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arXiv · 1702.02638

Wave-Turbulence Theory of four-wave nonlinear interactions

Sergio Chibbaro·Giovanni Dematteis·Christophe Josserand·Lamberto Rondoni
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Abstract

The Sagdeev-Zaslavski (SZ) equation for wave turbulence is analytically derived, both in terms of generating function and of multi-point pdf, for weakly interacting waves with initial random phases. When also initial amplitudes are random, the one-point pdf equation is derived. Such analytical calculations remarkably agree with results obtained in totally different fashions. Numerical investigations of the two-dimensional nonlinear Schroedinger equation (NLSE) and of a vibrating plate prove that: (i) generic Hamiltonian 4-wave systems rapidly attain a random distribution of phases independently of the slower dynamics of the amplitudes, vindicating the hypothesis of initially random phases; (ii) relaxation of the Fourier amplitudes to the predicted stationary distribution (exponential) happens on a faster timescale than relaxation of the spectrum (Rayleigh-Jeans distribution); (iii) the pdf equation correctly describes dynamics under different forcings: the NLSE has an exponential pdf corresponding to a quasi-gaussian solution, like the vibrating plates, that also show some intermittency at very strong forcings.

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Sergio Chibbaro, Giovanni Dematteis, Christophe Josserand, Lamberto Rondoni. 2017-07-26. Wave-Turbulence Theory of four-wave nonlinear interactions. https://doi.org/10.1103/physreve.96.021101

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