Precisely Soluble Model of the Parametrical Resonance

Authors

DOI:

https://doi.org/10.20535/1810-0546.2015.4.50335

Keywords:

Parametric resonance, Increments of parametric oscillation, Lame potentials, Exactly solvable model, Elliptic Weierstrass functions

Abstract

Background. Construction of the exactly solvable model of the parametric resonance with the solutions in analytical form. The study of the parametric resonance basic properties in the exactly solvable model in case that the periodic potential of the Hill equation is the Lame potential.

Objective. Using of the Lame one-dimensional finite-zone potentials for calculating of the parametric resonance parameters.

Methods. To achieve this purpose the methods of theoretical physics, the methods of the elliptic functions theory for use of the finite-zone potentials, the weak- and strong-binding approximations for studying of the limit cases of finite-zone potential are used.

Results. The study of exactly solvable models in case that the periodic potential of the Hill equation is the Lame potential proportional to the elliptic Weierstrass function is performed. These potentials correspond to a spectrum that consists of a finite number of zones. Zone approach for the analysis of the effect of parametric resonance is developed. The increments of growth of oscillations in the Lame potential model for external force applied to the oscillator are calculated. It is shown that the increment of growth reaches a maximum in the middle of zone (parametric resonance condition) and that derivatives of the increment of growth with respect to the natural frequency tend to infinity on the zone boundaries. This result corresponds to a conversion to infinity of derivatives of quasifrequency with respect to the natural frequency in case of the parametric oscillator fluctuations with constrained amplitude.

Conclusions. The Lame potential model gives an opportunity to explore the dependence of the increment of growth on the intensity of external influence in case that the weak- and strong-binding approximations are modulated by the potential periods.

Author Biographies

Віктор Григорович Бар'яхтар, National Academy of Sciences of Ukraine

Victor Baryakhtar.

Doctor of Physical and Mathematical Sciences, professor, academician of the National Academy of Sciences of Ukraine

Ганна Володимирівна Самар, National Technical University of Ukraine “KPI”

Ganna Samar.

Teaching fellow

References

I. Bloch et al., “Many-body physics with ultracold gases”, Rev. Mod. Phys, no. 80, p. 885, 2008. DOI: 10.1103/RevModPhys.80.

E.G. Galkina et al., “Dynamics of antiferromagnets exposed to ultrashort magnetic field pulses”, Pis’ma v ZETF, vol. 93, no. 12, pp. 792–796, 2011 (in Russian).

V.G. Baryakhtar et al., “A general method for solution of some problems of motion stabilization and destabilization”, in II Int. Conf. Quantum Electrodyn. Stat. Phys., Kharkiv, Ukraine, September 19–23, 2006, p. 179.

L.I. Mandel'shtam, “Lectures on oscillationsс”, in Complete Set of Works, vol. 4.Moscow,USSR: AN SSSR, 1955, 511 p. (in Russian).

A.A. Andronov et al., Vibration Theory.Moscow,USSR: Nauka, 1981, 918 p. (in Russian).

E.D. Courant and H.S. Snyder, “Theory of alternating – gradient synchrotron”, Ann. Phys., vol. 3, pp.1–123, 1958.

V.G. Baryakhtar et al., “A new method for calculating the electron spectrum of solids. Application to high-temperature superconductivity”, Phys. Stat. Sol. (b), vol. 169, pp. 105–114, 1992.

E.D. Belokolos, Mathematical Foundations of the Theory of Solids with a Quasi-Periodic Structure.Kyiv,USSR: ITF ANUSSR,1982 (in Russian).

E.D. Belokolos and V.Z. Enolskii, “Reduction of Abelian functions and Algebraically integrable systems”, J. Math. Sci., vol. 106, no. 6, pp. 3395–3486, 2001; Ibid, vol. 108, no. 3, pp. 295–374 (2002).

A.V. Turbiner, “Quasi-exactly-solvable problems and sl(2) algebra”, Comm. Math. Phys., vol. 118, pp. 467–474, 1988.

F. Correa et al., Hidden nonlinear supersymmetry of finite-gap Lame equation [Online]. Avaliable: arXiv:hep-th/0608096v2. DOI: 10.1016/j.physletb.2006.11.02.

E.D. Belokolos, “Spectra of the schrodinger operators with finite-gap potentials and integrable systems”, Proc. Institute of Mathematics of NAS of Ukraine, vol. 43, part I, pp. 273–280, 2002.

V.G. Baryakhtar et al., “On the electron motion theory in a crystal”, Naukovi Visti NTUU KPI, no. 4, pp. 7–13, 2006 (in Ukrainian).

I.M. Lifshic et al., Electron Theory of Metals.Moscow,USSR: Nauka, 1971, 415 p. (in Russian).

E.T. Uitteker and G.N. Vatson, А Course of Modern Analysis. Moscow: USSR , vol. 2, 2002, 515 p. (in Russian).

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Published

2015-09-18

Issue

No. 4 (2015): Physics and Mathematics

Section

Art

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