, 13, 1, 161-170 (2001) · Zbl 1027.35115
[22]	Wazwaz, A. M., General compactons solutions for the focusing branch of the nonlinear dispersive \(K(n,n)\) equations in higher dimensional spaces, Appl. Math. Comput, 133, 2/3, 213-227 (2002) · Zbl 1027.35117
[23]	Wazwaz, A. M., General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive \(K(n,n)\) equations in higher dimensional spaces, Appl. Math. Comput, 133, 2/3, 229-244 (2002) · Zbl 1027.35118
[24]	Wazwaz, A. M., A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. Comput. Simul, 56, 269-276 (2001) · Zbl 0999.65109
[25]	Wazwaz, A. M., Compactons dispersive structures for variants of the \(K(n,n)\) and the KP equations, Chaos Solitons Fract, 13, 5, 1053-1062 (2002) · Zbl 0997.35083
[26]	Wazwaz, A. M., Compactons and solitary patterns structures for variants of the KdV and the KP equations, Appl. Math. Comput, 139, 1, 37-54 (2003) · Zbl 1029.35200
[27]	Wazwaz, A. M., Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method, Chaos Solitons Fract, 12, 8, 1549-1556 (2001) · Zbl 1022.35051
[28]	Wazwaz, A. M., A computational approach to soliton solutions of the Kadomtsev-Petviashili equation, Appl. Math. Comput, 123, 2, 205-217 (2001) · Zbl 1024.65098
[29]	Wazwaz, A. M., An analytic study of compactons structures in a class of nonlinear dispersive equations, Math. Comput. Simul, 63, 1, 35-44 (2003) · Zbl 1021.35092
[30]	Wazwaz, A. M., Existence and construction of compacton solutions, Chaos Solitons Fract, 19, 3, 463-470 (2004) · Zbl 1068.35124
[31]	Wazwaz, A. M., A study on nonlinear dispersive partial differential equations of compact and noncompact solutions, Appl. Math. Comput, 135, 2-3, 399-409 (2003) · Zbl 1027.35120
[32]	Wazwaz, A. M., Compactons in a class of nonlinear dispersive equations, Math. Comput. Model, 37, 3/4, 333-341 (2003) · Zbl 1044.35078