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References

Published online by Cambridge University Press:  06 December 2010

Elena Kartashova
Elena Kartashova
Affiliation:
Johannes Kepler Universität Linz
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Nonlinear Resonance Analysis
Theory, Computation, Applications
 , pp. 209 - 220
Publisher: Cambridge University Press
Print publication year: 2010

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References

L. V., Abdurakhimov, Y. M., Brazhnikov, G. V., Kolmakov, and A. A., Levchenko. Study of high-frequency edge of turbulent cascade on the surface of He-II. J. Phys.: Conf. Ser. 150 (3) (2009), 032001.Google Scholar
R., Abramov, G., Kovac, and A. J., Majda. Hamiltonian structure and statistically relevant conserved quantities for the truncated Burgers–Hopf equation. Comm. Pure App. Math. 56 (1) (2003), 1–46.Google Scholar
M., Abramovitz and I. A., Stegun. Handbook of Mathematical Functions (Dover Publications, 1972).Google Scholar
A. D., Alexandrov. Uniqueness theorems for surfaces in general. Vestnik Leningrad Univ. Math. 11 (1956), 5–17.Google Scholar
F., Almonte, V. K., Jirsa, E. W., Large, and B., Tuller. Integration and segregation in auditory streaming. Physica D 212 (1–2) (2005), 137–59.Google Scholar
P. W., Anderson and H., Suhl. Instability in the motion of ferromagnets at high microwave power levels. Phys. Rev. 100 (1955), 1788–9.Google Scholar
V. I., Arnold. Proof of a theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surveys 18 (1963), 9–36.Google Scholar
V. I., Arnold, V. V., Kozlov, and A. I., Neishstadt. Mathematical Aspects of Classical and Celestial Mechanics (Springer, 1997).Google Scholar
A., Baker. Transcendental Number Theory (Cambridge University Press, 1975).Google Scholar
R., Baraka and W., Schreiner. Semantic querying of mathematical web service descriptions. LNCS 4184 (2006), 73–87.Google Scholar
J., Barrow-Green. Poincare and the Three Body Problem (American Mathematical Society, London Mathematical Society, 1997).Google Scholar
F., Bashforth and J. C., Adams. An Attempt to Test the Theories of Capillary Action (Cambridge University Press, 1883).Google Scholar
J. M., Basilla. On the solution of x2 + dy2 = m. Proc. Japan. Acad. Series A 80 (10) (2004), 40–1.Google Scholar
T. B., Benjamin. The threshhold classification of unstable disturbances in flexible surfaces bounding inviscid flows. Fluid Mech. 16 (1963), 436–50.Google Scholar
T. B., Benjamin and J. E., Feir. The disintegration of wavetrains in deep water, Part 1. Fluid Mech. 27 (1967), 417–31.Google Scholar
G. P., Berman and F. M., Israilev. The Fermi-Pasta-Ulam problem: fifty years of progress. Chaos 15 (1) (2005), 015104-1-18.Google Scholar
P., Bergé, Y., Pomeau, and Ch., Vidal. Order within Chaos: Towards a Deterministic Approach to Turbulence (Wiley-Interscience, 1987).Google Scholar
R. E., Berg and T. S., Marshall. Wilberforce pendulum oscillations and normal modes. Am. J. Phys. 59 (1) (1991), 32–7.Google Scholar
A. S., Besicovitch. On the linear independence of fractional powers of integers. J. Lond. Math. Soc. 15 (1) (1940), 3–6.Google Scholar
F. W., Bessel. Untersuchungen über die Länge des einfachen Secundenpendels. Abh. Berlin Akad. (Berlin, 1828; reprinted by H. Bruns, Leipzig, 1889).Google Scholar
L., Biferale and I., Procaccia. Anisotropy in turbulent flows and in turbulent transport. Phys. Rep. 414 (2006), 43–164.Google Scholar
L., Biven, S., Nazarenko, and A., Newell. Breakdown of wave turbulence and the onset of intermittency. Phys. Lett. A 280 (2001), 28–32.Google Scholar
L., Bourouiba. Discreteness and resolution effects in rapidly rotating turbulence. Phys. Rev. E 78 (5) (2008), 056309-1-12.Google Scholar
G. W., Brandstator. A striking example of the atmosphere's leading traveling pattern. J. Atm. Sci. 44 (1987), 2310–23.Google Scholar
M., Brazhnikov, G., Kolmakov, A., Levchenko, and L., Mezhov-Deglin. Observation of capillary turbulence on the water surface in a wide range of frequencies. Europhys. Lett. 58 (2002), 510–15.Google Scholar
B. M., Bredichin. Free number semigroups of power densities. Mat. Sborn. 46 (88) (1958), 143–58 [in Russian].Google Scholar
A. D., Bruno. Local Methods in Nonlinear Differential Equations (Springer, Berlin, 1989).Google Scholar
A., Bruno and V., Edneral. Normal forms and integrability of ODE systems. Prog. Comp. Soft. 32 (3) (2006), 139–44.Google Scholar
A., Bruno and V., Edneral. On integrability of the Euler–Poisson equations. J. Math. Sci. 152 (4) (2008), 479–89.Google Scholar
M. D., Bustamante and E., Kartashova. Dynamics of nonlinear resonances in Hamiltonian systems. EPL 85 (2009), 14004-1-5.Google Scholar
M. D., Bustamante and E., Kartashova. Effect of the dynamical phases on the nonlinear amplitudes' evolution. EPL 85 (2009), 34002-1-6.Google Scholar
F., Calogero. Classical Many-Body Problems Amendable to Exact Treatments (LNP: Monographs 66, Springer, 2001).Google Scholar
