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The Effective Equation Method

  • Sergei KuksinEmail author
  • Alberto Maiocchi
Part of the Lecture Notes in Physics book series (LNP, volume 908)

Abstract

In this chapter we present a general method of constructing the effective equation which describes the behavior of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behavior of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three- and four-wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanography.

Keywords

Effective Equation Dispersion Function Wave Turbulence Resonant Term Finite Cluster 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bogoljubov, N.N., Mitropol’skij, J.A.: Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon & Breach, New York (1961)Google Scholar
  2. 2.
    Cardy, J., Falkovich, G., Gawedzki, K.: Non-equilibrium Statistical Mechanics and Turbulence. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  3. 3.
    Faou, E., Germain, P., Hani, Z.: The weakly nonlinear large box limit of the 2d cubic nonlinear Schrödinger equation. E-print: arXiv:1308.6267 (2013)Google Scholar
  4. 4.
    Gérard, P., Grellier, S.: Effective integrable dynamics for a certain non-linear wave equation. Anal. PDE 5, 1139–1155 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Huang, G.: An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete Continuous Dyn. Syst. Ser. A 34(9), 3555–3574 (2014)CrossRefGoogle Scholar
  6. 6.
    Huang, G.: Long-time dynamics of resonant weakly nonlinear CGL equations. J. Dyn. Diff. Equat. 1–13 (2014). doi:10.1007/s10884-014-9391-0Google Scholar
  7. 7.
    Huang, G., Kuksin, S., Maiocchi, A.: Time-averaging for weakly nonlinear CGL equations with arbitrary potentials. In: Guyenne, P., Nicholls, D., Sulem, C. (eds.) Hamiltonian Partial Differential Equations and Applications, vol. 75, Fields Inst. Commun. (2015)Google Scholar
  8. 8.
    Kartashova, E.: Partitioning of ensembles of weakly interacting dispersing waves in resonators into disjoint classes. Phys. D 46, 43–56 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kartashova, E.: Nonlinear Resonance Analysis. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  10. 10.
    Kuksin, S.B.: Damped-driven KdV and effective equations for long-time behaviour of its solutions. GAFA Geom. Funct. Anal. 20, 1431–1463 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kuksin, S.B.: Weakly nonlinear stochastic CGL equations. Ann. Inst. Henri Poincaré Probab. Stat. 49(4), 1033–1056 (2013)MathSciNetCrossRefADSzbMATHGoogle Scholar
  12. 12.
    Kuksin, S., Maiocchi, A.: Resonant averaging for small solutions of stochastic NLS equations. E-print: arXiv:1311.6793 (2013)Google Scholar
  13. 13.
    Kuksin, S.B., Maiocchi, A.: The limit of small Rossby numbers for the randomly forced quasi-geostrophic equation on the β-plane. Nonlinearity 28, 2319–2341 (2015)CrossRefADSGoogle Scholar
  14. 14.
    Kuksin, S.B., Maiocchi, A.: Derivation of a wave kinetic equation from the resonant-averaged stochastic NLS equation. Phys. D. (2015, in press)Google Scholar
  15. 15.
    Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics: Mechanics of Turbulence, vol. II. Dover, New York (2007)Google Scholar
  16. 16.
    Nazarenko, S.: Wave Turbulence. Lecture Notes in Physics, vol. 825. Springer, Berlin (2011)Google Scholar
  17. 17.
    Newell, A.C., Rumpf, B.: Wave turbulence. Ann. Rev. Fluid Mech. 43(1), 59–78 (2011)MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Zakharov, V.E., L’vov, V.S., Falkovich, G.: Kolmogorov spectra of turbulence 1. Wave Turbulence. Springer, Berlin (1992)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.CNRS and I.M.JUniversité Paris-DiderotParis 7France
  2. 2.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItaly

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