Integrable turbulence developing from noise-induced modulational instability: numerical simulations and experimental results
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We examine turbulence development from noise-induced modulational instability of a plane wave (the condensate), in the framework of the (focusing) one-dimensional Nonlinear Schrodinger equation (1D-NLSE). The standard approach to such problems consists in derivation of the kinetic equation and studying of its stationary solutions. However, the 1D-NLSE is integrable in terms of the inverse scattering transform (IST), and for such systems the collision term in the kinetic equation vanishes exactly in any order. For this reason, we rely on direct numerical simulations and examine the basic statistical functions of the turbulence, averaged over realizations of random noise: the wave-action spectrum, the distribution of intensity and the autocorrelation of intensity. Among the most intriguing results are: damped oscillatory evolution towards the asymptotic stationary state, gradual appearance of long-distance correlations with increasing time, power-law divergence of the wave-action spectrum at small wavenumbers, the distribution of intensity that coincides with that for a purely linear system, and the imprint of the early modulational instability in the autocorrelation of intensity at the (final) asymptotic stationary state. A recent optical fiber experiment demonstrates that all these properties can be observed in real physical systems.