F., Calogero. Isochronous Systems (Oxford University Press, 2008).Google Scholar
F. D., Campello, J. M. B., Saraiva, and N., Krusche. Periodicity of atmospheric phenomena occurring in the extreme south of Brazil. Atm. Sci. Lett. 5 (2004), 65–76.Google Scholar
CENREC, http://cenrec.risc.uni-linz.ac.at/portal/
M., Cheney. Tesla Man out of Time (Dorset Press, 1989).Google Scholar
B. V., Chirikov. A universal instability of many-dimensional oscillator systems. Phys. Rep. 52 (5) (1979), 263–379.Google Scholar
Y., Choi, Y., Lvov, and S., Nazarenko. Wave turbulence. Recent Res. Devel. Fluid Dynamics 5 (2004), 1–33.Google Scholar
A., Chorin. Vorticity and Turbulence (Springer, 1994).Google Scholar
C. C., Chow, D., Henderson, and H., Segur. A generalized stability criterion for resonant triad interactions. Fluid Mech. 319 (1996), 67–76.Google Scholar
C., Connaughton, S. V., Nazarenko, and A. C., Newell, Dimensional analysis and weak turbulence. Physica D 184 (2003), 86.Google Scholar
A., Constantin. The trajectories of particles in Stokes waves. Invent. Math. 166 (2006), 523–35.Google Scholar
A., Constantin, M., Ehrnström, and E., Wahlén. Symmetry of steady periodic gravity water waves with vorticity. Duke Math. J. 140 (3) (2007), 591–603.Google Scholar
A., Constantin and E., Kartashova. Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves. EPL 86 (2009), 29001-1-6.Google Scholar
A., Constantin, E., Kartashova, and E., Wahlén. Discrete wave turbulence of rotational capillary water waves. E-print: arXiv. 1001.1497 (2010).
A., Constantin, D., Sattinger, and W., Strauss. Variational formulations for steady water waves with vorticity. Fluid Mech. 548 (2006), 151–63.Google Scholar
A., Constantin and W., Strauss. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004), 481–527.Google Scholar
A., Constantin and W., Strauss. Stability properties of steady water waves with vorticity. Comm. Pure Appl. Math. 60 (2007), 911–50.Google Scholar
D., Coutand and S., Shkoller. Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Amer. Math. Soc. 20 (2007), 829–930Google Scholar
W., Craig, D. M., Henderson, M., Oscamou, and H., Segur. Stable three-dimensional waves of nearly permanent form on deep water. Math. Comp. Simul. (2006), doi:10.1016/j.matcom.Google Scholar
A. D., Craik. Wave Interactions and Fluid Flows (Cambridge University Press, 1985).Google Scholar
D., Cumin and C., Unsworth. Generalising the Kuramoto model for the study of neuronal synchronisation in the brain. Physica D 226 (2007), 181–96.Google Scholar
Ch. A. C., Cunningham and I. F., De Albuquerque Cavalcanti. Intraseasonal modes of variability affecting the South Atlantic convergence zone. Int. J. Climatology 26 (2006), 1165–80.Google Scholar
A. F. T., Da Silva and D. H., Peregrine. Steep, steady surface waves on water of finite depth with constant vorticity. Fluid Mech. 195 (1988), 281–302.Google Scholar
H., Davenport. Multiplicative Number Theory (Markham Publishing Company, Chicago, 1967).Google Scholar
A., Delshamsa, R., de la Llaveb, and T. M., Seara. Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows. Advan. Math. 202 (1) (2006), 64–188.Google Scholar
P., Denissenko, S., Lukaschuk, and S., Nazarenko, Gravity surface wave turbulence in a laboratory flume. Phys. Rev. Lett. 99 (2007), 014501-1-4.Google Scholar
F., Diacu and P., Holmes. Celestial Encounters: The Origins of Chaos and Stability (Princeton University Press, 1996).Google Scholar
Ch., Doench, E., Kartashova, and L., Tec. Construction of optimal set of Manley–Rowe constants for resonance clustering in wave turbulent regimes. In preparation (2010).
A. E., Dolinko. From Newton's second law to Huygens's principle: visualizing waves in a large array of masses joined by springs. Eur. J. Phys. 30 (2009), 1217–28.Google Scholar
A. I., Dyachenko, A. O., Korotkevich, and V. E., Zakharov. Weak turbulence of gravity waves. JETP Lett. 77 (10) (2003), 546–50.Google Scholar
A. I., Dyachenko, Y. V., Lvov, and V. E., Zakharov, Five-wave interaction on the surface of deep fluid. Physica D 87 (1995), 233–61.Google Scholar
V. A., Dubinina, A. A., Kurkin, E. N., Pelinovsky, and O. E., Poluhina. Resonance three-wave interactions of Stokes edge waves. Izvestiya Atm. Ocean. Phys. 42(2) (2006), 254–61.Google Scholar
Elastic pendulum. http://academic.reed.edu/physics/courses/phys 100/Lab Manuals/Nonlinear Pendulum/nonlinear.pdf
M., Ehrnström. Uniquenes of steady symetric deep-water waves with vorticity. Nonlin. Math. Phys. 12(1) (2005), 27–30.Google Scholar
M., Ehrnström. Deep-water waves with vorticity: symmetry and rotational behaviour. DCDS Series A 19(3) (2007), 483–91.Google Scholar
J. K., Engelbrecht, V. E., Fridman, and E. N., Pelinovsky. Nonlinear Evolution Equations (Pitman Res. Not. Math. Ser. 180, Longman, London, 1988)Google Scholar
L., Euler. Theoria motuum planetarum et cometarum. Opera 25 (2) (1744), 105–251.
