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 result s about it. In those previous studies, in addition to the spectra of weak t urbulence (WT) theory, we also observed another distinct spectrum (the "MMT spectrum"). Our new results, presented here, include: (i) A detailed descr iption of coexistence of spectra at distinct spatial scales, and the transi tions between them at different temporal scales; (ii) The existence of a st able MMT front in k-space which separates the WT cascades from the dissipat ion range, for various forms of strong damping including "selective dissipa tion"; (iii) The existence of turbulent cycles in the one-dimensional model with focusing nonlinearity, induced by the interaction of spatially locali zed coherent structures with the resonant quartets of dispersive wave radia tion; (iv) The detailed composition of these turbulent cycles - including t he self-similar formation of focusing events (distinct in the forced and fr eely decaying cases), and the transport by the WT direct and inverse cascad es of excitations between spatial scales. This one-dimensional model admits a very precise and detailed realization of these turbulent cycles and thei r components. Our numerical experiments demonstrate that a complete theory of dispersive wave turbulence will require a full description of the turbul ent field over all spatial scales (including those of the forcing and dissi pation), and over extremely long times (as the nonlinear turnover time beco mes 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. (C) 2001 Elsevier Science B.V. All rig hts reserved.