波动湍流区的共振群聚：可积动力学
Resonance clustering in wave turbulent regimes: Integrable dynamics
摘要：
现代波动湍流理论的两个基本事实是：1）在$k$ -空间中存在功率能谱; 2）在该能谱中存在对应于共振成团的“间隙”。因此，三波湍流制度被挑出来：\emph{kinetic}，描述波动力学方程和功率能谱; \emph{discrete}，其特征在于共振聚类;和\emph{mesoscopic}，两种类型的波场时间演化共存。在本文中，我们研究了出现在离散和介观波动湍流制度的共振集团的可积动力学。使用一种新的方法的基础上的概念的动力学不变量，我们建立了一些经常遇到的集群是可积的求积为任意的初始条件和其他一些-只为特定的初始条件。我们还发现在某些情况下的混沌行为。所获得的结果的物理意义进行了讨论。
英文摘要：
Two fundamental facts of the modern wave turbulence theory are 1) existence of power energy spectra in $k$-space, and 2) existence of "gaps" in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: \emph{kinetic}, described by wave kinetic equations and power energy spectra; \emph{discrete}, characterized by resonance clustering; and \emph{mesoscopic}, where both types of wave field time evolution coexist. In this paper we study integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes. Using a novel method based on the notion of dynamical invariant we establish that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others -- only for particular initial conditions. We also identify chaotic behaviour in some cases. Physical implications of the results obtained are discussed.
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