C., Falcon, E., Falcon, U., Bortolozzo, and S., Fauve. Capillary wave turbulence on a spherical fluid surface in low gravity. EPL 86 (2009), 14002-1-6.Google Scholar
E., Falcon, C., Laroche, and S., Fauve. Observation of gravity-capillary wave turbulence. Phys. Rev. Lett. 98 (2007), 094503-1-4.Google Scholar
M., Francius and C., Kharif. Three-dimensional instabilities of periodic gravity waves in shallow water. Geoph. Res. Abst. 7 (2005), 08757.Google Scholar
U., Frisch. Turbulence (Cambridge University Press, 1995).Google Scholar
G., Galileo. Discorsi e Dimostrazioni Matematiche, Intorno a' due Nuove Scienze (Elsevier, Leiden, 1638).Google Scholar
A. N., Ganshin, V. B., Efimov, G. V., Kolmakov, L. P., Mezhov-Deglin, and P. V. E., McClintock. Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium. Phys. Rev. Lett. 101 (2008): 065303-1-4.Google Scholar
M., Ghil. Intra-seasonal oscillations in the extra-tropical atmosphere: observations, theory, and GCM experiments. In Proc. of Eighth Conference on Atmospheric and Oceanic Waves and Stabilities (American Meteorological Society, Boston, MA, 1992).Google Scholar
M., Ghil, D., Kondrashov, F., Lott, and A. W., Robertson. Intraseasonal oscillations in the mid-latitudes: observations, theory, and GCM results. In Proceedings ECMWF/CLIVAR Workshop on Simulations and Prediction of Intra-Seasonal Variability (Reading, UK, 2004).Google Scholar
M., Ghil and K. S., Mo. Intraseasonal oscillations in the global atmosphere. Part I: Northern hemisphere and tropics. Atmos. Sci. 48 (1991), 752–79.Google Scholar
N., Gold, A., Mohan, C., Knight, and M., Munro. Understanding service-oriented software. IEEE Software 21(2) (2004), 71–7.Google Scholar
H. L., Grant, R. W., Stuart, and A., Moilliet. Turbulence spectra from tidal channel. Fluid. Mech. 12 (1961), 241–63.Google Scholar
I. S., Grigoriev and E. Z., Meilikhov (eds.) Fisicheskie Velichiny (Physical Quantities) (Energoatomizdat, Moscow, 1991) [in Russian].Google Scholar
R., Grimshaw and Y., Skyrnnikov. Long-wave instability in a three-layer stratified shear flow. Stud. Appl. Math. 108 (2002), 77–88.Google Scholar
M., Guzzo and G., Benettin. A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. DCDS Series B 1 (2001), 1–28.Google Scholar
M., Guzzo, Z., Knezević, and A., Milani. Probing the Nekhoroshev stability of asteroids. Cel. Mech. Dyn. Astr. 83 (2002), 121–40.Google Scholar
J. L., Hammack and D. M., Henderson. Resonant interactions among surface water waves. Ann. Rev. Fluid Mech. 25 (1993), 55–96.Google Scholar
J. L., Hammack and D. M., Henderson. Experiments on deep water waves with two-dimensional surface patterns. J. Offshore Mech. & Artic Eng., 125 (2003), 48–53.Google Scholar
J. L., Hammack, D. M., Henderson, and H., Segur. Progressive waves with persistent, two-dimensional surface patterns in deep water. Fluid Mech. 532 (2005), 1–51.Google Scholar
K., Hasselmann. On nonlinear energy transfer in gravity-wave spectrum. Part 1: General theory. Fluid Mech. 12 (1962), 481–500.Google Scholar
K., Hasselmann. A criterion for nonlinear wave stability. Fluid Mech. 30 (1967), 737–39.Google Scholar
D., Hilbert. Mathematical problems. Bull. Amer. Math. Soc. 8 (1902), 437–79.Google Scholar
D. M., Henderson, M. S., Patterson, and H., Segur. On the laboratory generation of two-dimensional, progressive, surface waves of nearly permanent form on deep water. Fluid Mech. 559 (2006), 413–37.Google Scholar
K., Horvat, M., Miskovic, and O., Kuljaca. Avoidance of nonlinear resonance jump in turbine governor positioning system using fuzzy controller. Industr. Techn. 2 (2003), 881–5.Google Scholar
T. R. N., Jansson, M. P., Haspang, K. H., Jensen, P., Hersen, and T., Bohr. Polygons on a rotating fluid surface. Phys. Rev. Lett. 96 (2006), 174502-1-4.Google Scholar
C., Jacobi. Fundamenta Nova Theoriae Functionum Ellipticarum (Koenigsberg, 1829) [in Latin].Google Scholar
