Classical WTT allows to approximate original nonlinear evolution equations by corresponding kinetic equations which are easier to study, to model numerically, to compute Kolmogorov's spectra for them, etc. It is well known that kinetic equations do not work in large scale wave systems (so-called finite-size effects in resonators), i.e. in systems with discrete spectra. We are going to present self-consistent theory of wave turbulence for these systems and illustrate it with examples for waves in water, ocean, atmosphere, plasma. Mathematical part of this theory is based on the general results and methods of number theory which are used to construct reductions of the original evolution nonlinear PDE to a few small systems of ODEs describing resonances of the wave system.
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Institute of Theoretical Physics Russian Academy of Sciences, Chernogolovka