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Resonance Clustering in Wave Turbulent Regimes: Integrable Dynamics

Published online by Cambridge University Press:  20 August 2015

Miguel D. Bustamante and
Elena Kartashova
Miguel D. Bustamante*
Affiliation:
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
Elena Kartashova*
Affiliation:
IFA, J. Kepler University, Linz 4040, Austria
*
Email:miguel.bustamante@ucd.ie
Corresponding author.Email:elena.kartaschova@jku.at
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Abstract

Two fundamental facts of the modern wave turbulence theory are 1) existence of power energy spectra in k-space, and 2) existence of “gaps” in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: kinetic, described by wave kinetic equations and power energy spectra; discrete, characterized by resonance clustering; and mesoscopic, where both types of wave field time evolution coexist. In this review paper we present the results on integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes. Using a novel method based on the notion of dynamical invariant we show that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others-only for particular initial conditions. We also identify chaotic behaviour in some cases. Physical implications of the results obtained are discussed.

Keywords

47.10.Df47.10.Fg02.70.DhResonance clusteringHamiltonian formulationwave turbulent regimesNR-diagramintegrable dynamics

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Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011, pp. 1211 - 1240
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Almonte, F., Jirsa, V. K., Large, E. W. and Tuller, B., Integration and segregation in auditory streaming, Phys. D, 212(1-2) (2005), 137–159.CrossRefGoogle Scholar
[2]Arnold, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, Grundleheren der mathematischen Wissenschaften 250, A Series of Comprehensive Studies in Mathematics, New York Heidelberg Berlin, Springer-Verlag, 1983.Google Scholar
[3]Bolsino, A. V. and Fomenko, A. T., Integrable Hamiltonian Systems, Chapmann & Hall/CRC, 2004.Google Scholar
[4]Bustamante, M. D. and Kartashova, E., Dynamics of nonlinear resonances in Hamiltonian systems, Europhys. Lett., 85 (2009), 14004–1–14004-5.Google Scholar
[5]Bustamante, M. D. and Kartashova, E., Effect of the dynamical phases on the nonlinear amplitudes’ evolution, Europhys. Lett., 85 (2009), 34002–1–34002-6.Google Scholar
[6]Calogero, F., Why are certain nonlinear PDEs both widely applicapable and intergable? in book: Zakharov, V. E. (ed.), What is Integrability?, 1 (1991), 1–62, Springer Series in Nonlinear Dynamics, Springer Verlag.Google Scholar
[7]Chow, C. C., Henderson, D. and Segur, H., A generalized stability criterion for resonant triad interactions, Fluid Mech., 319 (1996), 67–76.Google Scholar
[8]Cretin, B. and Vernier, D., Quantized amplitudes in a nonlinear resonant electrical circuit, E-print: arxiv.org/abs/0801.1301 (2008).CrossRefGoogle Scholar
[9]Constantin, A. and Kartashova, E., Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves, Europhys. Lett., 86 (2009), 29001–1–29001-6.Google Scholar
[10] A.Constantin, Kartashova, E. and Wahlén, E., Discrete wave turbulence of rotational capillary water waves, E-print: arXiv:1001.1497 (2010).Google Scholar
