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Intermittency in generalized NLS equation with focusing six-wave interactions

We study numerically the statistics of waves for generalized one-dimensional Nonlinear Schrodinger (NLS) equation that takes into account focusing six-wave interactions, dumping and pumping terms. We demonstrate the universal behavior of this system for the region of parameters when six-wave interactions term affects significantly only the largest waves. In particular, in the statistically steady state of this system the probability density function (PDF) of wave amplitudes turns out to be strongly non-Rayleigh one for large waves, with characteristic "fat tail" decaying with amplitude $|Ψ|$ close to $\propto\exp(-γ|Ψ|)$, where $γ>0$ is constant. The corresponding non-Rayleigh addition to the PDF indicates strong intermittency, vanishes in the absence of six-wave interactions, and increases with six-wave coupling coefficient.