In a recent answer (here: The equipartition theorem in momentum space ), I suggested that if you have an appropriate first order equation (in the answer I used a second order equation, but it is more likely to be true for a first order equation) of the form:
$$ i\partial_t \phi = (\nabla^2)^N \phi + \lambda \phi^3 + f + D $$
(EDIT: I first had $|\phi|^2\phi$, which is the nonlinear Schrodinger equation, but this has the same frequency shifting property as the relativistic version--- the frequencies are not completely perturbatively decoupled even with a resonance condition. The only reason I went to first order was to fix this annoying problem, and there is no condition of being physical. Although this will not conserve the total $|\phi|^2$, it has a conserved energy.)
Where f is a driving force with only long-wavlength components, and D is a damping force which only affects short wavelength components. The field $\phi$ is a complex scalar. Then if N is large and $\lambda$ is sufficiently small (by scaling, $\phi$'s magnitude can substitute for $\lambda$, so the limit of small $\lambda$ is really the limit of small $|\phi|$), the cascade due to the nonlinear term is a thermal gradient draining energy from the hot low frequency modes to the cold high-frequency modes.
I don't know if this model is in the literature, of if the thermal analysis is accurate (I am pretty sure it is correct, because the resonance matching does lead to local flow of energy in |k|, but I am not sure if there isn't a block to thermalization due to some problem with mixing in each |k| modes separately).
In the original answer, I used a second order equation, which has frequencies of both signs. This allows some extra mixing between distant modes, which shifts the frequency of low modes based on the occupation of high modes. While I don't think this changes the cascade significantly, I wanted a clean example, so I used a first order equation here. (EDIT: The clean example does not use the absolute value, so that the frequencies have to strictly additively combine.)
Is this model analyzed in the literature? Does it thermalize into a thermal gradient cascade?