[11]Denissenko, P., Lukaschuk, S. and Nazarenko, S., Gravity surface wave turbulence in a laboratory flume, Phys. Rev. Lett., 99 (2007), 014501–1–014501-4.CrossRefGoogle Scholar
[12]Dubrovin, B. and Zhang, Y., Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, E-print: arXiv:math/0108160v1 (2001).Google Scholar
[13]Eissa, M., El-Ganainia, W. A. A. and Hameda, Y. S., Saturation, stability and resonance of non-linear systems, Phys. A: Stat. Mech. Appl., 356(2-4) (2005), 341–358.Google Scholar
[14]Evans, N. W., Superintegrability in classical mechanics, Phys. Rev. A, 41(10) (1990), 5666–5676.Google Scholar
[15]Ghil, M., Kondrashov, D., Lott, F. and Robertson, A. W., Intraseasonal oscillations in the mid-latitudes: observations, theory, and GCM results, in: Proc. ECMWF/CLIVAR Workshop on Simulations and prediction of Intra-Seasonal Variability, Reading, UK, 2004.Google Scholar
[16]Haenggi, P., Stochastic resonance in biology how noise can enhance detection of weak signals and help improve biological information processing, ChemPhysChem, 3(3) (2002), 285–290.Google Scholar
[17]Hasselmann, K., A criterion for nonlinear wave stability, Fluid Mech., 30 (1967), 737–739.Google Scholar
[18]Horvat, K., Miskovic, M. and Kuljaca, O., Avoidance of nonlinear resonance jump in turbine governor positioning system using fuzzy controller, Industrial Technology, 2 (2003), 881–885.Google Scholar
[19]Johnson, R. S., A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, 1997.Google Scholar
[20]Kartashova, E. A., Piterbarg, L. I. and Reznik, G. M., Weakly nonlinear interactions between Rossby waves on a sphere, Oceanology, 29 (1990), 405–411.Google Scholar
[21]Kartashova, E., Partitioning of ensembles of weakly interacting dispersing waves in resonators into disjoint classes, Phys. D, 46 (1990), 43–56.CrossRefGoogle Scholar
[22]Kartashova, E., Weakly nonlinear theory of finite-size effects in resonators, Phys. Rev. Lett., 72 (1994), 2013–2016.Google Scholar
[23]Kartashova, E., Wave resonances in systems with discrete spectra, in book: Nonlinear Waves and Weak Turbulence, ed., V. E., ZakharovSeries: Advances in the MathematicalSciences, AMS Transl., 2 (1998), 95–129.Google Scholar
[24]Kartashova, E., A model of laminated turbulence, JETP Lett., 83(7) (2006), 341–345.CrossRefGoogle Scholar
[25]Kartashova, E., Fast computation algorithm for discrete resonances among gravity waves, Low Temp. Phys., 145(1-4) (2006), 286–295.Google Scholar
[26]Kartashova, E., Exact and quasi-resonances in discrete water-wave turbulence, Phys. Rev. Lett., 98(21) (2007), 214502–1–214502-4.Google Scholar
[27]Kartashova, E., Discrete wave turbulence, Europhys. Lett., 87 (2009), 44001–1–44001-6.Google Scholar
[28]Kartashova, E., Nonlinear Resonance Analysis: Theory, Computation, Applications, Cambridge University Press, 2010.Google Scholar
[29]Kartashova, E. and Kartashov, A., Laminated wave turbulence: generic algorithms I, Int. J. Mod. Phys. C, 17(11) (2006), 1579–1596.Google Scholar
[30]Kartashova, E. and Kartashov, A., Laminated wave turbulence: generic algorithms II, Commun. Comput. Phys., 2(4) (2007), 783–794.Google Scholar
[31]Kartashova, E. and Kartashov, A., Laminated wave turbulence: generic algorithms III, Phys. A: Stat. Mech. Appl., 380 (2007), 66–74.Google Scholar
[32]Kartashova, E. and L’vov, V. S., A model of intra-seasonal oscillations in the Earth atmosphere, Phys. Rev. Lett., 98(19) (2007), 198501–1–198501-4.Google Scholar
