Kolmogorov-Zakharov湍流的随机矩阵模型
Random matrix model of Kolmogorov-Zakharov turbulence
AI总结：
本文提出随机矩阵模型研究有限自由度非线性系统中的Kolmogorov-Zakharov湍流，揭示直接级联在混沌边界以上出现符合KZ理论的稳态能量流与代数衰减范数分布，并指出该动力学湍流可推广至无序晶格模型及多模光纤。
摘要：
我们引入并研究了一个非线性纯动力学有限尺寸多自由度系统中Kolmogorov-Zakharov湍流的随机矩阵模型。对于直接级联的情形，能量和范数的泵浦发生在低能量尺度，并在高能量处被吸收。当泵浦强度超过某一混沌边界时，出现一个全局混沌吸引子，并伴随有穿过哈密顿惯性能量区间的稳态能量流。在该区域中，稳态范数分布由代数衰减描述，其指数与Kolmogorov-Zakharov理论一致。在混沌边界以下，系统处于类似于Kolmogorov-Arnold-Moser理论的准可积区域，湍流被抑制。对于逆级联，系统迅速进入强非线性区域，此时弱湍流描述不再适用。我们认为这种动力学湍流具有普适性，并证明它存在于其他具有无序和Anderson局域化的晶格模型中。我们指出，这类动力学模型可以在多模光纤中实现。
英文摘要：
We introduce and study a random matrix model of Kolmogorov-Zakharov turbulence in a nonlinear purely dynamical finite size system with many degrees of freedom. For the case of a direct cascade the energy and norm pumping takes place at low energy scales with absorption at high energies. For a pumping strength above a certain chaos border a global chaotic attractor appears with a stationary energy flow through a Hamiltonian inertial energy interval. In this regime, the steady-state norm distribution is described by an algebraic decay with an exponent in agreement with the Kolmogorov-Zakharov theory. Below the chaos border the system is located in the quasi-integrable regime similar to the Kolmogorov-Arnold-Moser theory and the turbulence is suppressed. For the inverse cascade the system rapidly enters a strongly nonlinear regime where the weak turbulence description is invalid. We argue that such a dynamical turbulence is generic showing that it is present in other lattice models with disorder and Anderson localization. We point out that such dynamical models can be realized in multimode optical fibers.
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