Integrable turbulence and formation of rogue waves
Integrable turbulence and formation of rogue waves
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DOI：
10.1088/0951-7715/28/8/2791
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发表时间：
2015-08-01
期刊：
影响因子：
1.7
通讯作者：
Zakharov, V. E.
中科院分区：
文献类型：
--
作者：
Agafontsev, D. S.;Zakharov, V. E.
在聚焦非线性薛定谔方程的框架中，我们对凝聚态调制不稳定性（MI）的非线性阶段进行了数值研究。 MI 的发展导致“可积湍流”的形成（Zakharov 2009 Stud. Appl. Math. 122 219-34）。我们研究了其主要特征的时间演化，这些特征在初始数据的实现中平均——由具有固定统计特性的小随机噪声产生的凝聚解。我们观察到系统渐近地接近稳态可积湍流，然而这是一个漫长的过程。在此过程中，动量以及动能和势能围绕其渐近值振荡。这些振荡的幅度随时间 t 衰减为 t(-3/2)，相位包含衰减为 t(-1/2) 的非线性相移，振荡频率等于 MI 的双最大增长率。波作用谱的演化也是振荡的，其特征是在指数a接近2/3的零次谐波k = 0的小附近形成类似于竖条k竖条(-α)的幂律区域。相应的模态形成“准凝聚态”，获得非常显着的波动作用和宏观势能。波幅的概率密度函数以振荡方式渐近逼近瑞利分布。然而，在非线性阶段开始时，MI 会稍微增加异常波的发生。这发生在势能模量最小值的时刻，此时 PDF 获得“大故事”，并且异常波发生的概率比渐近稳态大大约两倍。所提出的事实需要理论解释。
In the framework of the focusing nonlinear Schrodinger 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 similar to vertical bar k vertical bar(-alpha) in the small vicinity of the zeroth harmonic k = 0 with exponent a 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.