A., Jarmén, L., Stenflo, H., Wilhelmsson, and F., Engelmann. Effect of dissipation on nonlinear interaction. Phys. Lett. 28A (11) (1969), 748–9.Google Scholar
F.-F., Jin and M., Ghil. Intraseasonal oscillations in the extratropics: Hofp bifurcation and topographic instabilities. Atm. Sci. 47 (1990), 3007–22.Google Scholar
R. S., Johnson. A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge University Press, 1997).Google Scholar
I. G., Jonsson. Wave–current interactions. In The Sea (Wiley, New York, 1990), 65–120.Google Scholar
B. B., Kadomtsev. Plasma Turbulence (Acad. Press, London, 1965).Google Scholar
P. B., Kahn and Y., Zarmi. Nonlinear Dynamics: Exploration Through Normal Forms (Wiley, New York, 1998).Google Scholar
V. M., Kamenkovich and A. S., Monin. Small fluctuations in the ocean. In Physics of the Ocean. Vol. 2: Hydrodynamics of the Ocean (Nauka, Moscow, 1978) [in Russian].Google Scholar
V. M., Kamenkovich and G. M., Reznik. Rossby waves. In Physics of the Ocean. Vol. 2: Hydrodynamics of the Ocean (Nauka, Moscow, 1978) [in Russian].Google Scholar
E., Kartashova. Partitioning of ensembles of weakly interacting dispersing waves in resonators into disjoint classes. Physica D 46 (1990), 43–56.Google Scholar
E., Kartashova. On properties of weakly nonlinear wave interactions in resonators. Physica D 54 (1991), 125–34.Google Scholar
E., Kartashova. Resonant interactions of the water waves with discrete spectra. In Proc. of Nonlinear Water Waves Workshop, ed. D. H., Peregrine (University of Bristol, UK, 1992), 43–53.Google Scholar
E., Kartashova. Clipping – a new investigation method for PDEs in compact domains. Theor. Math. Phys. 99 (1994), 675–80.Google Scholar
E., Kartashova. Weakly nonlinear theory of finite-size effects in resonators. Phys. Rev. Lett. 72 (1994), 2013–16.Google Scholar
E., Kartashova. Towards H-mode discharge explanation? In Current Topics in Astrophysical and Fusion Plasma, eds. M. F., Heyn, W., Kernbichler, and K., Biernat (Verlag der Technische Universität Graz, 1994), 179–84.Google Scholar
E., Kartashova. Applicability of weakly nonlinear theory for planetary-scale flows. Scientic report WR 95-03, KNMI, 1995, 1–29.
E., Kartashova. On large-scale dynamics of weakly nonlinear wave systems. In Advanced Series in Nonlinear Dynamics 7, eds. A., Mielke and K., Kirchgaessner (World Scientific, 1995), 282–90.Google Scholar
E., Kartashova. Wave resonances in systems with discrete spectra. In Nonlinear Waves and Weak Turbulence, ed. V. E., Zakharov (AMS Trans. 2, 1998), 95–129.Google Scholar
E., Kartashova. Fast computation algorithm for discrete resonances among gravity waves. Low Temp. Phys. 145 (2006), 286–95.Google Scholar
E., Kartashova. A model of laminated turbulence. JETP Lett. 83 (2006), 341–45.Google Scholar
E., Kartashova. Exact and quasi-resonances in discrete water-wave turbulence. Phys. Rev. Lett. 98 (2007), 214502-1-4.Google Scholar
E., Kartashova. Nonlinear resonances of water waves. DCDS Series B 12(3) (2009), 607–21.Google Scholar
E., Kartashova. Discrete wave turbulence. EPL 87 (2009), 44001-1–5.Google Scholar
E., Kartashova and M. D., Bustamante. Resonance clustering in wave turbulent regimes: integrable dynamics. E-print: arXiv:1002.4994 (2010).
E., Kartashova and A., Kartashov. Laminated wave turbulence: generic algorithms I. Int. J. Mod. Phys. C 17 (2006), 1579–96.Google Scholar
E., Kartashova and A., Kartashov. Laminated wave turbulence: generic algorithms II. Comm. Comp. Phys. 2 (2007), 783–94.Google Scholar
E., Kartashova and A., Kartashov. Laminated wave turbulence: generic algorithms III. Physica A: Stat. Mech. Appl. 380 (2007), 66–74.Google Scholar
E., Kartashova and A., Kartashov. Resonance clustering of rotational capillary waves. Comm. Comp. Phys. submitted (2010).Google Scholar
E., Kartashova and A., Kartashov. Exact and quasi-resonances among gravity-capillary waves. (In preparation, 2010).
