Theoretical and Mathematical Physics

, Volume 167, Issue 3, pp 762–771 | Cite as

Asymptotic analysis of a model of nuclear magnetic autoresonance

  • L. A. KalyakinEmail author
  • O. A. Sultanov
  • M. A. Shamsutdinov


We study the system of three first-order differential equations arising when averaging the Bloch equations in the theory of nuclear magnetic resonance. For the averaged system, we construct an asymptotic series for the stable solution with an infinitely increasing amplitude. This result gives a key to understanding the autoresonance in weakly dissipative magnetic systems as a phenomenon of significant growth of the magnetization initiated by a small external pumping.


nonlinear equation perturbation small parameter asymptotic behavior autoresonance dissipation stability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. I. Kurkin and E. A. Turov, NMR in Magnetically Ordered Materials and Its Application [in Russian], Fizmatlit, Moscow (1990).Google Scholar
  2. 2.
    A. S. Borovik-Romanov, Yu. M. Bun’kov, B. S. Dumesh, M. I. Kurkin, M. P. Petrov, and V. P. Chekmarev, Sov. Phys. Usp., 27, 235 (1984).ADSCrossRefGoogle Scholar
  3. 3.
    A. I. Neishtadt, J. Appl. Math. Mech., 39, 594–605 (1975).CrossRefMathSciNetGoogle Scholar
  4. 4.
    V. I. Arnol’d, V. V. Kozlov, and A. I. Neuishtadt, Mathematical Aspects of Classical and Celestial Mechanics [in Russian] (Dynamical Systems 3, Sovrem. Probl. Mat. Fund. Naprav.), VINITI, Moscow (1985); English transl. (Encycl. Math. Sci., Vol. 3), Springer, Berlin (2006).Google Scholar
  5. 5.
    A. P. Itin, A. I. Neishtadt, and A. A. Vasiliev, Phys. D, 141, 281–296 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    V. V. Kozlov and S. D. Furta, Asymptotic Expansions of Solutions of Strongly Nonlinear Systems of Differential Equations [in Russian], Moscow State Univ. Press, Moscow (1996).zbMATHGoogle Scholar
  7. 7.
    A. D. Bruno, Power Geometry in Algebraic and Differential Equations [in Russian], Nauka, Moscow (1998); English transl. (North-Holland Math. Lib., Vol. 57), North-Holland, Amsterdam (2000).Google Scholar
  8. 8.
    L. A. Kalyakin, Theor. Math. Phys., 137, 1476–1484 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Fajans and L. Frièdland, Amer. J. Phys., 69, 1096–1102 (2001).ADSCrossRefGoogle Scholar
  10. 10.
    L. A. Kalyakin, Russ. Math. Surveys, 63, 791–857 (2008).ADSzbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    L. A. Kalyakin and M. A. Shamsutdinov, Proc. Inst. Math. Mech., 259(Suppl. 2), S124–S140 (2007).zbMATHGoogle Scholar
  12. 12.
    L. A. Kalyakin and M. A. Shamsutdinov, Theor. Math. Phys., 160, 960–967 (2009).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    N. N. Bogoliubov and Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations [in Russian], Nauka, Moscow (1974); English transl. prev. ed., Gordon and Breach, New York (1961).Google Scholar
  14. 14.
    R. N. Garifullin, Dokl. Math., 70, 799–802 (2004).Google Scholar
  15. 15.
    A. N. Kuznetsov, Funct. Anal. Appl., 23, No. 4, 308–317 (1989).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    O. A. Sultanov, Ufimsk. Mat. Zh., 2, No. 4, 88–98 (2010).zbMATHGoogle Scholar
  17. 17.
    I. G. Malkin, Theory of Stability of Motion [in Russian], GITTL, Moscow (1952); English transl., R. Oldenbourg, München (1959).zbMATHGoogle Scholar
  18. 18.
    N. N. Krasovskii, Certain Problems in the Theory of Stability of Motion [in Russian], Fizmatlit, Moscow (1959).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  • L. A. Kalyakin
    • 1
    Email author
  • O. A. Sultanov
    • 2
  • M. A. Shamsutdinov
  1. 1.Institute of MathematicsRASUfaRussia
  2. 2.Ufa State Aircraft Technical UniversityUfaRussia

Personalised recommendations