Theoretical and Mathematical Physics

, Volume 172, Issue 1, pp 1001–1016 | Cite as

Rolling in the Higgs model and elliptic functions

  • I. Ya. Arefeva
  • I. V. Volovich
  • E. V. Piskovskiy
Article

Abstract

Asymptotic methods in nonlinear dynamics such as, for example, the Krylov-Bogoliubov averaging method and the KAM theory are commonly used to improve perturbation theory results in the regime of small oscillations. But for a series of problems in nonlinear dynamics, in particular, for the Higgs equation in field theory, not only the small-oscillation regime but also the rolling regime is of interest. Both slow- and fast-rolling regimes are important in the Friedmann cosmology. We present an asymptotic method for solving the Higgs equation in the rolling regime. We show that to improve the perturbation theory in the rolling regime, expanding a solution known in terms of elliptic functions not in trigonometric functions (as with the averaging method in the small-oscillation regime) but in hyperbolic functions turns out to be effective. We estimate the accuracy of the second approximation. We also investigate the Higgs equation with damping.

Keywords

asymptotic methods in nonlinear dynamics rolling Higgs model 

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References

  1. 1.
    N. M. Krylov and N. N. Bogoliubov, Introduction to Non-linear Mechanics [in Russian], Acad. Sci. Ukr. SSR, Kiev (1937); English transl. (Annals Math. Stud., Vol. 11), Princeton Univ. Press, Princeton, N. J. (1943).Google Scholar
  2. 2.
    N. N. Bogolyubov and Yu. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (2005); English transl., Gordon and Breach, New York (1961).MATHGoogle Scholar
  3. 3.
    V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial mechanics [in Russian], VINITI, Moscow (1985); English transl., Springer, Berlin (1997).Google Scholar
  4. 4.
    V. V. Kozlov and S. D. Furta, Asymptotic Expansions of Solutions of Strongly Nonlinear Systems of Differential Equations [in Russian], Regular and Chaotic Dynamics, Izevsk (2009).Google Scholar
  5. 5.
    V. A. Rubakov, Classical Theory of Gauge Fields [in Russian], URSS, Moscow (1999); English transl., Princeton Univ. Press, Princeton, N. J. (2002).Google Scholar
  6. 6.
    V. F. Mukhanov, Physical Foundations of Cosmology, Cambridge Univ. Press, Cambridge (2005).MATHCrossRefGoogle Scholar
  7. 7.
    D. S. Gorbunov and V. A. Rubakov, Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory [in Russian], Inst. Nucl. Res., Russ. Acad. Sci., Moscow (2009); English transl., World Scientific, Hackensack, N. J. (2011).Google Scholar
  8. 8.
    I. Ya. Aref’eva and I. V. Volovich, JHEP, 1108, 102 (2011); arXiv:1103.0273v2 [hep-th] (2011).ADSCrossRefGoogle Scholar
  9. 9.
    I. Ya. Aref’eva, N. V. Bulatov, and R. V. Gorbachev, “FRW cosmology with non-positively defined Higgs potentials,” arXiv:1112.5951v3 [hep-th] (2011).Google Scholar
  10. 10.
    I. V. Volovich, Theor. Math. Phys., 164, 1128–1135 (2010).CrossRefGoogle Scholar
  11. 11.
    E. V. Piskovskiy and I. V. Volovich, “On the correspondence between Newtonian and functional mechanics,” in: Quantum Bio-Informatics IV: From Quantum Information to Bio-informatics (Quantum Prob. White Noise Anal., Vol. 28, L. Accardi, W. Freudenberg, and M. Ohaya, eds.), World Scientific, Hackensack, N. J. (2011), p. 363–372.Google Scholar
  12. 12.
    A. M. Zhuravskii, Handbook of Elliptic Functions [in Russian], Nauka, Moscow (1941).Google Scholar
  13. 13.
    N. I. Akhiezer, Elements of the Theory of Elliptic Functions [in Russian], Nauka, Moscow (1970); (Transl. Math. Monogr., Vol. 79), Amer. Math. Soc., Providence, R. I. (1990).Google Scholar
  14. 14.
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Natl. Bur. Stds. Applied Math. Series, Vol. 55), U.S. Gov. Printing Office, Washington, D. C. (1964).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • I. Ya. Arefeva
    • 1
  • I. V. Volovich
    • 1
  • E. V. Piskovskiy
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

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