Geodesic
Parametric resonance in integrable systems and averaging on Riemann surfaces
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 65-80.

Voir la notice de l'article provenant de la source Math-Net.Ru

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, algebraic-geometric method, finite-gap solution, theta function, parametric resonance, Whitham deformation, synchronization, phase capture.
Mots-clés : Lax pair, invariant torus 
@article{INTO_2019_163_a4,
     author = {V. Yu. Novokshenov},
     title = {Parametric resonance in integrable systems and averaging on {Riemann} surfaces},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {65--80},
     publisher = {mathdoc},
     volume = {163},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_163_a4/}
}
TY  - JOUR
AU  - V. Yu. Novokshenov
TI  - Parametric resonance in integrable systems and averaging on Riemann surfaces
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2019
SP  - 65
EP  - 80
VL  - 163
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2019_163_a4/
LA  - ru
ID  - INTO_2019_163_a4
ER  - 
%0 Journal Article
%A V. Yu. Novokshenov
%T Parametric resonance in integrable systems and averaging on Riemann surfaces
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2019
%P 65-80
%V 163
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2019_163_a4/
%G ru
%F INTO_2019_163_a4
V. Yu. Novokshenov. Parametric resonance in integrable systems and averaging on Riemann surfaces. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 65-80. http://geodesic.mathdoc.fr/item/INTO_2019_163_a4/

[1] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1967

[2] Bikbaev R. F., “Uravnenie Kortevega—de Friza s konechnozonnymi granichnymi usloviyami i uizemovskie deformatsii rimanovykh poverkhnostei”, Funkts. anal. prilozh., 23:4 (1989), 1–10 | MR | Zbl

[3] Bikbaev R. F., “Konechnozonnye attraktory i perekhodnye protsessy tipa udarnykh voln v integriruemykh sistemakh”, Zap. nauchn. sem. POMI, 199, 1992, 25–36 | Zbl

[4] Bikbaev R. F., Sharipov R. A., “Asimptotika pri $t \to \infty$ resheniya zadachi Koshi dlya uravneniya Kortevega—de Friza v klasse potentsialov s konechnozonnym povedeniem pri $x \to \pm\infty$”, Teor. mat. fiz., 78:3 (1989), 345–356 | MR | Zbl

[5] Bobenko A. I., Integrirovanie klassicheskikh volchkov metodom obratnoi zadachi, Preprint LOMI No 10-87, 1987

[6] Borisov A. V., Mamaev I. S., Sovremennaya teoriya integriruemykh sistem, RKhD, M., 2003

[7] Veksler V. I., “Novyi metod uskoreniya relyativistskikh chastits”, ZhETF., 9:3 (1945), 153–158

[8] Veselov A. P., “Konechnozonnye potentsialy i integriruemye sistemy na sfere s kvadratichnym potentsialom”, Funkts. anal. prilozh., 14:1 (1980), 48–50 | MR | Zbl

[9] Dubrovin B. A., “Teta-funktsii i nelineinye uravneniya”, Usp. mat. nauk., 36:2 (1981), 11–92 | MR | Zbl

[10] Dubrovin B. A., Krichever I. M., Novikov S. P., “Integriruemye sistemy. 1”, Dinamicheskie sistemy–4, Itogi nauki i tekhniki, VINITI, M., 1985, 179–285 | MR

[11] Dubrovin B. A., Matveev V. B., Novikov S. P., “Nelineinye uravneniya tipa Kortevega—de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, Usp. mat. nauk., 31:1 (187) (1976), 55–136 | MR | Zbl

[12] Krichever I. M., “Metod usredneniya dlya dvumernykh integriruemykh uravnenii”, Funkts. anal. prilozh., 22:3 (1989), 200–213

[13] Kuzmak G. A., “Asimptoticheskie resheniya nelineinykh differentsialnykh uravnenii vtorogo poryadka s peremennymi koeffitsientami”, Prikl. mat. mekh., 23 (1959), 515–526 | Zbl

[14] Mitropolskii Yu. A., Nestatsionarnye protsessy v nelineinykh kolebatelnykh sistemakh, Izd-vo AN USSR, Kiev, 1955 | MR

[15] Novokshenov V. Yu., “Uizemovskie deformatsii integriruemykh dinamicheskikh sistem tipa volchkov”, Funkts. anal. prilozh., 27:2 (1993), 50–62 | MR | Zbl

[16] Novokshenov V. Yu., “Vremennýe asimptotiki dlya solitonnykh uravnenii so stupenchatymi nachalnymi usloviyami”, Sovr. mat. prilozh., 5 (2003), 138–168 | Zbl

[17] Oden M., Vraschayuschiesya volchki: kurs integriruemykh sistem, RKhD, M., 1999

[18] Pikovskii A., Rozenblyum M., Kurts Yu., Sinkhronizatsiya: Fundamentalnoe nelineinoe yavlenie, Tekhnosfera, M., 2003

[19] Rabinovich M. I., Trubetskov D. I., Vvedenie v teoriyu kolebanii i voln, Nauka, M., 1984 | MR

[20] Uizem Dzh., Lineinye i nelineinye volny, Mir, M., 1977 | MR

[21] Tsarev S. P., “Geometriya gamiltonovykh sistem gidrodinamicheskogo tipa. Obobschennyi metod godografa”, Izv. AN SSSR. Ser. mat., 54:5 (1990), 1048–1068 | MR | Zbl

[22] Chirikov B. V., Nelineinyi rezonans, Izd-vo NGU, Novosibirsk, 1977

[23] Bobenko A. I., Reyman A. G, Semenov-Tian-Shansky M. A., “The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions”, Commun. Math. Phys., 122:2 (1989), 321–354 | DOI | MR

[24] Bohm D., Foldy L., “The Theory of the Synchrotron”, Phys. Rev. A., 70 (1946) | DOI

[25] Fay J. D., Theta functions on Riemann surfaces, Lect. Notes Math., 352, 1973 | DOI | MR | Zbl

[26] Flaschka H., Forest M. G., McLaughlin D. W., “Multiphase averaging and the inverse spectral solution of the Korteweg–de Vries equation”, Commun. Pure Appl. Math., 33:6 (1980), 732–784 | DOI | MR

[27] Fajans J., Gilson E., Friedland L., “Autoresonant excitation of diocotron waves”, Phys. Rev. Lett., 82 (1999), 4444 | DOI

[28] Novokshenov V. Yu., “Whitham deformations of two-dimensional Liouville tori”, Singular Limits of Dispersive Waves, eds. Ercolani N., Levermore D., World Scientific, 1996, 105–116 | MR