Integrable turbulence and formation of rogue waves

Abstract

In the framework of the focusing nonlinear Schrödinger equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. The development of the MI leads to the formation of ‘integrable turbulence’ (Zakharov 2009 Stud. Appl. Math. 122 219–34). We study the time evolution of its major characteristics averaged across realizations of initial data—the condensate solution seeded by small random noise with fixed statistical properties.We observe that the system asymptotically approaches to the stationary integrable turbulence, however this is a long process. During this process momenta, as well as kinetic and potential energies, oscillate around their asymptotic values. The amplitudes of these oscillations decay with time t as t−3/2, the phases contain the nonlinear phase shift that decays as t−1/2, and the frequency of the oscillations is equal to the double maximum growth rate of the MI. The evolution of wave-action spectrum is also oscillatory, and characterized by formation of the power-law region ∼|k|−α in the small vicinity of the zeroth harmonic k = 0 with exponent α close to 2/3. The corresponding modes form ‘quasi-condensate’, that acquires very significant wave action and macroscopic potential energy.The probability density function of wave amplitudes asymptotically approaches the Rayleigh distribution in an oscillatory way. Nevertheless, in the beginning of the nonlinear stage the MI slightly increases the occurrence of rogue waves. This takes place at the moments of potential energy modulus minima, where the PDF acquires ‘fat tales’ and the probability of rogue waves occurrence is by about two times larger than in the asymptotic stationary state.Presented facts need a theoretical explanation.