Asymptotic expansion of solutions in a rolling problem

  • I. Ya. Aref’eva
  • I. V. Volovich


Asymptotic methods in the theory of differential equations and in nonlinear mechanics are commonly used to improve perturbation theory in the small oscillation regime. However, in some problems of nonlinear dynamics, in particular for the Higgs equation in field theory, it is important to consider not only small oscillations but also the rolling regime. In this article we consider the Higgs equation and develop a hyperbolic analogue of the averaging method. We represent the solution in terms of elliptic functions and, using an expansion in hyperbolic functions, construct an approximate solution in the rolling regime. An estimate of accuracy of the asymptotic expansion in an arbitrary order is presented.


Asymptotic Expansion STEKLOV Institute Average Method Elliptic Function Relaxation Oscillation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. F. Mishchenko, “Asymptotic Theory of Relaxation Oscillations Described by Second-Order Systems,” Mat. Sb. 44(4), 457–480 (1958).MathSciNetGoogle Scholar
  2. 2.
    E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations (Nauka, Moscow, 1975; Plenum Press, New York, 1980).Google Scholar
  3. 3.
    D. S. Gorbunov and V. A. Rubakov, Introduction to the Theory of the Early Universe: Hot Big Bang Theory (LKI, Moscow, 2008; World Sci., Singapore, 2010); Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory (Krasand, Moscow, 2010; World Sci., Singapore, 2011).Google Scholar
  4. 4.
    I. Ya. Aref’eva, N. V. Bulatov, and R. V. Gorbachev, “FRW Cosmology with Non-positively Defined Higgs Potentials,” arXiv: 1112.5951v3 [hep-th].Google Scholar
  5. 5.
    V. A. Rubakov, Classical Theory of Gauge Fields (URSS, Moscow, 1999; Princeton Univ. Press, Princeton, NJ, 2002).Google Scholar
  6. 6.
    N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics (Izd. Akad. Nauk Ukr. SSR, Kiev, 1937) [in Russian].Google Scholar
  7. 7.
    N. N. Bogoliubov and Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations (Nauka, Moscow, 1974; Gordon and Breach, New York, 1961).Google Scholar
  8. 8.
    D. V. Anosov, “Averaging in Systems of Ordinary Differential Equations with Rapidly Oscillating Solutions,” Izv. Akad. Nauk SSSR, Ser. Mat. 24(5), 721–742 (1960).MathSciNetMATHGoogle Scholar
  9. 9.
    V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (VINITI, Moscow, 1985), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 3; Engl. transl. in Dynamical Systems III (Springer, Berlin, 2006), Encycl. Math. Sci. 3.Google Scholar
  10. 10.
    V. V. Kozlov and S. D. Furta, Asymptotics of Solutions for Strongly Nonlinear Systems of Differential Equations (Regular and Chaotic Dynamics, Izhevsk, 2009) [in Russian].Google Scholar
  11. 11.
    I. Ya. Aref’eva, L. V. Joukovskaya, and A. S. Koshelev, “Time Evolution in Superstring Field Theory on Non-BPS Brane. 1: Rolling Tachyon and Energy-Momentum Conservation,” J. High Energy Phys., No. 9, 012 (2003).Google Scholar
  12. 12.
    V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (Nauka, Moscow, 1994; World Sci., Singapore, 1994).CrossRefGoogle Scholar
  13. 13.
    V. S. Vladimirov and Ya. I. Volovich, “Nonlinear Dynamics Equation in p-adic String Theory,” Teor. Mat. Fiz. 138(3), 355–368 (2004) [Theor. Math. Phys. 138, 297–309 (2004)]MathSciNetCrossRefGoogle Scholar
  14. 14.
    V. S. Vladimirov, “The Equation of the p-adic Open String for the Scalar Tachyon Field,” Izv. Ross. Akad. Nauk, Ser. Mat. 69(3), 55–80 (2005) [Izv. Math. 69, 487–512 (2005)].MathSciNetGoogle Scholar
  15. 15.
    I. Ya. Aref’eva and I. V. Volovich, “Cosmological Daemon,” J. High Energy Phys., No. 8, 102 (2011).Google Scholar
  16. 16.
    I. V. Volovich, “Bogolyubov Equations and Functional Mechanics,” Teor. Mat. Fiz. 164(3), 354–362 (2010) [Theor. Math. Phys. 164, 1128–1135 (2010)].CrossRefGoogle Scholar
  17. 17.
    E. V. Piskovskiy and I. V. Volovich, “On the Correspondence between Newtonian and Functional Mechanics,” in Quantum Bio-Informatics IV, Ed. by L. Accardi, W. Freudenberg, and M. Ohya (World Sci., Singapore, 2011), pp. 363–372.CrossRefGoogle Scholar
  18. 18.
    I. Ya. Aref’eva, I. V. Volovich, and E. V. Piskovskiy, “Rolling in the Higgs Model and Elliptic Functions,” Teor. Mat. Fiz. (in press).Google Scholar
  19. 19.
    L. Accardi, Y. G. Lu, and I. Volovich, Quantum Theory and Its Stochastic Limit (Springer, Berlin, 2002).MATHGoogle Scholar
  20. 20.
    S. P. Suetin, Numerical Analysis of Some Characteristics of Limiting Cycle of the Free Van der Pol Equation (Steklov Math. Inst., Moscow, 2010), Sovrem. Probl. Mat. 14.Google Scholar
  21. 21.
    A. M. Zhuravskii, Handbook of Elliptic Functions (Izd. Akad. Nauk SSSR, Moscow, 1941) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • I. Ya. Aref’eva
    • 1
  • I. V. Volovich
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations