Effective dispersion relation and waveaction spectrum for fully nonlinear MMT model

Abstract

We investigate a version of the Majda-McLaughlin-Tabak model of dispersive wave turbulence with no linear dispersion. In particular, we make predictions for the waveaction spectrum and the effective dispersion relation, and we test these predictions using time-dynamic simulations. We consider driven-damped and undriven, undamped cases of the model. In the undriven, undamped cases, we make predictions using the ideas of ergodicity and ensemble equivalence. In the driven, damped cases, we make predictions using two different methods. One of these uses the effective dispersion relation of the model, along with weak turbulence methods. The other method uses scaling symmetries of the model as well as assumptions about the dependence of the waveaction spectrum on dissipation rates of waveaction and linear energy dissipation. We find that the results of these predictions agree with each other. For both the undriven, undamped, and driven, damped, cases of our model, we find that some of our predictions agree with numerical results, while others do not. It is possible that these disagreements are the result of choosing parameters from a regime where our predictions do not hold.