E., Kartashova and V. S., L'vov. A model of intraseasonal oscillations in Earth's atmosphere. Phys. Rev. Lett. 98 (2007), 198501-1-4.Google Scholar
E., Kartashova and V. S., L'vov. Cluster dynamics of planetary waves. Europhys. Lett. 83 (2008), 50012-1-6.Google Scholar
E., Kartashova and G., Mayrhofer. Cluster formation in mesoscopic systems. Physica A: Stat. Mech. Appl. 385 (2007), 527–42.Google Scholar
E., Kartashova, S., Nazarenko, and O., Rudenko, Resonant interactions of nonlinear water waves in a finite basin. Phys. Rev. E 98 (2008), 0163041-1-9.Google Scholar
E. A., Kartashova, L. I., Piterbarg, and G. M., Reznik. Weakly nonlinear interactions between Rossby waves on a sphere. Oceanology 29 (1990), 405–11.Google Scholar
E., Kartashova, C., Raab, Ch., Feurer, G., Mayrhofer, and W., Schreiner, Symbolic computations for nonlinear wave resonances. In Extreme Ocean Waves, eds. E., Pelinovsky and Ch., Kharif (Springer, 2008), 97–128.Google Scholar
E. A., Kartashova and G. M., Reznik. Interactions between Rossby waves in bounded regions. Oceanology 31 (1992), 385–89.Google Scholar
A. L., Karuzskii, A. N., Lykov, A. V., Perestoronin, and A. I., Golovashkin. Microwave nonlinear resonance incorporating the helium heating effect in superconducting microstrip resonators. Phys. C: Supercond. 408–410 (2004), 739–40.Google Scholar
R. E., Kelly. The stability of an unsteady Kelvin–Helmholz flow. Fluid. Mech. 22 (1965), 547–60.Google Scholar
W., Kluzniak. Quasi-periodic oscillations and the possibility of an observational distinction between neutron and quark stars. Acta Phys. Polon. B 37 (4) (2006), 1361–65.Google Scholar
Z., Knezević and R., Pavlović. Application of the Nekhoroshev theorem to the real dynamical system. Novi Sad J. Math. 38 (3) (2008), 181–8.Google Scholar
G., Kolmakov, A., Levchenko, M., Braznikov, L., Mezhov-Deglin, A., Slichenko, and P., McClintock, Formation of a direct Kolmogorov-like cascade of second-sound waves in He II. Phys. Rev. Lett. 93 (2004), 074501-1-4.Google Scholar
A. N., Kolmogorov. The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30 (1941), 301–5; reprinted: Proc. R. Soc. Lond. A434 (1991), 9–13.Google Scholar
A. N., Kolmogorov. On the conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl. Akad. Nauk SSSR 98 (1954), 527–30.Google Scholar
D. A., Kovriguine and G. A., Maugin. Multiwave nonlinear couplings in elastic structures. Math. Prob. Engin. (2006), doi:10.1155/MPE/2006/76041.Google Scholar
E. V., Kozik and B. V., Svistunov. Kelvin-wave cascade and decay of superfluid turbulence. Phys. Rev. Lett. 92 (2004), 035301-1-4.Google Scholar
U., Köpf. Wilberforce's pendulum revisitedJ. Am. Phys. 58 (9) (1990), 833–7.Google Scholar
V. P., Krasitskii. On reduced equations in the Hamiltonian theory of weakly non-linear surface waves. Fluid Mech. 272 (1994), 1–20.Google Scholar
S., Kuksin. Analysis of Hamiltonian PDEs (Oxford University Press, 2000).Google Scholar
S., Kuksin. Fifteen years of KAM for PDE, AMS Transl. 2 (212) (2004), 237–58.Google Scholar
S., Kuksin. Hamiltonian PDEs. In Handbook on Dynamical Systems 1B, eds. B., Hasselblatt and A., Katok (Elsevier, 2005), 1087–133.Google Scholar
M., Kundu and D., Bauer. Nonlinear resonance absorption in the laser-cluster interaction. Phys. Rev. Lett. 96 (2005), 123401-1-4.Google Scholar
Y., Kuramoto. Chemical Oscillations, Waves and Turbulence (Springer, New York, 1984).Google Scholar
S. V., Kuznetsov. The motion of the elastic pendulum. Reg. Chaot. Dyn. 4 (3) (1999), 3–12.Google Scholar
J., Lagrange. Oeuvres 6 (Gauthier-Villars, Paris, 1873).Google Scholar
B. M., Lake and H. C., Yuen. A note on some water-wave experiments and the comparision of data with the theory. Fluid Mech. 83 (1977), 75–81.Google Scholar
E., Landau. Handbuch für Lehre von der Verteilung der Primazahlen. II (Teubner, Leipzig, 1909).Google Scholar
W., Lee, G., Kovacic and D., Cai. Renormalized resonance quartets in dispersive wave turbulence. Phys. Rev. Lett. 103 (2009), 024502-1-4.Google Scholar
J., Lighthill. Waves in Fluids (Cambridge University Press, 1978).Google Scholar
F., Lindemann. Über die Zahl π. Math. Annal. 20 (1882), 213–25.Google Scholar
M. S., Longuet-Higgins and A. E., Gill. Resonant interactions between planetary waves. Proc. Roy. Soc. Lond. A299 (1967), 120–40.Google Scholar
P., Lochak and C., Meunier. Multiphase Averaging for Classical Systems (Appl. Math. Sci. Series 72, Springer, 1988)Google Scholar
Y. V., Lvov, S., Nazarenko and B., Pokorni. Discreteness and its effect on water-wave turbulence. Physica D 218 (2006), 24–35.Google Scholar
P., Lynch. Resonant motions of the three-dimensional elastic pendulum. Int. J. Nonl. Mech. 37 (2002), 258–64.Google Scholar
P., Lynch. The swinging spring: a simple model of amospheric balance. in Large-Scale Atmosphere-Ocean Dynamics. Vol II: Geometric Methods and Models, eds. J., Norbury and I., Roulstone (Cambridge University Press, 2002), 64–108.Google Scholar
P., Lynch. On resonant Rossby–Haurwitz triads. Tellus 61A (2009), 438–45.Google Scholar
P., Lynch and C., Houghton. Pulsation and precession of the resonant swinging spring. Physica D 190 (2004), 38–62.Google Scholar
R. A., Madden and P. R., Julian. Detection of a 40–50 day oscillation in the zonal wind in the tropical Pacific. Atm. Sci. 28 (1971), 702–8.Google Scholar
R. A., Madden and P. R., Julian. Description of global-scale circulation cells in the tropics with a 40–50 day period. Atm. Sci. 29 (1972), 1109–23.Google Scholar
A. J., Majda and A. L., Bertozzi. Vorticity and Incompressible Flow (Cambridge University Press, 2002).Google Scholar
Yu. I., Manin and A., Panchishkin. Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Springer, 2005).Google Scholar
J. M., Manley and H. E., Rowe. Some general properties of non-linear elements – Part 1: General energy relations. Proc. Inst. Rad. Engrs. 44 (1956), 904–13.Google Scholar
MathBroker II: Brokering Distributed Mathematical Services. Reseach Institute for Symbolic Computation (RISC), 2007. http://www.risc.uni-linz.ac.at/projects/mathbroker2.
