Abstract
In this paper, using the exp-function method we obtain some new exact solutions for (1+1)-dimensional and (2+1)-dimensional Kaup–Kupershmidt (KK) equations. We show figures of some of the new solutions obtained here. We conclude that the exp-function method presents a wider applicability for handling nonlinear partial differential equations.
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Biswas A., Milovic D., Ranasinghe A.: Solitary waves of Boussinesq equation in a power law media. Commun. Nonlinear Sci. Numer. Simul. 14(11), 3738–3742 (2009)
Biswas A., Milovic D.: Chiral solitons with Bohm potential by He’s variational principle. Phys. Atom. Nuclei 74(5), 781–783 (2011)
Girgis L., Biswas A.: A study of solitary waves by He’s semi-inverse variational principle. Waves Random Compl Med. 21(1), 96–104 (2011)
Triki H., Wazwaz A.M.: Bright and dark soliton solutions for a K(m, n) equation with t-dependent coefficients. Phys. Lett. A 373, 2162–2165 (2009)
Wazwaz A.M.: New solitary wave solutions to the modified Kawahara equation. Phys. Lett. A 360, 588–592 (2007)
Wazwaz A.-M.: Compactons and solitary wave solutions for the Boussinesq wave equation and its generalized form. Appl. Math. Comput. 182(1), 529–535 (2006)
Yildirim A., Mohyud-Din S.T.: A variational approach to soliton solutions of good Boussinesq equation. J. King Saud Univ. Sci. 22(4), 205–208 (2010)
Ma W.X., Maruno K.: Complexiton solutions of the Toda lattice equation. Phys. A 343, 219–237 (2004)
Ma W.X.: Soliton, positon and negaton solutions to a Schrodinger self consistent source equation. J. Phys. Soc. Jpn. 72, 3017–3019 (2003)
Ma W.X.: Complexiton solutions of the Korteweg–de Vries equation with self consistent sources. Chaos Solitons Fractals 26, 1453–1458 (2005)
Ling L.H., Qiang L.X.: Exact Solutions to (2+1)-dimensional Kaup–Kupershmidt equation. Commun. Theor. Phys. 52, 795–800 (2009)
Reyes E.G.: Nonlocal symmetries and the Kaup–Kupershmidt equation. J. Math. Phys. 46, 073507 (2005)
Kupershmidt B.A.: A super Korteweg–de Vries equation:an integrable system. Phys. Lett. A 102, 213 (1994)
Zait R.A.: Bäcklund transformations, cnoidal wave and travelling wave solutions of the SK and KK equations. Chaos Solitons Fractals 15, 673 (2003)
He J.H., Wu X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700 (2006)
Yıldırım A., Pınar Z.: Application of Exp-function method for nonlinear reaction-diffusion equations arising in mathematical biology. Comput. Math. Appl. 60, 1873–1880 (2010)
Mohyud-din S.T., Noor M.A., Noor K.I.: Exp-function method for solving higher-order boundary value problems. Bull. Inst. Math. Acad. Sinica (New Series) 4(2), 219–234 (2009)
Zhang S.: Exp-function method for solving Maccari’s system. Phys. Lett. A 371, 65–71 (2007)
He J.H., Abdou M.A.: New periodic solutions for nonlinear evolution equations using exp-function method. Chaos Solitons Fractals 34, 1421–1429 (2007)
Ma W.X., You Y.: Solving the Korteweg–de Vries equation by its bilinear form: Wronskian solutions. Trans. Am. Math. Soc. 357, 1753–1778 (2005)
Ma W.X., Li C.X., He J.S.: A second Wronskian formulation of the Boussinesq equation. Nonlinear Anal. Theory Methods Appl. 70, 4245–4258 (2009)
Ma W.X., Lee J.-H.: A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation. Chaos Solitons Fractals 42, 1356–1363 (2009)
Ma W.X., Huang T.W., Zhang Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scripta 82, 065003 (2010)
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Bhrawy, A.H., Biswas, A., Javidi, M. et al. New Solutions for (1+1)-Dimensional and (2+1)-Dimensional Kaup–Kupershmidt Equations. Results. Math. 63, 675–686 (2013). https://doi.org/10.1007/s00025-011-0225-7
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DOI: https://doi.org/10.1007/s00025-011-0225-7