Relation to the Wigner Transformation

One can also derive waveaction transport equation (6) directly from equation of motion (25), without obtaining first the Hamiltonian structure (5). To do this we define the Wigner waveaction by using the Wigner transformation:

$${n}^{\rm W}_{\textbf{k}\textbf{x}}\equiv\int e^{i\textbf{m}\cdot\... ...k}+\frac{1}{2}\textbf{m}} \hat{a}_{\textbf{k}-\frac{1}{2}\textbf{m}}^* \rangle.$$

Then the Wigner waveaction ${n}^{\rm W}_{\textbf{k}\textbf{x}}$ obeys the same kinetic equation (6). The prove can be found, for example, in [18]. To prove this formula, one calculates the time evolution of waveaction ${n}^{\rm W}$ by using definition (48) and equation of motion (25). Then one uses transformation similar to (29,30), and expands ${n}^{\rm W}$ using the smallness of $m$ , and finally uses integration by parts to obtain (6) - see [18] for details.

To address the question on how waveaction ${n}^{\rm W}_{\textbf{k},\textbf{x}}$ , defined through the Wigner transformation, is related to the wave action $\check{n}_{\textbf{k},\textbf{x}}$ , defined through the Gabor variables, we substitute (21) into the definition of (48) to write

$${n}^{\rm W}_{\textbf{k},\textbf{x}}= (\frac{\varepsilon^*}{(\sqr... ...\rangle e^{i \textbf{m}\cdot\textbf{x}} d\textbf{x}'d\textbf{x}'' d\textbf{m}.$$

Taking into account that $\check{a}_{\textbf{k}+\textbf{m}/2\textbf{x}'}$ and $\check{a}_{\textbf{k}-\textbf{m}/2,x''}^*$ are slow functions of $\textbf{x}'$ and $\textbf{x}''$ , one can neglect this slow coordinate dependence relative to fast $\textbf{x}$ dependence in the exponent. This allows to perform $\textbf{x}'$ and $\textbf{x}''$ integrations to obtain a delta-function with respect to the $\textbf{m}$ argument. Consequently we obtain that these two wave-actions, (48) and (47) are approximately proportional to each other:

$${n}^{\rm W}_{\textbf{k},\textbf{x}}\propto \check{n}_{\textbf{k},\textbf{x}}.$$

There are several important advantages of our method. First, it allows to rigorously right the equation of motion for the field variable $a_{\textbf{k},\textbf{x}}$ in addition to the transfer equation of (6). In addition, our approach shows how to derive the kinetic equation (6) to a much broader class of nonlinear systems, those described by equation (4) with nonzero value of $B(\textbf{q},\textbf{q}_1)\equiv 0$ , as we show in the next section. Lastly, Hamiltonian formulation helps significantly to establish a wave turbulence theory, which takes into account nonlinear wave-wave interactions.