Weak turbulence and collapses in the Majda-McLaughlin-Tabak equation: Fluxes in wavenumber and in amplitude space

The turbulent energy flow of the one-dimensional Majda-McLaughlin-Tabak equation (MMT) is studied numerically. The system exhibits weak turbulence for weak driving forces, while weak turbulence coexists with strongly nonlinear intermittent collapses when the system is strongly driven. These two types of dynamics can be distinguished by their energy and particle fluxes. The weakly turbulent process can be characterized by fluxes in wavenumber space, while additional fluxes in amplitude space emerge in the intermittent process. The particle flux is directed from low amplitudes towards high amplitudes, and the energy flows in the opposite direction.