Evolution equation for the Wigner distribution

The aim of this section is the derivation of the evolution equation for the Wigner distribution. This equation governs the wave field that is considered as a phase-space representation, that is, a description of the wave as a function of both position $\vec{x}$ and wave number $\vec{k}$. The key to the approach to follow is the back and forth Fourier transformation between the physical space $\vec{x}$ and the phase space $(\vec{x},\vec{k})$ and the underlying formalism is Weyl symbol calculus (Weyl, 1931; McDonald, 1988; Cohen, 2012). Basically, we have ordinary functions living in phase space called symbols and operators or kernels acting in physical space. The correspondence between symbols and operators was introduced by Weyl (1931) and appeared to be useful in deriving an equation governing the phase-space representation of the wave field.


Before proceeding let us recall the following property of the Fourier transforms which will be used frequently later on. The Fourier transform of a derivative yields the Fourier transform of the function itself multiplied by the Fourier variable. For instance, the Fourier transform of $\nabla_{\vec{\xi}}$ is given by

  $$\frac{1}{4\pi^2} \int \nabla_{\vec{\xi}} f(\vec{\xi}) \, e^{-{\rm... ...{-{\rm i}\vec{k}\cdot\vec{\xi}}d\vec{\xi} = {\rm i}\vec{k}\, {\hat f}(\vec{k})$$ (2.201)

where we have employed an integration by parts, and $f(\vec{\xi})$ and ${\hat f}(\vec{k})$ is a compactly supported smooth function and its Fourier transform, respectively. Here, we say that the derivative $\nabla_{\vec{\xi}}$ is associated with ${\rm i}\vec{k}$, denoted by $-{\rm i}\nabla_{\vec{\xi}} \leftrightarrow \vec{k}$ with $\leftrightarrow$ representing the correspondence symbol. Likewise, ${\rm i}\nabla_{\vec{k}} \leftrightarrow \vec{\xi}$.


The starting point for the derivation of the equation governing the evolution of the Wigner distribution in phase space is the following dispersive wave equation (e.g., Bremmer, 1973; Besieris and Tappert, 1976)

  $$\frac{\partial \eta}{\partial t} = -{\rm i}\Omega \left (\vec{x},-{\rm i}\nabla_{\vec{x}} \right )\,\eta$$ (2.202)

where $\eta(\vec{x},t)$ is the complex-valued surface elevation^2.6and $\Omega(\vec{x},-{\rm i}\nabla_{\vec{x}})$ is the linear pseudo-differential wave operator associated with the dispersion relation $\omega(\vec{x},\vec{k})$ which is given by^2.7

  $$\omega(\vec{x},\vec{k}) = \sqrt{g\vert\vec{k}\vert\tanh(\vert\vec{k}\vert d)} + \vec{k}\cdot\vec{u}$$ (2.203)

with $d(\vec{x})$ the water depth and $\vec{u}(\vec{x})$ the mean current. Eq. (2.203) is valid for linear waves propagating over slowly varying medium for which the local dispersion relation (2.204) can be employed.


Since $\Omega(\vec{x},-{\rm i}\nabla_{\vec{x}})$ is the coordinate space representation of the operator that acts on functions living in physical space, we invoke the Weyl rule to obtain the correspondence with phase-space functions. This rule thus assigns operators acting on physical space to functions defined on the phase space (Cohen, 2012). First, we define the Fourier transform of $\omega(\vec{x},\vec{k})$ by

  $${\hat \omega} (\vec{q},\vec{p}) = \frac{1}{4\pi^2} \int \int \ome... ...,e^{-{\rm i}\vec{q}\cdot\vec{x}-{\rm i}\vec{p}\cdot\vec{k}} \,d\vec{x}d\vec{k}$$ (2.204)

and the inverse

  $$\omega (\vec{x},\vec{k}) = \int \int {\hat \omega}(\vec{q},\vec{p})\,e^{{\rm i}\vec{q}\cdot\vec{x}+{\rm i}\vec{p}\cdot\vec{k}} \,d\vec{q}d\vec{p}$$ (2.205)

Next, the Weyl operator $\Omega(\vec{x},-{\rm i}\nabla_{\vec{x}})$ associated with $\omega(\vec{x},\vec{k})$ is defined by the substitution of the operator $-{\rm i}\nabla_{\vec{x}}$ for $\vec{k}$ in Eq. (2.206),

  $$\Omega(\vec{x},-{\rm i}\nabla_{\vec{x}}) = \int \int {\hat \omega... ...e^{{\rm i}\vec{q}\cdot\vec{x}+\vec{p}\cdot\nabla_{\vec{x}}} \,d\vec{q}d\vec{p}$$ (2.206)

