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Pramana

, 92:1 | Cite as

Group classification, conservation laws and Painlevé analysis for Klein–Gordon–Zakharov equations in (3\(+\)1)-dimension

  • Manjit Singh
  • R K Gupta
Article
  • 72 Downloads

Abstract

In this paper, we study Klein–Gordon–Zakharov equations which describe the propagation of strong turbulence of the Langmuir wave in a high-frequency plasma. Using the symbolic manipulation tool Maple, the classifications of symmetry algebra are carried out, and the construction of several local non-trivial conservation laws based on a direct method of Anco and Bluman is illustrated. Starting with determination of symmetry algebra, the one- and two-dimensional optimal systems are constructed, and optimality is also established using various invariant functions of full adjoint action. Apart from classification and construction of several conservation laws, the Painlevé analysis is also performed in a symbolic manner which describes the non-integrability of equations.

Keywords

Klein–Gordon–Zakharov equations optimal systems conservation laws Painlevé analysis 

PACS Nos

02.20.Qs 02.20.Sv 02.30.Jr 11.30.j 

Notes

Acknowledgements

Rajesh Kumar Gupta thanks the University Grants Commission for sponsoring this research under the Research Award Scheme (F. 30-105 / 2016 (SA-II)).

References

  1. 1.
    N H Ibragimov, Russ. Math. Surv.  47(4), 89 (1992)CrossRefGoogle Scholar
  2. 2.
    A Stubhaug, The mathematician Sophus Lie: It was the audacity of my thinking (Springer Science & Business Media, Berlin, 2013)zbMATHGoogle Scholar
  3. 3.
    L V Ovsiannikov, Group analysis of differential equations (Academic Press, New York, 1982)Google Scholar
  4. 4.
    P J Olver, Applications of Lie groups to differential equations (Springer-Verlag Inc., New York, 1986) Vol. 107zbMATHGoogle Scholar
  5. 5.
    S Lie, Theorie der tranformationsgruppen (B.G. Teubner, Leipzig, 1888)Google Scholar
  6. 6.
    L Boza, E M Fedriani, J Nunez and A F Tenorio, Rev. Union Math. Argentina  54(2), 75 (2013)Google Scholar
  7. 7.
    J Patera, P Winternitz and H Zassenhaus, J. Math. Phys.  16(8), 1597 (1975)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    J Patera, P Winternitz and H Zassenhaus, J. Math. Phys.  16(8), 1615 (1975)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    J Patera, R T Sharp, P Winternitz and H Zassenhaus, J. Math. Phys.  18(12), 2259 (1977)ADSCrossRefGoogle Scholar
  10. 10.
    J Patera and P Winternitz, J. Math. Phys.  18(7), 1449 (1977)ADSCrossRefGoogle Scholar
  11. 11.
    F Galas and E W Richter, Physica D  50(2), 297 (1991)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    K S Chou, G X Li and C Qu, J. Math. Anal. Appl.  261(2), 741 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    K S Chou and C Qu, Acta Appl. Math.  83(3), 257 (2004)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    S G Thornhill and D Ter Haar, Phys. Rep.  43(2), 43 (1978)ADSCrossRefGoogle Scholar
  15. 15.
    T Ozawa, K Tsutaya and Y Tsutsumi, Ann. l’IHP Anal. Non-Linéaire  12, 459 (1995)CrossRefGoogle Scholar
  16. 16.
    K Tsutaya, Nonlinear Anal: Theory, Methods Appl.  27(12), 1373 (1996)Google Scholar
  17. 17.
    T Ozawa, K Tsutaya and Y Tsutsumi, Math. Ann.  313(1), 127 (1999)MathSciNetCrossRefGoogle Scholar
  18. 18.
    J Li, Chaos Solitons Fractals  34(3), 867 (2007)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Y Shang, Y Huang and W Yuan, Comput. Math. Appl.  56(5), 1441 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    M Ismail and A Biswas, Appl. Math. Comput.  217(8), 4186 (2010)MathSciNetGoogle Scholar
  21. 21.
    M Dehghan and A Nikpour, Comput. Phys. Commun.  184(9), 2145 (2013)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    H L Zhen, B Tian, Y Sun, J Chai and X Y Wen, Phys. Plasmas (1994-present)  22(10), 102304 (2015)ADSCrossRefGoogle Scholar
  23. 23.
    G Bluman and S C Anco, Symmetry and integration methods for differential equations (Springer-Verlag Inc., New York, 2002) Vol. 154zbMATHGoogle Scholar
  24. 24.
    R K Gupta and K Singh, Commun. Nonlinear Sci. Numer. Simul.  16(11), 4189 (2011)ADSCrossRefGoogle Scholar
  25. 25.
    R K Gupta and M Singh, Nonlinear Dyn.  87(3), 1543 (2016)CrossRefGoogle Scholar
  26. 26.
    E S Cheb Terrab and K Von Bülow, Comput. Phys. Commun.  90(1), 102 (1995)ADSCrossRefGoogle Scholar
  27. 27.
    S V Coggeshall and J M Vehn, J. Math. Phys.  33(10), 3585 (1992)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    X Hu, Y Li and Y Chen, J. Math. Phys.  56(5), 053504 (2015)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    I I Ryzhkov, Commun. Nonlinear Sci. Numer. Simul.  11(2), 172 (2006)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    H Koetz, Z. Naturf. A  48(4), 535 (1993)ADSGoogle Scholar
  31. 31.
    R Naz, F M Mahomed and D P Mason, Appl. Math. Comput.  205(1), 212 (2008)MathSciNetGoogle Scholar
  32. 32.
    N H Ibragimov, J. Math. Anal. Appl.  333(1), 311 (2007)MathSciNetCrossRefGoogle Scholar
  33. 33.
    S C Anco, Symmetry  9(3), 33 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    S C Anco and G Bluman, Phys. Rev. Lett.  78(15), 2869 (1997)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    S C Anco and G Bluman, Eur. J. Appl. Math.  13(05), 545 (2002)Google Scholar
  36. 36.
    S C Anco and G Bluman, Eur. J. Appl. Math.  13(05), 567 (2002)Google Scholar
  37. 37.
    G Bluman, A F Cheviakov and S C Anco, Applications of symmetry methods to partial differential equations (Springer, New York, 2010) Vol. 168zbMATHGoogle Scholar
  38. 38.
    D Poole and W Hereman, Appl. Anal.  89(4), 433 (2010)MathSciNetCrossRefGoogle Scholar
  39. 39.
    J Weiss, M Tabor and G Carnevale, J. Math. Phys.  24(3), 522 (1983)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    X Gui Qiong and L Zhi Bin, Comput. Phys. Commun.  161(1–2), 65 (2004)ADSGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Yadavindra College of EngineeringPunjabi UniversityTalwandi SaboIndia
  2. 2.Centre for Mathematics and Statistics, School of Basic and Applied SciencesCentral University of PunjabBathindaIndia

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