Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory
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Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory, 2019, Volume 163, Pages 65–80 (Mi into451)  
Parametric resonance in integrable systems and averaging on Riemann surfaces

V. Yu. Novokshenov

Institution of Russian Academy of Sciences Institute of Mathematics with Computer Center, Ufa
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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 quasi-periodic 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.
Keywords: integrable system, Lax pair, algebraic-geometric method, finite-gap solution, theta function, invariant torus, parametric resonance, Whitham deformation, synchronization, phase capture.
Bibliographic databases:
Document Type: Article
UDC: 517.928, 517.933, 517.984.54
MSC: 37J35, 37K15, 37K20
Language: Russian
Citation: V. Yu. Novokshenov, “Parametric resonance in integrable systems and averaging on Riemann surfaces”, Differential Equations, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 163, VINITI, Moscow, 2019, 65–80
Citation in format AMSBIB
\Bibitem{Nov19}
\by V.~Yu.~Novokshenov
\paper Parametric resonance in integrable systems and averaging on Riemann surfaces
\inbook Differential Equations
\serial Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz.
\yr 2019
\vol 163
\pages 65--80
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/into451}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4014976}
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