Dispersive wave turbulence in one dimension

In this article, we study numerically a one-dimensional model of dispersive wave turbulence. The article begins with a description of the model which we introduced earlier, followed by a concise summary of our previous results about it. In those previous studies, in addition to the spectra of weak turbulence (WT) theory, we also observed another distinct spectrum (the "MMT spectrum"). Our new results, presented here, include: (i) A detailed description of coexistence of spectra at distinct spatial scales, and the transitions between them at different temporal scales; (ii) The existence of a stable MMT front in k-space which separates the WT cascades from the dissipation range, for various forms of strong damping including "selective dissipation"; (iii) The existence of turbulent cycles in the one-dimensional model with focusing nonlinearity, induced by the interaction of spatially localized coherent structures with the resonant quartets of dispersive wave radiation; (iv) The detailed composition of these turbulent cycles - including the self-similar formation of focusing events (distinct in the forced and freely decaying cases), and the transport by the WT direct and inverse cascades of excitations between spatial scales. This one-dimensional model admits a very precise and detailed realization of these turbulent cycles and their components. Our numerical experiments demonstrate that a complete theory of dispersive wave turbulence will require a full description of the turbulent field over all spatial scales (including those of the forcing and dissipation), and over extremely long times (as the nonlinear turnover time becomes very long in the weakly nonlinear limit). And, in the focusing case, a complete theory must also incorporate the interaction of localized coherent structures with resonant radiation.