Results in Mathematics

, Volume 63, Issue 1–2, pp 675–686 | Cite as

New Solutions for (1+1)-Dimensional and (2+1)-Dimensional Kaup–Kupershmidt Equations

  • A. H. Bhrawy
  • Anjan Biswas
  • M. Javidi
  • Wen Xiu Ma
  • Zehra Pınar
  • Ahmet Yıldırım
Article

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.

Mathematics Subject Classification (2010)

35D99 65N99 

Keywords

(1+1)-dimensional and (2+1)-dimensional Kaup–Kupershmidt (KK) equations Exact solutions Exp-function method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)MATHCrossRefGoogle Scholar
  2. 2.
    Biswas A., Milovic D.: Chiral solitons with Bohm potential by He’s variational principle. Phys. Atom. Nuclei 74(5), 781–783 (2011)CrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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)MATHCrossRefGoogle Scholar
  5. 5.
    Wazwaz A.M.: New solitary wave solutions to the modified Kawahara equation. Phys. Lett. A 360, 588–592 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    Ma W.X., Maruno K.: Complexiton solutions of the Toda lattice equation. Phys. A 343, 219–237 (2004)MathSciNetGoogle Scholar
  9. 9.
    Ma W.X.: Soliton, positon and negaton solutions to a Schrodinger self consistent source equation. J. Phys. Soc. Jpn. 72, 3017–3019 (2003)MATHCrossRefGoogle Scholar
  10. 10.
    Ma W.X.: Complexiton solutions of the Korteweg–de Vries equation with self consistent sources. Chaos Solitons Fractals 26, 1453–1458 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Ling L.H., Qiang L.X.: Exact Solutions to (2+1)-dimensional Kaup–Kupershmidt equation. Commun. Theor. Phys. 52, 795–800 (2009)MATHCrossRefGoogle Scholar
  12. 12.
    Reyes E.G.: Nonlocal symmetries and the Kaup–Kupershmidt equation. J. Math. Phys. 46, 073507 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kupershmidt B.A.: A super Korteweg–de Vries equation:an integrable system. Phys. Lett. A 102, 213 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zait R.A.: Bäcklund transformations, cnoidal wave and travelling wave solutions of the SK and KK equations. Chaos Solitons Fractals 15, 673 (2003)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    He J.H., Wu X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700 (2006)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    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)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    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)MathSciNetMATHGoogle Scholar
  18. 18.
    Zhang S.: Exp-function method for solving Maccari’s system. Phys. Lett. A 371, 65–71 (2007)MATHCrossRefGoogle Scholar
  19. 19.
    He J.H., Abdou M.A.: New periodic solutions for nonlinear evolution equations using exp-function method. Chaos Solitons Fractals 34, 1421–1429 (2007)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    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)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    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)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    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)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    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)CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • A. H. Bhrawy
    • 1
    • 2
  • Anjan Biswas
    • 3
  • M. Javidi
    • 4
  • Wen Xiu Ma
    • 5
  • Zehra Pınar
    • 6
  • Ahmet Yıldırım
    • 5
    • 6
  1. 1.Faculty of ScienceBeni-Suef UniversityBeni-SuefEgypt
  2. 2.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Department of Mathematical SciencesDelaware State UniversityDoverUSA
  4. 4.Department of Mathematics, Faculty of scienceRazi UniversityKermanshahIran
  5. 5.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  6. 6.Department of MathematicsEge UniversityBornovaTurkey

Personalised recommendations