Fetching the paper…

Reading the bibliography…

2015

The effective equation method

Kuksin, Sergei, Maiocchi, Alberto

Original
PDF
Post on X

Understand

In this chapter we present a general method of constructing the effective equation which describes the behaviour 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 behaviour 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.

The effective equation method · Around

Built on

N.N. Bogoljubov, J.A. Mitropol’skij, Asymptotic Methods in the Theory of Non-Linear Oscillations (Gordon & Breach, 1961)
1961
Earlier work this paper cites.
V. Zakharov, V. L’vov, G. Falkovich, Kolmogorov Spectra of Turbulence (Springer, Berlin, 1992)
1992
Earlier work this paper cites.
E. Kartashova, Phys. Rev. Lett. 72
1994
Earlier work this paper cites.
A.S. Monin, A.M. Yaglom, Statistical fluid mechanics: mechanics of turbulence. Vol. II. (Dover, New York, 2007)
2007
Earlier work this paper cites.
J. Cardy, G. Falkovich, K. Gawedzki, Non-equilibrium Statistical Mechanics and Turbulence (Cambridge University Press, Cambridge, 2008)
2008
Earlier work this paper cites.
S.B. Kuksin, GAFA 20
2010
Earlier work this paper cites.

Similar

E. Kartashova, Nonlinear Resonance Analysis (Cambridge University Press, Cambridge, 2010)
2010
Cited alongside, same era.
S. Nazarenko, Wave Turbulence (Springer, Berlin, 2011)
2011
Cited alongside, same era.
A. Newell, B. Rumpf, Annu. Rev. Fluid Mech. 43
2011
Cited alongside, same era.
P. Gérard, S. Grellier, Analysis and PDE 5
2012
Cited alongside, same era.
S. Kuksin, Annales de l’insitut Henri Poincaré-Probabilité et Statistiques 49
2013
Cited alongside, same era.
ArXiv:1311.6793
Original
S. Kuksin, A. Maiocchi, (2013) · 2013
Cited alongside, same era.

Then

ArXiv:1311.6794
Original
S. Kuksin, A. Maiocchi, (2013) · 2013
Later among the works it cites.
E. Faou, P. Germain, Z. Hani, Preprint (2013)
2013
Later among the works it cites.
ArXiv:1409:0652
S. Kuksin, A. Maiocchi, (2014) · 2014
Later among the works it cites.
G. Huang, DCDS-A 34
2014
Later among the works it cites.
Doi:10.1007/s10884-014-9391-0
G. Huang, JDDE pp. 1–13 (2014) · 2014
Later among the works it cites.
G. Huang, S. Kuksin, A. Maiocchi, Fields Institute Communications, to appear (2015)
2015
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…