[33]Kartashova, E. and L’vov, V. S., Cluster dynamics of planetary waves, Europhys. Lett., 83 (2008), 50012–1–50012-6.CrossRefGoogle Scholar
[34]Kartashova, E. and Mayrhofer, G., Cluster formation in mesoscopic systems, Phys. A: Stat. Mech. Appl., 385 (2007), 527–542.Google Scholar
[35]Kartashova, E., Raab, C., Feurer, Ch., Mayrhofer, G. and Schreiner, W., Symbolic computations for nonlinear wave resonances, in: Extreme Ocean Waves: 97–128, Eds: Pelinovsky, E. and Harif, Ch., Springer, 2008.Google Scholar
[36]Kartashova, E. and Shabat, A., Computable Integrability, Chapter 1: General notions and ideas, RISC Report Series, N0002-04-2005, 2005.Google Scholar
[37]Karuzskii, A. L., Lykov, A. N., Perestoronin, A. V. and Golovashkin, A. I., Microwave nonlinear resonance incorporating the helium heating effect in superconducting microstrip resonators, Phys. C: Superconduct., 408-410 (2004), 739–740.Google Scholar
[38]Kovriguine, D. A. and Maugin, G. A., Multiwave nonlinear couplings in elastic structures, Math. Prob. Eng., doi:10.1155/MPE/2006/76041, 2006.CrossRefGoogle Scholar
[39]Kundu, M. and Bauer, D., Nonlinear resonance absorption in the laser-cluster interaction, Phys. Rev. Lett., 96 (2006), 123401–1–123401-4.Google Scholar
[40]Leyvraz, F., personal communication, 2008.Google Scholar
[41]Longuet-Higgins, M. S. and Gill, A. E., Resonant interactions between planetary waves, Proc. Roy. Soc. Lond., A299 (1967), 120–140.Google Scholar
[42]L’vov, V. S., Pomyalov, A., Procaccia, I. and Rudenko, O., Finite-dimensional turbulence of planetary waves, Phys. Rev. E, 80 (2009), 066319–43.Google Scholar
[43]Lynch, P., Resonant motions of the three-dimensional elastic pendulum, Int. J. Nonlinear Mech., 37 (2002), 258–264.CrossRefGoogle Scholar
[44]Lynch, P., The swinging spring: a simple model of amospheric balance, in book: Large-Scale Atmosphere-Ocean Dynamics: Vol II: Geometric Methods and Models: 64–108, Eds. Norbury, J. and Roulstone, I., Cambridge University Press, 2002.Google Scholar
[45]Lynch, P., private communication, 2009.Google Scholar
[46]Lynch, P. and Houghton, C., Pulsation and pecession of the resonant swinging spring, Phys. D, 190 (2004), 38–62.Google Scholar
[47]McGoldrick, L. F., Resonant interactions among capillary-gravity waves, Fluid Mech., 21 (1967), 305–331.Google Scholar
[48]Menyuk, C. R., Chen, H. H. and Lee, Y. C., Restricted multiple three-wave interactions: Pan-levè analysis, Phys. Rev. A, 27 (1983), 1597–1611.Google Scholar
[49]Menyuk, C. R., Chen, H. H. and Lee, Y. C., Restricted multiple three-wave interactions: integrable cases of this system and other related systems, J. Math. Phys., 24 (1983), 1073–1079.Google Scholar
[50]Mikhailov, A. V., Shabat, A. B. and Sokolov, V. V., The symmetry approach to classification of integrable equations, in book: What is integrability?, 115, Ed. Zakharov, V. E., Springer Series in Nonlinear Dynamics, Springer Verlag, 1991.Google Scholar
[51]Moser, J., On invariant curvesof area preserving mappings of anannulus, Nachr. Akad. Wiss. Goett, Math. Phys. Kl., 1 (1962), 1–20.Google Scholar
[52]Murdock, J., Normal Forms and Unfoldings for Local Dynamical Systems, Springer-Verlag, New York, 2003.CrossRefGoogle Scholar
[53]Musher, S. L., Rubenchik, A. M. and Zakharov, V. E., Hamiltonian approach to the description of nonlinear plasma phenomena, Phys. Rep., 129 (1985), 285–366.Google Scholar
