Nonlinear Dynamics

, Volume 92, Issue 2, pp 781–792 | Cite as

Some group-invariant solutions of potential Kadomtsev–Petviashvili equation by using Lie symmetry approach

  • Mukesh Kumar
  • Atul Kumar Tiwari
Original Paper


A variety of closed-form solutions such as multiple-front wave, kink wave, waves interaction, curve-shaped multisoliton, parabolic and stationary wave solutions have been obtained by using invariance of the concerned potential Kadomtsev–Petviashvili (PKP) equation under the one-parameter Lie group of transformations. Lie symmetry transformations have been applied to generate various forms of invariant solutions of the PKP equation. The solutions provide extensive rich physical structure due to the existence of various arbitrary constants and functions. Results have been traced in context to spatiotemporal dynamics. Dynamic behavior of the results have been analyzed in terms of various wave propagations. Numerical simulation has been performed to obtain appropriate visual appearance of the traced solutions. The nature of solutions is investigated both analytically and physically through their evolutionary profiles by considering adequate choices of arbitrary functions and constants.


PKP equation Similarity method Soliton Multiple-front wave Invariant solutions 



The authors sincerely acknowledge the inputs provided by Dr. Tanuj Nandan, Associate Professor, School of Management Studies, MNNIT, Allahabad. One of the authors, Atul Kumar Tiwari, is grateful to CSIR-UGC, New Delhi, for the award of Senior Research Fellowship for writing this manuscript.


  1. 1.
    Senthilvelan, M.: On the extended applications of homogenous balance method. Appl. Math. Comput. 123, 381–388 (2001)MathSciNetMATHGoogle Scholar
  2. 2.
    Li, D.S., Zhang, H.Q.: New soliton-like solutions to the potential Kadomstev–Petviashvili (PKP) equation. Appl. Math. Comput. 146, 381–384 (2003)MathSciNetMATHGoogle Scholar
  3. 3.
    Batiha, B., Batiha, K.: An analytic study of the (2 + 1)-dimensional potential Kadomtsev–Petviashvili equation. Adv. Theor. Appl. Mech. 3, 513–520 (2010)MATHGoogle Scholar
  4. 4.
    Inan, I.E., Kaya, D.: Some exact solutions to the potential Kadomtsev–Petviashvili equation and to a system of shallow water wave equations. Phys. Lett. A 355, 314–318 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dai, Z., Liu, J., Liu, Z.: Exact periodic kink-wave and degenerative soliton solutions for potential Kadomtsev–Petviashvili equation. Commun. Nonlinear Sci. Numer. Simul. 15, 2331–2336 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Rosenhaus, V.: On conserved densities and asymptotic behaviour for the potential Kadomtsev–Petviashvili equation. J. Phys. A Math. Gen. 39, 7693–7703 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Li, D.S., Zhang, H.Q.: Symbolic computation and various exact solutions of potential Kadomstev–Petviashvili equation. Appl. Math. Comput. 145, 351–359 (2003)MathSciNetMATHGoogle Scholar
  8. 8.
    Jawad, A.J.M., Petković, M.D., Biswas, A.: Soliton solutions for nonlinear Calaogero–Degasperis and potential Kadomtsev–Petviashvili equations. Comput. Math. Appl. 62, 2621–2628 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Pohjanpelto, J.: The cohomology of the variational bicomplex invariant under the symmetry algebra of the potential Kadomtsev–Petviashvili equation. J. Nonlinear Math. Phys. 4, 364–376 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ren, B., Yu, J., Liu, X.Z.: Nonlocal symmetries and interaction solutions for potential Kadomtsev–Petviashvili equation. Commun. Theor. Phys. 65, 341–346 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Wazwaz, A.M.: Multiple-soliton solutions for a (3 + 1)-dimensional generalized KP equation. Commun. Nonlinear Sci. Numer. Simul. 17, 491–495 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Wazwaz, A.M.: Variants of a (3+1)-dimensional generalized BKP equation: multiple-front waves solutions. Comput. Fluids 97, 164–167 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wazwaz, A.M., El-Tantawy, S.A.: A new integrable (3+1)-dimensional KdV-like model with its multiple-soliton solutions. Nonlinear Dyn. 83, 1529–1534 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Wazwaz, A.M., El-Tantawy, S.A.: A new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation. Nonliear Dyn. 84, 1107–1112 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Bluman, G.W., Cole, J.D.: Similarity Methods for Differential Equations. Springer, New York (1974)CrossRefMATHGoogle Scholar
  16. 16.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)CrossRefMATHGoogle Scholar
  17. 17.
    Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)CrossRefMATHGoogle Scholar
  18. 18.
    Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)MATHGoogle Scholar
  19. 19.
    Kumar, M., Kumar, R.: On some new exact solutions of incompressible steady state Navier–Stokes equations. Meccanica 49, 335–345 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kumar, M., Kumar, R.: On new similarity solutions of the Boiti–Leon–Pempinelli system. Commun. Theor. Phys. 61, 121–126 (2014)CrossRefMATHGoogle Scholar
  21. 21.
    Kumar, M., Kumar, R., Kumar, A.: Some more similarity solutions of the (2 + 1)-dimensional BLP system. Comput. Math. Appl. 70, 212–221 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kumar, M., Kumar, R.: Soliton solutions of KD system using similarity transformations method. Comput. Math. Appl. 73, 701–712 (2017)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sahoo, S., Garai, G., Ray, S.S.: Lie symmetry analysis for similarity reduction and exact solutions of modified KdV–Zakharov–Kuznetsov equation. Nonlinear Dyn. 87, 1995–2000 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Johnpillai, A.G., Kara, A.H., Biswas, A.: Symmetry solutions and reductions of a class of generalized (2 + 1)-dimensional Zakharov–Kuznetsov equation. Int. J. Nonlinear Sci. Numer. Simul. 12, 45–50 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kumar, S., Hama, A., Biswas, A.: Solutions of Konopelchenko–Dubrovsky equation by traveling wave hypothesis and Lie symmetry approach. Appl. Math. Inf. Sci. 8, 1533–1539 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Özer, T.: An application of symmetry groups to nonlocal continuum mechanics. Comput. Math. Appl. 55, 1923–1942 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Özer, T.: New exact solutions to the CDF equations. Chaos Solitons Fractals 39, 1371–1385 (2009)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Sekhar, T.R., Sharma, V.D.: Similarity analysis of modified shallow water equations and evolution of weak waves. Commun. Nonlinear Sci. Numer. Simul. 17, 630–636 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Bira, B., Sekhar, T.R., Zeidan, D.: Application of Lie groups to compressible model of two-phase flows. Comput. Math. Appl. 71, 46–56 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ndogmo, J.C.: Symmetry properties of a nonlinear acoustics model. Nonlinear Dyn. 55, 151–167 (2009)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Springer, Berlin (2009)CrossRefMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMotilal Nehru National Institute of Technology AllahabadAllahabadIndia

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