Journal of Mathematical Sciences

, Volume 208, Issue 2, pp 222–228 | Cite as

The Singular Hill Equation and Generalized Lindemann–Stieltjes Method

  • V. A. Koutvitsky
  • E. M. Maslov

Based on the Lindemann–Stieltjes method, we propose an approach to the solution of a singular Hill equation. We consider Hill equations with logarithmic and fractional power singularities. In the space of parameters, we find resonance zones and compute the Floquet exponent.


Parametric Resonance Real Scalar Fractional Power Hill Equation Parametric Instability 
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  1. 1.
    J.J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, New York (1950).Google Scholar
  2. 2.
    E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge, (1927).Google Scholar
  3. 3.
    P. B. Greene, L. Kofman, A. Linde, and A. A. Starobinsky, “Structure of resonance in preheating after inflation,” Phys. Rev. D 56, 6175 (1997).CrossRefGoogle Scholar
  4. 4.
    D. I. Kaiser, “Resonance structure for preheating with massless fields,” Phys. Rev. D 57, 702 (1998).CrossRefGoogle Scholar
  5. 5.
    F. Finkel, A. González-López, A. L. Maroto, and M. A. Rodríguez, “The Lamé equation in parametric resonance after inflation,” Phys. Rev. D 62, 103515 (2000).CrossRefGoogle Scholar
  6. 6.
    E. M. Maslov and A. G. Shagalov, “Dynamics of first-order phase transitions in the φ 4 -φ 6 model caused by the parametric instability of the metastable vacuum,” Physica D 152-153, 769–778 (2001).MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. A. Koutvitsky and E. M. Maslov, “Instability of coherent states of a real scalar field,” J. Math. Phys. 47, 022302 (2006).MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. A. Koutvitsky and E. M. Maslov, “Parametric instability of the real scalar pulsons,” Phys. Lett., A 336, No. 1, 31–36 (2005).MATHCrossRefGoogle Scholar
  9. 9.
    J. D. Barrow, P. Parsons, “Inflationary models with logarithmic potentials,” Phys. Rev. D 52, 5576 (1995).CrossRefGoogle Scholar
  10. 10.
    K. Enqvist and J. McDonald, “Q-balls and baryogenesis in the MSSM,” Phys. Lett., B 425, 309-321 (1998).CrossRefGoogle Scholar
  11. 11.
    S. D. H. Hsu, “Cosmology of nonlinear oscillations,” Phys. Lett., B 567, 9–11 (2003).MATHCrossRefGoogle Scholar
  12. 12.
    X. Dong, B. Horn, E. Silverstein, A. Westphal, “Simple exercises to flatten your potential,” Phys. Rev. D 84, 026011 (2011).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation RASTroitskRussia

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