Yu. V., Matijasevich. Hilbert's Tenth Problem (MIT Press, Cambridge, MA, 1993).Google Scholar
M. R., Matthews, C. F., Gauld, and A., Stinner (eds.). The Pendulum Scientific, Historical, Philosophical and Educational Perspectives (Springer, 2005).Google Scholar
L. F., McGoldrick. Resonant interactions among capillary-gravity waves. Fluid Mech. 21 (1967), 305–31.Google Scholar
N. W., McLachlan. Theory and Application of Mathieu Functions (Clarendon Press, Oxford, 1947).Google Scholar
W. H., Meeks. The topology and geometry of embedded surfaces of constant mean curvature. J. Differential Geometry 27 (3) (1988), 539–52.Google Scholar
C. R., Menyuk, H. H., Chen, and Y. C., Lee. Restricted multiple three-wave interactions: Panlevé analysis. Phys. Rev. A 27 (1983), 1597–611.Google Scholar
C. R., Menyuk, H. H., Chen, and Y. C., Lee. Restricted multiple three-wave interactions: integrable cases of this system and other related systems. J. Math. Phys. 24 (1983), 1073–9.Google Scholar
L., Merkine and L., Shtilman. Explosive instability of baroclinic waves. Proc. R. Soc. Lond. A 395 (1984), 313–39.Google Scholar
MONET – Mathematics on the Web. The MONET Consortium, 2004. http://monet.nag.co.uk.
J., Moser. On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Goett., Math. Phys. Kl. (1962), 1–20.Google Scholar
T., Murakami. Intraseasonal atmospheric teleconnection patterns during the Nothern Hemisphere winter. Climate 1 (1988), 117–31.Google Scholar
J., Murdock. Normal Forms and Unfoldings for Local Dynamical Systems (Springer-Verlag, New York, 2003).Google Scholar
S. L., Musher, A. M., Rubenchik and V. E., Zakharov. Hamiltonian approach to the description of nonlinear plasma phenomena. Phys. Rep. 129 (1985), 285–366.Google Scholar
A. H., Nayfeh. Introduction to Perturbation Techniques (Wiley-Interscience, NY, 1981).Google Scholar
A. H., Nayfeh. Method of Normal Forms (Wiley-Interscience, NY, 1993).Google Scholar
S., Nazarenko. Sandpile behaviour in discrete water-wave turbulence. J. Stat. Mech.: Theor. Exp. (2006), L02002, doi:10.1088/1742-5468/2006/02/L02002.Google Scholar
S., Nazarenko. Wave Turbulence (In preparation, 2010).Google Scholar
N. N., Nekhoroshev. An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Russ. Math. Surv. (Usp. Mat. Nauk) 32 (6) (1977), 1–65.Google Scholar
R. A., Nelson and M. G., Olsson. The pendulum – rich physics from a simple system. Am. J. Phys. 54 (2) (1986), 112–21.Google Scholar
A., Newell, S., Nazarenko and L., Biven. Wave turbulence and intermittency. Phys. D, 152–153 (2001), 520–50.Google Scholar
I., Newton. Philosophiae Naturalis Principia Mathematica I–III. Royal Soc. Lond. (1687).Google Scholar
H., Ocamoto and M., Shoji. The Mathematical Theory of Permanent Progressive Water Waves (World Scientific, 2001).Google Scholar
P. J., Olver. Applications of Lie Groups to Differential Equations (Graduated texts in Mathematics 107, Springer, 1993).Google Scholar
U., Omar, 2001. http://bulbphotography.com/pendulum/gallery.php
V. N., Oraevsky and R. Z., Sagdeev. On stability of steady longitudinal oscillations in plasma. Zh. Tekh. Fiz. 32 (1962), 1291–96 [in Russian].Google Scholar
L. A., Ostrovskii, S. A., Rybak, and S. L., Tsimring. Negative energy waves in hydrodynamics. Sov. Phys. Uspekhi 29 (11) (1986), 1040–52.Google Scholar
R., Pavlović and M., Guzzo. Fulfillment of the conditions for the application of the Nekhoroshev theorem to the Koronis and Veritas asteroid families. Mon. Not. R. Astron. Soc. 384 (2008), 1575–82.Google Scholar
J., Pedlosky. Geophysical Fluid Dynamics (Springer-Verlag, New York, 1987).Google Scholar
M., Perlin and W. M., Schultz, 2000. Capillary effects on surface waves. Annu. Rev. Fluid. Mech. 32, 241–74.Google Scholar
O. M., Phillips. On the dynamics of unsteady gravity waves of infinite amplitude. Fluid Mech. 9 (1960), 193–217.Google Scholar
O. M., Phillips. Theoretical and experimental studies of gravity wave interactions. Proc. Roy. Soc. Lond. A299 (1967), 104–19.Google Scholar
O. M., Phillips. Wave interactions – evolution of an idea. Fluid Mech. 106 (1981), 215–27.Google Scholar
A., Pikovsky and M., Rosenblum. Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators. Physica D 238 (2009), 27–37.Google Scholar
A., Pikovsky, M., Rosenblum, and J., Kurths. Sinchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, 2001).Google Scholar
A., Pikovsky and Yu., Maistrenko. Synchronization: Theory and Application NATO Science Series II: Mathematics, Physics and Chemistry 109 (Kluwer Academic Publishers, Dodrecht, Boston, London, 2003).Google Scholar
