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Parametric Resonance in Integrable Systems and Averaging on Riemann Surfaces

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Abstract

In this paper, we consider adiabatic deformations of Riemann surfaces that preserve the integrability of the corresponding dynamic system, which leads to the appearance of modulated quasiperiodic motions, similar to the effect of parametric resonance. We show that in this way it is possible to control the amplitude and frequency of nonlinear modes. We consider several examples of the dynamics of top-type systems.

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References

  1. V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  2. M. Audin, Spinning Tops. A Course on Integrable Systems, Cambridge Univ., Cambridge (1999).

  3. R. F. Bikbaev, “Korteweg–de Vries equation with finite-gap boundary conditions and Whitham deformations of Riemannian surfaces,” Funkts. Anal. Prilozh., 23, No. 4, 1–10 (1989).

    MathSciNet  MATH  Google Scholar 

  4. R. F. Bikbaev, “Finite-gap attractors and transition processes of the shock wave type in integrable systems,” Zap. Nauch. Sem. POMI, 199, 25–36 (1992).

    MathSciNet  MATH  Google Scholar 

  5. R. F. Bikbaev and R. A. Sharipov, “Asymptotics as t→∞ of solutions to the Cauchy problem for the Korteweg–de Vries equation in the class of potentials with finite-gap behavior as x → ±∞,” Teor. Mat. Fiz., 78, No. 3, 345–356 (1989).

    Article  MathSciNet  Google Scholar 

  6. A. I. Bobenko, Integration of classical tops by the inverse scattering method [in Russian], preprint LOMI No. 10-87 (1987).

  7. A. I. Bobenko, A. G. Reyman, and M. A. Semenov-Tian-Shansky, “The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions,” Commun. Math. Phys., 122, No. 2, 321–354 (1989).

    Article  MathSciNet  Google Scholar 

  8. D. Bohm and L. Foldy, “249-258,” Phys. Rev., 70 (1946).

  9. A. V. Borisov and I. S. Mamaev, Modern Theory of Integrable Systems [in Russian], Moscow (2003).

  10. B. V. Chirikov, Nonlinear Resonance [in Russian], Novosibirsk (1977).

  11. B. A. Dubrovin, “Theta functions and nonlinear equations,” Usp. Mat. Nauk, 36, No. 2, 11–92 (1981).

    MathSciNet  MATH  Google Scholar 

  12. B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable Systems. Part 1,” Itogi Nauki Tekhn. Dynam. Sist., 4, 179–285 (1985).

    Google Scholar 

  13. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Korteweg–de Vries-type nonlinear equations, finite-gap linear operators, and Abelian varieties,” Usp. Mat. Nauk, 31, No. 1 (187), 55–136 (1976).

  14. J. Fajans, E. Gilson, L. Friedland, “Autoresonant excitation of diocotron waves,” Phys. Rev. Lett., 82, 4444 (1999).

    Article  Google Scholar 

  15. J. D. Fay, “Theta Functions on Riemann Surfaces,” Lect. Notes Math., 352 (1973).

  16. H. Flaschka, M. G. Forest, and D. W. McLaughlin, “Multiphase averaging and the inverse spectral solution of the Korteweg–de Vries equation,” Commun. Pure Appl. Math., 33, No. 6, 732–784 (1980).

    Article  MathSciNet  Google Scholar 

  17. I. M. Krichever, “Averaging method for two-dimensional integrable equations,” Funkts. Anal. Prilozh., 22, No. 3, 200–213 (1989).

    Article  MathSciNet  Google Scholar 

  18. G. A. Kuzmak, “Asymptotic solutions of second-order nonlinear differential equations with variable coefficients,” Prikl. Mat. Mekh., 23, 515–526 (1959).

    MATH  Google Scholar 

  19. Yu. A. Mitropol’sky, Nonstationary Processes in Nonlinear Oscillation Systems [in Russian], Akad Nauk Ukr. SSR, Kiev (1955).

  20. V. Yu. Novokshenov, “Whitham deformations of top-type integrable dynamical systems,” Funkts. Anal. Prilozh., 27, No. 2, 50–62 (1993).

    MATH  Google Scholar 

  21. V. Yu. Novokshenov, “Whitham deformations of two-dimensional Liouville tori,” in: Singular Limits of Dispersive Waves (N. Ercolani and D. Levermore, eds.), World Scientific (1996), pp. 105–116.

  22. V. Yu. Novokshenov, “Time asymptotics for soliton equations with step-like initial conditions,” Sovr. Mat. Prilozh., 5, 138–168 (2003).

    Google Scholar 

  23. A. Pikovsky, M. Rosenblum, and J. Kurths Synchronization — A Universal Concept in Nonlinear Sciences (2001).

  24. M. I. Rabinovich and D. I. Trubetskov, Introduction to the Theory of Oscillations and Waves [in Russian], Nauka, Moscow (1984).

  25. S. P. Tsarev, “Geometry of hydrodynamical Hamiltonian systems. Generalized hodograph method,” Izv. Akad. Nauk SSSR. Ser. Mat., 54, No. 5, 1048–1068 (1990).

    Google Scholar 

  26. V. I. Veksler, “A new method of acceleration of relativistic particles,” Zh. Eksp. Teor. Fiz., 9, No. 3, 153–158 (1945).

    Google Scholar 

  27. A. P. Veselov, “Finite-gap potentials and integrable systems on the sphere with a quadratic potential,” Funkts. Anal. Prilozh., 14, No. 1, 48–50 (1980).

    Article  Google Scholar 

  28. G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974).

    MATH  Google Scholar 

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  1. Institute of Mathematics with Computing Center, Ufa Federal Research Center of the Russian Academy of Sciences, Ufa, Russia

    V. Yu. Novokshenov

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  1. V. Yu. Novokshenov
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Correspondence to V. Yu. Novokshenov.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 163, Differential Equations, 2019.

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Novokshenov, V.Y. Parametric Resonance in Integrable Systems and Averaging on Riemann Surfaces. J Math Sci 258, 65–80 (2021). https://doi.org/10.1007/s10958-021-05536-7

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  • DOI: https://doi.org/10.1007/s10958-021-05536-7

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