On new symmetry, series solution and conservation laws of nonlinear coupled Higgs field equation

Abstract

The work presents systematic investigations on invariant analysis and the analytic solution of the second order Higgs equation. On employing Lie classical approach, new symmetry and the corresponding reduction of the system are obtained. Explicit convergent infinite series solution of the reduced system is obtained. Local conservation laws of the system are derived by the multiplier approach.

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References

  1. 1.

    S. Abbasbandy, A. Shirzadi, The first integral method for modified Benjamin-Bona-Mahony equation. Commun. Nonlinear Sci. Numer. Simul. 15(7), 1759–1764 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    S.C. Anco, G. Bluman, Direct construction method for conservation laws of partial differential equations Part I: examples of conservation law classifications. Eur. J. Appl. Math. 13(5), 545–566 (2002)

    Article  Google Scholar 

  3. 3.

    G. Bluman, S. Anco, Symmetry and Integration Methods for Differential Equations (Springer, Berlin, 2008)

    Google Scholar 

  4. 4.

    G.W. Bluman, A.F. Cheviakov, S.C. Anco, Construction of conservation laws: how the direct method generalizes Noether’s theorem. In Proceedings of 4th Workshop “Group Analysis of Differential Equations and Integrability, vol. 1 (2009) pp. 1–23

  5. 5.

    W. George, J.D. Cole, Similarity Methods for Differential Equations (Springer, Berlin, 2012)

    Google Scholar 

  6. 6.

    W. George, S. Bluman, Symmetries and Differential Equations (Springer, Kumei, 2013)

    Google Scholar 

  7. 7.

    P.A. Clarkson, E.L. Mansfield, Algorithms for the nonclassical method of symmetry reductions. SIAM J. Appl. Math. 54(6), 1693–1719 (1994)

    MathSciNet  Article  Google Scholar 

  8. 8.

    E. Fan, J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys. Lett. A 305(6), 383–392 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    Z. Feng, The first-integral method to study the Burgers-Korteweg-de Vries equation. J. Phys. Math. General 35(2), 343 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Z. Fu, S. Liu, S. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A 290(1–2), 72–76 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    M.L. Gandarias, M.S. Bruzon, Classical and nonclassical symmetries of a generalized Boussinesq equation. J. Nonlinear Math. Phys. 5(1), 8–12 (1998)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    N.H. Ibragimov, A new conservation theorem. J. Math. Anal. Appl. 333(1), 311–328 (2007)

    MathSciNet  Article  Google Scholar 

  13. 13.

    H. Jafari, M. Zabihi, M. Saidy, Application of homotopy perturbation method for solving gas dynamics equation. Appl. Math. Sci. 2(48), 2393–2396 (2008)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    L. Kaur, R.K. Gupta, Kawahara equation and modified Kawahara equation with time dependent coefficients: Symmetry analysis and generalized-expansion method. Math. Methods Appl. Sci. 36(5), 584–600 (2013)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    K. Khan, M.A. Akbar, Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified Kdv-Zakharov-Kuznetsov equations using the modified simple equation method. Ain Shams Eng. J. 4(4), 903–909 (2013)

    Article  Google Scholar 

  16. 16.

    S. Kumar, K. Singh, R.K. Gupta, Coupled Higgs field equation and Hamiltonian amplitude equation: Lie classical approach and (G’/G)-expansion method. Pramana 79(1), 41–60 (2012)

    ADS  Article  Google Scholar 

  17. 17.

    R. Naz, Conservation laws for some systems of nonlinear partial differential equations via multiplier approach. J. Appl. Math. (2012). https://doi.org/10.1155/2012/871253

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    E. Noether, Invariante Variationsprobleme Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, math-phys. klasse, 235-257 (1918). Translated by MA Travel. Transp. Theory Stat. Phys., 1(3):183–207, (1971)

  19. 19.

    P.J. Olver, Applications of Lie groups to Differential Equations (Springer, Berlin, 2000)

    Google Scholar 

  20. 20.

    W. Rudin, Principles of Mathematical Analysis (China Machine Press, Beijing, 2004)

    Google Scholar 

Download references

Acknowledgements

It is hereby acknowledged that the author (Pinki Kumari) is grateful to the University Grants Commission for assisting her financially (Reference ID 19/06/2016(i)EU-V).

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Correspondence to Sachin Kumar.

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Kumari, P., Gupta, R.K. & Kumar, S. On new symmetry, series solution and conservation laws of nonlinear coupled Higgs field equation. Eur. Phys. J. Plus 135, 476 (2020). https://doi.org/10.1140/epjp/s13360-020-00460-2

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