L. I., Piterbarg. Hamiltonian formalism for Rossby waves. In Nonlinear Waves and Weak Turbulence, ed. V. E., Zakharov (American Mathematical Society Trans. 2, 1998), 131–66.Google Scholar
Jü., Pöschel. Nonlinear partial differential equations, Birkhoff normal forms and KAM theoryProgr. Math. 169 (1998), 167–86.Google Scholar
H., Poincaré. Oeuvres (Paris, 1951).Google Scholar
A. N., Pushkarev. On the Kolmogorov and frozen turbulence in numerical simulation of capillary waves. Eur. J. Mech. – B/Fluids 18(3) (1999), 345–51.Google Scholar
A. N., Pushkarev and V. E., Zakharov. Turbulence of capillary waves – theory and numerical simulation. Physica D 135(1–2) (2000), 98–116.Google Scholar
H., Punzmann, M. G., Shats, and H., Xia. Phase randomization of three-wave interactions in capillary waves. Phys. Rev. Lett. 103 (2009), 064502-1-4.Google Scholar
K., Rajendran and A., Kitoh. Modulation of tropical intraseasonal oscillations by ocean–atmosphere coupling. J. Climate 19 (2006), 366–91.Google Scholar
M., Rosenblum and A., Pikovsky. Self-organized quasiperiodicity in oscillator ensembles with global nonlinear coupling. Phys. Rev. Lett. 98 (2007), 064101-1-4.Google Scholar
O., Rudenko. Nonlinear wave resonances. Wolfram Demonstrations Project, 2008; http://demonstrations.wolfram.com/NonlinearWaveResonances/.
B. R., Safdi and H., Segur. Explosive instability due to four-wave mixing. Phys. Rev. Lett. 99 (2007), 245004-1-4.Google Scholar
R. Z., Sagdeev and A. A., Galeev. Nonlinear Plasma Theory (Benjamin, New York, 1969).Google Scholar
J. A., Sanders, F., Verhulst, and J., Murdock. Averaging Methods in Nonlinear Dynamical Systems. Appl. Math. Sci. 59 (Springer, 2007).Google Scholar
H., Segur and D. M., Henderson. The modulation instability revisited. Euro. Phys. J. – Spec. Topics 1147 (2007), 25–43.Google Scholar
H., Segur, D., Henderson, J., Hammack, C. -M., Li, D., Pheiff, and K., Socha. Stabilizing the Benjamin–Feir instability. Fluid Mech. 539 (2005), 229–71.Google Scholar
D. C., Schmidt. Model-driven engineering. IEEE Comp. 39(2) (2006), 25–31.Google Scholar
W. M., Schmidt. Diophantine Approximations (Math. Lec. Not. 785, Springer, Berlin, 1980).Google Scholar
W., Schreiner. Web Service for computing of nonlinear resonances http://www.risc.uni-linz.ac.at/people/schreine/Wave.
M., Shats, H., Punzmann, and H., Xia. Capillary rogue waves. Phys. Rev. Lett. 104 (2010), 104503-1-4.Google Scholar
V., Shrira, V., Voronovich, and N., Kozhelupova. Explosive instability of vorticity waves. Phys. Oceanogr. 27 (1997), 542–54.Google Scholar
I., Silberman. Planetary waves in atmosphere. Meteorology 11 (1954), 27–34.Google Scholar
L., Stenflo. Resonant three-wave interactions in plasmas. Phys. Scr. T50 (1994), 15–9.Google Scholar
L., Stenflo, J., Weiland, and H., Wilhelmsson. A solution of equations describing explosive instabilities. Phys. Scr. 1 (1970), 46.Google Scholar
L., Stenflo and H., Wilhelmsson. Stabilization of nonlinear instabilities by means of dissipation. Phys. Lett 29A (5) (1969), 217–18.Google Scholar
M., Stiassnie, and L., Shemer. On the interactions of four water waves. Wave motion 41 (2005), 307–28.Google Scholar
S., Strogatz. Nonlinear Dynamics and Chaos (Reading, MA: Addison Wesley, 1994).Google Scholar
S., Strogatz. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143 (2000), 1–20.Google Scholar
C., Sulem and P. -L., Sulem. Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse (App. Math. Sci. 139, New York, Springer, 1999).Google Scholar
Tacoma video: http://www.youtube.com/watch?v=3mclp9QmCGs.
K., Takaya and H., Nakamura. Geographical dependence of upper–level blocking formation associated with intraseasonal amplification of the Siberian high. Atmos. Sci. 62 (2005), 4441–9.Google Scholar
M., Tanaka and N., Yokoyama. Effects of discretization of the spectrum in water-wave turbulence. Fluid Dyn. Res. 34 (2004), 199–216.Google Scholar
M., Tanaka. On the role of resonant interactions in the short-term evolution of deep-water ocean spectra. J. Phys. Oceanogr. 37 (2007), 1022–36.Google Scholar
R., Treumann and W., Baumjohann. Advanced Space Plasma Physics (Imperial College Press, London, 2001).Google Scholar
V. N., Tsytovich. Nonlinear Effects in Plasma (Plenum, New York, 1970).Google Scholar
J. M., Tuwankotta and F., Verhulst. Symmetry and resonance in Hamiltonian system. Preprint, Utrecht University, 2000, 1–21.
