Agafontsev D S, Randoux S, Suret P
P. P. Shirshov Institute of Oceanology of RAS, 36 Nakhimovsky prospekt, Moscow 117997, Russia.
Skolkovo Institute of Science and Technology, 121205 Moscow, Russia.
Phys Rev E. 2021 Mar;103(3-1):032209. doi: 10.1103/PhysRevE.103.032209.
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.
在聚焦一维非线性薛定谔方程的框架下，我们对由部分相干波（PCW）发展而来的可积湍流进行了数值研究，部分相干波代表不相关线性波的叠加。从这些初始条件出发的长时间演化的特征是出现具有重尾（非高斯）统计特性的 rogue 波，并且如先前已确定的那样，对于更窄的初始频谱，观察到与高斯性的偏差更强（即 rogue 波的频率更高）。我们研究了非常窄的初始频谱的基本极限情况，发现运动开始后不久，湍流进入准稳态（QSS），其特征是统计量的演化非常缓慢，并且在到达渐近稳态之前会持续很长时间。在 QSS 开始时，强度的概率密度函数（PDF）结果几乎与初始频谱无关，并且可以很好地由某个贝塞尔函数近似，该贝塞尔函数表示两个指数分布乘积的积分。该 PDF 对应于幅度的四阶矩的最大可能稳态值 κ₄ = 4，并产生高于 rogue 波阈值的强度出现概率，该概率比线性波的随机叠加高 1.5 个数量级。我们经常观察到幅度比平均值大十倍的 rogue 波，并且我们研究的所有最大波都可以很好地由一阶（佩雷格林呼吸子）或二阶幅度缩放的有理呼吸子解近似。
