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Applied
Math Seminar |
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Wave Turbulence,the long time statistical behavior of a sea of weakly nonlinear, dispersive waves in the presence of sources and sinks,is one of the few examples of a far-from-equilibrium many-body system with a natural closure. The closure of the BBGKY hierarchy is due to a combination of phase mixing by linear dispersive wavetrains and the manner in which the nonlinear couplings rebuild departure from joint Gaussianity. The result is a set of kinetic equations describing the redistribution of the spectral densities of formally conserved integrals such as energy and particle number and a frequency renormalization. The kinetic equations have a family of Kolmogorov-like stationary solutions far richer than the traditional thermodynamic equilibria (Rayleigh-Jeans,Bose-Einstein,Fermi-Dirac) which carry finite fluxes of the conseved densities from the sources to the sinks.It turns out, however,that on these solutions the weak turbulence approximation almost always breaks down at very small or very large scales and this breakdown leads to fully nonlinear intermittent behavior. I will describe this breakdown and illustrate its manifestation in the contexts of wind driven deep water waves and optical waves of diffraction in nonlinear media. |