C., Vedruccio, E., Mascia, and V., Martines. Ultra high frequency and microwave non-linear interaction device for cancer detection and tissue characterization, a military research approach to prevent health diseases. Int. Rev. Armed Forces Med. Serv. 78 (2) (2005), 120–30.Google Scholar
F., Verheest. Proof of integrability for five-wave interactions in a case with unequal coupling constants. Phys. A: Math. Gen. 21 (1988), L545–49.Google Scholar
F., Verheest. Integrability of restricted multiple three-wave interactions. II: Coupling constants with ratios 1 and 2. J. Math. Phys. 29 (1988), 2197–201.Google Scholar
F., Verhulst. Hamiltonian normal forms. Scholarpedia 2 (8) (2007), 2101.Google Scholar
I. M., Vinogradov. The Foundations of Number Theory (Pergamon, London, 1955).Google Scholar
B. L., van der Waerden. Modern Algebra I (Springer, 2003).Google Scholar
W., Wang and J. -J. E., Slotine. On partial contraction analysis for coupled nonlinear oscillators. Biol. Cybern. 92 (2005), 38–53.Google Scholar
E., Wahlén. A Hamiltonian formulation of water waves with constant vorticity. Lett. Math. Phys. 79 (2007), 303–15.Google Scholar
E., Wahlén. On rotational water waves with surface tension. Philos. Trans. Roy. Soc. Lond. Ser. A 365 (2007), 2215–25.Google Scholar
K., Watson and J., Bride. Excitation of capillary waves by longer waves. Fluid. Mech. 250 (1993), 103–19.Google Scholar
K. M., Watson, B. J., West, and B. I., Cohen. Coupling of surface and internal gravity waves: a mode coupling model. Fluid Mech. 77 (1976), 185–208.Google Scholar
K. M., Weickmann, G. R., Lussky, and J. E., Kutzbach. Intraseasonal (30–60 day) fluctuations of ongoing longwave radiation and 250 mb stream function during northern winter. Mon. Wea. Rev. 113 (1985), 941–61.Google Scholar
T. P., Weissert. The Genesis of Simulations in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem (Springer, 1997).Google Scholar
G. B., Whitham. Lectures on Wave Propagation (Springer for TATA Institute of fundamental research, Bombay, 1979).Google Scholar
G. B., Whitham. Linear and Nonlinear Waves (Wiley Series in Pure and Applied Mathematics, 1999).Google Scholar
E. T., Whittaker. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambrige University Press, 1937).Google Scholar
L. R., Wilberforce. On the vibrations of a loaded spiral spring. Phil. Magaz. 38 (1896), 386–92.Google Scholar
H., Wilhelmsson, L., Stenflo, and F., Engelmann. Explosive instabilities in the well-defined phase description. J. Math. Phys. 11 (1970), 1738–42.Google Scholar
H., Wilhelmsson and V.P., Pavlenko. Five-wave interaction – a possibility for enhancement of optical or microwave radiation by nonlinear coupling of explosively unstable plasma waves. Phys. Scripta 7 (1972), 213–16.Google Scholar
W. B., Wright, R., Budakian, and S. J., Putterman. Diffusing light photography of fully developed isotropic ripple turbulence. Phys. Rev. Lett. 76 (1996), 4528–31.Google Scholar
H. W., Wyld. Formulation of the theory of turbulence in an incompressible fluid. Ann. Phys. 14 (1961), 134–65.Google Scholar
V. E., Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Tekh. Fiz. 9 (1968), 86–94.Google Scholar
V. E., Zakharov. Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. Eur. J. Mech. B: Fluids 18 (1999), 327–44.Google Scholar
V. E., Zakharov, ed. Nonlinear Waves and Weak Wave Turbulence (American Mathematical Society Trans. 2, 182, 1998).Google Scholar
V., Zakharov, F., Dias, and A., Pushkarev. One-dimensional wave turbulence. Phys. Rep. 398 (2004), 1–65.Google Scholar
V. E., Zakharov and N. N., Filonenko. Weak turbulence of capillary waves. Appl. Mech. Tech. Phys. 4 (1967), 500–15.Google Scholar
V. E., Zakharov, V. S., L'vov, and G., Falkovich. Kolmogorov Spectra of Turbulence (Series in Nonlinear Dynamics, Springer-Verlag, New York, 1992).Google Scholar
V. E., Zakharov, A. O., Korotkevich, A. N., Pushkarev, and A. I., Dyachenko. Mesoscopic wave turbulence. JETP Lett. 82 (2005), 487–91.Google Scholar
Sh. E., Zimring. Special Functions and Definite Integrals, Algorithms, Mini-computer Programs (Radio i svjaz, Moscow, 1988).Google Scholar

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  • References
  • Elena Kartashova, Johannes Kepler Universität Linz
  • Book: Nonlinear Resonance Analysis
  • Online publication: 06 December 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511779046.011
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  • References
  • Elena Kartashova, Johannes Kepler Universität Linz
  • Book: Nonlinear Resonance Analysis
  • Online publication: 06 December 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511779046.011
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  • References
  • Elena Kartashova, Johannes Kepler Universität Linz
  • Book: Nonlinear Resonance Analysis
  • Online publication: 06 December 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511779046.011
Available formats
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