[54]Nayfeh, A. H., Method of Normal Forms, Wiley-Interscience, NY, 1993.Google Scholar
[55]Oliveira, H. P. de, Soares, I. Damia o and Tonini, E. V., Nonlinear resonance of Kolmogorov-Arnold-Moser tori in bouncing universes, Cosmology and Astroparticle Physics, doi: 10.1088/1475-7516/2006/02/015, 2006.Google Scholar
[56]Olver, P. J., Applications of Lie groups to differential equations, Graduated Texts in Mathematics, 107, Springer, 1993.Google Scholar
[57]Ostrovskii, L. A., Rybak, S. A. and Tsimring, S. L., Negative energy waves in hydrodynamics, Sov. Phys. Uspekhi., 29(11) (1986), 1040–1052.Google Scholar
[58]Pedlosky, J., Geophysical Fluid Dynamics, Second Edition, Springer, 1987.Google Scholar
[59]Phillips, O. M., On the dynamics of unsteady gravity waves of infinite amplitude, Fluid Mech., 9 (1960), 193–217.Google Scholar
[60]Piterbarg, L. I., Hamiltonian formalism for Rossby waves, AMS Trans., 2 (1998), 131–166.Google Scholar
[61]Punzmann, H., Shats, M. G. and Xia, H., Phase randomization of three-wave interactions in capillary waves, Phys. Rev. Lett., 103 (2009), 064502–1–064502-4.Google Scholar
[62]Pushkarev, A. N., On the Kolmogorov and frozen turbulence in numerical simulation of capillary waves, Euro. J. Mech. B: Fluids, 18(3) (1999), 345–351.Google Scholar
[63]Syvokon, V. E., Kovdrya, Yu. Z. and Nasyedkin, K. A., Nonlinear features of phonon-ripplon modes in the electron crystal over liquid Helium, Low Temp. Phys., 144(1-3) (2006), 35–48.Google Scholar
[64]Treumann, R. and Baumjohann, W., Advanced Space Plasma Physics, Imperial Colledge Press, London, 2001.Google Scholar
[65]Tsytovich, V. N., Nonlinear effects in plasma, Plenum, New York, 1970.Google Scholar
[66]Verheest, F., Proof of integrability for five-wave interaactions in a case with unequal coupling constants, J. Phys. A: Math. Gen., 21 (1988), L545–L549.CrossRefGoogle Scholar
[67]Verheest, F., Integrability of restricted multiple three-wave interactions II, coupling constants with ratios 1 and 2, J. Math. Phys., 29 (1988), 2197–2201.Google Scholar
[68]Whittaker, E. T. and Watson, G. N., A Course in Modern Analysis, 4th ed., Cambridge, England, Cambridge University Press, 1990.Google Scholar
[69]Wright, W. B., Budakian, R. and Putterman, S. J., Diffusing light photography of fully developed isotropic ripple turbulence, Phys. Rev. Lett., 76 (1996), 4528–4531.Google Scholar
[70]Zakharov, V. E., Lvov, V. S. and Falkovich, G., Kolmogorov Spectra of Turbulence, Springer-Verlag, Berlin, 1992.Google Scholar
[71]Zakharov, V. E., Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid, Euro. J. Mech. B: Fluids, 18 (1999), 327–344.Google Scholar
[72]Zakharov, V. E., Stability of periodic waves of finite amplitude on the surface of deep fluid, Zh. Prikl. Mekh. Tekh. Fiz., 9 (1968), 86–94.Google Scholar
[73]Zakharov, V. E. and Filonenko, N.N., Weak turbulence of capillary waves, Appl. Mech. Tech. Phys., 4 (1967), 500–515.Google Scholar
[74]Zakharov, V. E., Korotkevich, A. O., Pushkarev, A. N. and Dyachenko, A. I., Mesoscopic wave turbulence, JETP Lett., 82 (2005), 487–491.Google Scholar
[75]Zakharov, V. E. and Shulman, E. I., Integrability of nonlinear systems and perturbation theory, in book: What is Integrability? Ed. Zakharov, V. E., Springer, 185, 1990.Google Scholar