Extreme rogue wave generation from narrowband partially coherent waves

In the framework of the focusing one-dimensional nonlinear Schrödinger equation, we study numerically the integrable turbulence developing from partially coherent waves (PCW), which represent superposition of uncorrelated linear waves. The long-time evolution from these initial conditions is characterized by emergence of rogue waves with heavy-tailed (non-Gaussian) statistics, and, as was established previously, the stronger deviation from Gaussianity (i.e., the higher frequency of rogue waves) is observed for narrower initial spectrum. We investigate the fundamental limiting case of very narrow initial spectrum and find that shortly after the beginning of motion the turbulence enters a quasistationary state (QSS), which is characterized by a very slow evolution of statistics and lasts for a very long time before arrival at the asymptotic stationary state. In the beginning of the QSS, the probability density function (PDF) of intensity turns out to be nearly independent of the initial spectrum and is very well approximated by a certain Bessel function that represents an integral of the product of two exponential distributions. The PDF corresponds to the maximum possible stationary value of the fourth-order moment of amplitude κ_{4}=4 and yields a probability to meet intensity above the rogue wave threshold that is higher by 1.5 orders of magnitude than that for a random superposition of linear waves. We routinely observe rogue waves with amplitudes ten times larger than the average one, and all of the largest waves that we have studied are very well approximated by the amplitude-scaled rational breather solutions of either the first (Peregrine breather) or the second orders.