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2007 | 5 | 3 | 351-366

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Compacton and periodic wave solutions of the non-linear dispersive Zakharov-Kuznetsov equation



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In this paper, the nonlinear dispersive Zakharov-Kuznetsov equation is solved by using the sine-cosine method. As a result, compactons, periodic, and singular periodic wave solutions are found.



  • [1] W. Hereman and M. Takaoka: “Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA”, J. Phys. A, Vol. 23, (1990), pp. 4805–4822. http://dx.doi.org/10.1088/0305-4470/23/21/021[Crossref]
  • [2] W. Hereman: “Exact solitary wave solutions of coupled nonlinear evolution equations using MACSYMA”, Comp. Phys. Commun., Vol. 65, (1991), pp. 143–150. http://dx.doi.org/10.1016/0010-4655(91)90166-I[Crossref]
  • [3] E.J. Parkes and B.R. Duffy: “An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations”, Comp. Phys. Commun., Vol. 98, (1996), pp. 288–300. http://dx.doi.org/10.1016/0010-4655(96)00104-X[Crossref]
  • [4] B.R. Duffy and E.J. Parkes: “Travelling solitary wave solutions to a seventh-order generalized KdV equation”, Phys. Lett. A, Vol. 214, (1996), pp. 271–272. http://dx.doi.org/10.1016/0375-9601(96)00184-3[Crossref]
  • [5] P. Rosenau and J.M. Hyman: “Compactons: solitons with finite wavelengths”, Phys. Rev. Lett., Vol. 70, (1993), pp. 564–567. http://dx.doi.org/10.1103/PhysRevLett.70.564[Crossref]
  • [6] P.J. Olver and P. Rosenau: “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support”, Phys. Rev. E, Vol. 53, (1996), pp. 1900–1906. http://dx.doi.org/10.1103/PhysRevE.53.1900[Crossref]
  • [7] A.M. Wazwaz: “Special types of the nonlinear dispersive Zakharov-Kuznetsov equations with compactons, solitons, and periodic solutions”, Int. J. Computer Math., Vol. 81, (2004), pp. 1107–1119. http://dx.doi.org/10.1080/00207160410001684253[Crossref]
  • [8] A.M. Wazwaz: “Nonlinear dispersive special type of the Zakharov-Kuznetsov equation ZK(n,n) with compact and noncompact structures”, Appl. Math. Comput., Vol. 161, (2005), pp. 577–590. http://dx.doi.org/10.1016/j.amc.2003.12.050[Crossref]
  • [9] A.M. Wazwaz: “Exact solutions with solitons and periodic structures for the Zakharov-Kuznetsov (ZK) equation and its modified form”, Commun. Nonlinear Sci. Numer. Simul., Vol. 10, (2005), pp. 597–606. http://dx.doi.org/10.1016/j.cnsns.2004.03.001[WoS][Crossref]
  • [10] A.M. Wazwaz: “Explicit travelling wave solutions of variants of the K(n, n) and the ZK(n, n) equations with compact and noncompact structures”, Appl. Math. Comput., Vol. 173, (2006), pp. 213–230. http://dx.doi.org/10.1016/j.amc.2005.02.050[Crossref]
  • [11] M.S. Ismail and T. Taha: “A numerical study of compactons”, Math. Comput. Simul., Vol. 47, (1998), pp. 519–530. http://dx.doi.org/10.1016/S0378-4754(98)00132-3[Crossref]
  • [12] A. Ludu and J.P. Draayer: “Patterns on liquid surfaces: cnoidal waves, compactons and scaling”, Physica D, Vol. 123, (1998), pp. 82–91. http://dx.doi.org/10.1016/S0167-2789(98)00113-4[Crossref]
  • [13] A.M. Wazwaz: “A study of nonlinear dispersive equations with solitary-wave solutions having compact support”, Math. Comput. Simul., Vol. 56, (2001), pp. 269–276. http://dx.doi.org/10.1016/S0378-4754(01)00291-9[Crossref]
  • [14] V.E. Zakharov and E.A. Kuznetsov: “On three-dimensional solitons”, Sov. Phys., Vol. 39, (1974), pp. 285–288.
  • [15] S. Monro and E.J. Parkes: “The derivation of a modified Zakharov-Kuznetsov equation and the stability of its solutions”, J. Plasma Phys., Vol. 62, (1999), pp. 305–317. http://dx.doi.org/10.1017/S0022377899007874[Crossref]
  • [16] S. Monro and E.J. Parkes: “Stability of solitary-wave solutions to a modified Zakharov-Kuznetsov equation”, J. Plasma Phys., Vol. 64, (2000), pp. 411–426.
  • [17] Z. Yan: “Modified nonlinearly dispersive mK(m,n,k) equations: I. New compacton solutions and solitary pattern solutions”, Comp. Phys. Commun., Vol. 152, (2003), pp. 25–33. http://dx.doi.org/10.1016/S0010-4655(02)00794-4[Crossref]
  • [18] A.M. Wazwaz: “General compactons solutions and solitary patterns solutions for modified nonlinear dispersive equations mK(n,n) in higher dimensional spaces”, Math. Comput. Simul., Vol. 59, (2002), pp. 519–531. http://dx.doi.org/10.1016/S0378-4754(01)00439-6[Crossref]
  • [19] P. Bracken: “Specific solutions of the generalized Korteweg-de Vries equation with possible physical applications”, Cent. Eur. J. Phys, Vol. 3, (2005), pp. 127–138. http://dx.doi.org/10.2478/BF02476511[Crossref]
  • [20] Z. Yan: “New families of solitons with compact support for Boussinesq-like B(m,n) equations with fully nonlinear dispersion”, Chaos, Solitons and Fractals, Vol. 14, (2002), pp. 1151–1158. http://dx.doi.org/10.1016/S0960-0779(02)00062-0[Crossref]
  • [21] A. Das: Integrable Models, World Scientific Notes in Physics, World Scientific, Singapore, 1989.
  • [22] J. Dalibard, J.M. Raimond and J. Zinn-Justin: Fundamental Systems in Quantum Optics, Les Houches, Session LIII, North-Holland, 1992.
  • [23] A.M. Wazwaz: Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002.

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