Integrable Turbulence with Nonlinear Random Optical Waves

Abstract

The field of integrable turbulence deals with the general question of statistical changes that are experienced by ensembles of nonlinear random waves propagating in systems ruled by integrable equations. In this chapter, we specifically focus on the one-dimensional nonlinear Schrodinger equation that describes quantitatively very well experiments performed with single mode fibers and optical waves randomly fluctuating in time. As a result of nonlinear propagation, the power spectrum of the random wave broadens and takes exponential wings both in focusing and in defocusing propagation regimes. Heavy-tailed deviations from gaussian statistics are observed in focusing regime while low-tailed deviations from gaussian statistics are observed in defocusing regime. After some transient evolution, the wave system is found to exhibit a statistically stationary state in which neither the probability density function of the wave field nor the spectrum change with the evolution variable. Separating fluctuations of small scale from fluctuations of large scale both in focusing and defocusing regime, we evidence the phenomenon of intermittency; i.e., small scales are characterized by large heavy-tailed deviations from Gaussian statistics, while the large ones are almost Gaussian.