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Compact and noncompact physical structures for the ZK-BBM equation. (English) Zbl 1078.35527
Summary: A variety of exact solutions for the (2 + 1) dimensional ZK-BBM equation are developed by means of the tanh method and the sine-cosine methods. Generalized forms of the ZK-BBM equation are studied. The tanh and the sine-cosine methods are reliable to derive solutions of distinct physical structures: compactons, solitons, solitary patterns and periodic solutions.

MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 37K40 Soliton theory, asymptotic behavior of solutions 35B10 Periodic solutions of PDE
MACSYMA
Full Text:
References:
 [1] Benjamin, R. T.; Bona, J. L.; Mahony, J. J.: Model equations for long waves in nonlinear dispersive systems. Philos. trans. Roy. soc. London 272, 47-78 (1972) · Zbl 0229.35013 [2] Ablowitz, M. J.; Clarkson, P. A.: Solitons, nonlinear evolution equations and inverse scattering. (1991) · Zbl 0762.35001 [3] 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 [4] Hereman, W.; Korpel, A.; Banerjee, P. P.: A general physical approach to solitary wave construction from linear solutions. Wave motion 7, 283-289 (1985) [5] Kadomtsev, B. B.; Petviashvili, V. I.: On the stability of solitary waves in weakly dispersive media. Sov. phys. Dokl. 15, 539-541 (1970) · Zbl 0217.25004 [6] Kivshar, Y. S.; Pelinovsky, D. E.: Self-focusing and transverse instabilities of solitary waves. Phys. rep. 331, 117-195 (2000) [7] 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 [8] Ludu, A.; Draayer, J. P.: Patterns on liquid surfaces: cnoidal waves, compactons and scaling. Physica D 123, 82-91 (1998) · Zbl 0952.76008 [9] Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, No. 7, 650-654 (1992) · Zbl 1219.35246 [10] Malfliet, W.; Hereman, W.: The tanh method: II. Perturbation technique for conservative systems. Phys. scr. 54, 569-575 (1996) · Zbl 0942.35035 [11] 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) [12] 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) [13] Parkes, E. J.; Duffy, B. R.: An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. phys. Commun. 98, 288-300 (1996) · Zbl 0948.76595 [14] Rosenau, P.; Hyman, J. M.: Compactons: solitons with finite wavelengths. Phys. rev. Lett. 70, No. 5, 564-567 (1993) · Zbl 0952.35502 [15] Wadati, M.: Introduction to solitons. Pramana: journal of physics 57, No. 5-6, 841-847 (2001) [16] Wadati, M.: The modified kortweg-de Vries equation. J. phys. Soc. jpn. 34, 1289-1296 (1973) [17] Wazwaz, A. M.: Partial differential equations: methods and applications. (2002) · Zbl 1079.35001 [18] Wazwaz, A. M.: New solitary-wave special solutions with compact support for the nonlinear dispersive $K(m,n)$ equations. Chaos, solitons and fractals 13, No. 2, 321-330 (2002) · Zbl 1028.35131 [19] 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, No. 2/3, 229-244 (2002) · Zbl 1027.35118 [20] Wazwaz, A. M.: A study of nonlinear dispersive equations with solitary-wave solutions having compact support. Math. comput. Simulat. 56, 269-276 (2001) · Zbl 0999.65109 [21] Wazwaz, A. M.: Compactons dispersive structures for variants of the $K(n,n)$ and the KP equations. Chaos solitons and fractals 13, No. 5, 1053-1062 (2002) · Zbl 0997.35083 [22] 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 [23] 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 [24] Wazwaz, A. M.: An analytic study of compactons structures in a class of nonlinear dispersive equations. Math. comput. Simulat. 63, No. 1, 35-44 (2003) · Zbl 1021.35092 [25] Wazwaz, A. M.; Taha, T.: Compact and noncompact structures in a class of nonlinearly dispersive equations. Math. comput. Simulat. 62, No. 1-2, 171-189 (2003) · Zbl 1013.35072 [26] Wazwaz, A. M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. math. Comput. 154, No. 3, 713-723 (2004) · Zbl 1054.65106 [27] Zabusky, N. J.; Kruskal, M. D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. rev. Lett. 15, 240-243 (1965) · Zbl 1201.35174 [28] Zakharov, V. E.; Kuznetsov, E. A.: On three-dimensional solitons. Soviet phys. 39, 285-288 (1974)