Starting from action-angle variables and using a standard asymptotic expansion, we present an original and coincise derivation of the Wave Kinetic equation for a resonant process of the type ##IMG## [http://ej.iop.org/images/2399-6528/4/9/095016/jpcoabb4b7ieqn1.gif] {$2\leftrightarrow 2$} . Despite not being more rigorous than others, our procedure has the merit of being straightforward; it allows for a direct control of the random phases and random action of the initial wave field. We show that the Wave Kinetic equation can be derived assuming only initial random phases. The random action approximation has to be taken only after the weak nonlinearity and large box limits are taken. The reason is that the oscillating terms in the evolution equation for the action contain, as an argument, the action-dependent nonlinear corrections which is dropped, using the large box limit. We also show that a discrete version of the Wave Kinetic Equation can be obtained for...

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作用角变量中四波动力学方程的直接推导

从作用角变量开始，并使用标准渐近展开，我们提出了波动动力学方程的原始且精确的推导，用于类型为## IMG ##的谐振过程[http://ej.iop.org/images/ 2399-6528 / 4/9/095016 / jpcoabb4b7ieqn1.gif] {$ 2 \ leftrightarrow 2 $}。尽管没有比其他方法更严格，但我们的程序具有简单明了的优点。它允许直接控制初始波场的随机相位和随机作用。我们表明，仅假设初始随机相位，就可以推导波动动力学方程。仅在采取了弱的非线性和大的盒极限之后才需要采取随机作用近似。原因是动作的演化方程中的振动项包含作为参数的与动作有关的非线性校正，该校正被丢弃，使用大盒子的限制。我们还表明，可以针对以下情况获得波动动力学方程的离散形式：