Growing of integrable turbulence

Abstract: We study numerically the integrable turbulence in the framework of the focusing one-dimensional nonlinear Schrodinger equation using a new method -- the "growing of turbulence". We add to the equation a weak controlled pumping term and start adiabatic evolution of turbulence from statistically homogeneous Gaussian noise. After reaching a certain level of average intensity, we switch off the pumping and realize that the "grown up" turbulence is statistically stationary. We measure its Fourier spectrum, the probability density function (PDF) of intensity and the autocorrelation of intensity. Additionally, we show that, being adiabatic, our method produces stationary states of the integrable turbulence for the intermediate moments of pumping as well. Presently, we consider only the turbulence of relatively small level of nonlinearity; however, even this "moderate" turbulence is characterized by enhanced generation of rogue waves.