We investigate the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The statistical properties of the noise-induced MI, observed previously in numerical simulations and in experiments, have not been explained theoretically. In this Letter, using the inverse scattering transform (IST) formalism, we propose a theoretical model of the asymptotic stage of the noise-induced MI based on N-soliton solutions of the focusing one-dimensional nonlinear Schrödinger equation. Specifically, we use ensembles of N-soliton bound states having a special semiclassical distribution of the IST eigenvalues, together with random phases for norming constants. To verify our model, we employ a recently developed numerical approach to construct an ensemble of N-soliton solutions with a large number of solitons, N∼100. Our investigation reveals a remarkable agreement between spectral (Fourier) and statistical properties of the long-term evolution of the MI and those of the constructed multisoliton, random-phase bound states. Our results can be generalized to a broad class of strongly nonlinear integrable turbulence problems.

The interaction of the external donor (propyltrimethoxysilane - PTMS) with titanium-magnesium catalysts (TMCs) containing dibutylphthalate (DBP) as internal donor, which were prepared in different ways, was studied by chemical analysis and infrared diffuse reflectance spectroscopy (DRIFTS). The chemical composition of the catalysts after their interaction with heptane solutions of PTMS, PTMS/AlEt3 or AlEt3 during 1h at 70°C showed that this interaction led to removal of both TiCl4 and DBP from the catalysts. The fractions of DBP and Ti extracted, as well as the amounts of PTMS and AlEt3 bound, depended on the method of synthesizing the catalysts. DRIFT spectroscopy data concerning the state of DBP in the catalysts, before and after treatment with heptane solutions of PTMS or PTMS/AlEt3 during 1h at 70°C, showed that PTMS could substitute both TiCl4 and DBP, while adsorbing on coordinatively unsaturated Ti and Mg ions in the catalyst. The presence of AlEt3 played a key role in the interaction of PTMS with the catalyst. Activity data for propylene polymerization showed that treatment of TMC catalysts with PTMS before polymerization led to a sharp activity decrease due to deactivation of active sites, while the interaction of the catalyst with PTMS in the presence of AlEt3 led only to a slight decrease of activity, probably due to deactivation of non-stereospecific active centers.