We call the function $\omega$ corresponding to the operator $\Omega$ the Weyl symbol of $\Omega$; notation: symb $[\Omega](\vec{x},\vec{k}) = \omega(\vec{x},\vec{k})$. Note further that the exponential function here is to be understood as its Taylor series expansion, that is,

  $$e^{{\rm i}\vec{q}\cdot\vec{x}+\vec{p}\cdot\nabla_{\vec{x}}} = \su... ...{n!} \Bigl [ {\rm i}\vec{q}\cdot\vec{x}+\vec{p}\cdot\nabla_{\vec{x}} \Bigr ]^n$$ (2.207)

In a similar vein, the function $\Gamma$ lives on the physical space. Hence, in the context of symbol formalism and considering Eq. (2.196), $\Gamma$ is interpreted as a coordinate space (kernel) operator and its Weyl symbol is the Wigner distribution (McDonald, 1988).


In the following step, we consider the correlation function $\Gamma (\vec{x},\vec{\xi})$ and derive its evolution equation as follows. First, referring to Eq. (2.195), we have^2.8

  $$\frac{\partial \Gamma}{\partial t} = \biggl < \frac{\partial \eta... ...< \eta(\vec{x}_1)\, \frac{\partial \eta^\star(\vec{x}_2)}{\partial t} \biggr >$$ (2.208)

with $\vec{x}_1 = \vec{x}+\vec{\xi}/2$ and $\vec{x}_2 = \vec{x}-\vec{\xi}/2$. Next, we substitute Eq. (2.203) into the above equation, resulting in

  $$\frac{\partial \Gamma}{\partial t} = -{\rm i} \Bigl < \Omega \lef... ...t (\vec{x}_2,{\rm i}\nabla_{\vec{x}_2} \right )\,\eta^\star(\vec{x}_2) \Bigr >$$ (2.209)

By applying the transformation $(\vec{x}_1,\vec{x}_2) \rightarrow (\vec{x},\vec{\xi})$ (using the inverse of the Jacobian gives $\nabla_{\vec{x}_1} = \nabla_{\vec{x}}/2 + \nabla_{\vec{\xi}}$ and $\nabla_{\vec{x}_2} = \nabla_{\vec{x}}/2 - \nabla_{\vec{\xi}}$), one obtains

  $$\frac{\partial \Gamma}{\partial t}(\vec{x},\vec{\xi}) = -{\rm i} ... ...{x}}/2 - {\rm i}\nabla_{\vec{\xi}} \right ) \right ] \Gamma(\vec{x},\vec{\xi})$$ (2.210)

Since symb $[\Gamma](\vec{x},\vec{k}) = W(\vec{x},\vec{k})$, the phase-space equation for the Wigner distribution can be found using the Fourier transformation, and is given by

  $$\frac{\partial W}{\partial t}(\vec{x},\vec{k}) = -{\rm i} \left [... ...c{k}}/2,\vec{k}+{\rm i}\nabla_{\vec{x}}/2 \right ) \right ] W(\vec{x},\vec{k})$$ (2.211)

or written in shorthand as

  $$\frac{\partial W}{\partial t}(\vec{x},\vec{k}) = -{\rm i} \omega ... ...2,\vec{k}-{\rm i}\nabla_{\vec{x}}/2 \right )\, W(\vec{x},\vec{k}) + \rm {c.c.}$$ (2.212)

where c.c. stands for complex conjugate.


It should be noted that we could have derived an evolution equation for the correlation function in the Fourier space instead of physical space, that is, ${\hat \Gamma}(\vec{k},\vec{u})$, and subsequently an equation governing the Wigner distribution, $W = {\rm symb}[{\hat \Gamma}]$, which is then exactly Eq. (2.213); see Smit and Janssen (2013).


Eq. (2.213) contains the product of two symbols $\omega$ and $W$ in phase space. The purpose of what follows is to find an explicit expression for symbol $\omega(\vec{x}+{\rm i}\nabla_{\vec{k}}/2,\vec{k}-{\rm i}\nabla_{\vec{x}}/2)$. Its Weyl operator is given by

  $$\Omega(\vec{x}+\vec{\xi}/2,-{\rm i}\nabla_{\vec{\xi}}-{\rm i}\nab... ...xi}/2)+\vec{p}\cdot(\nabla_{\vec{\xi}}+\nabla_{\vec{x}}/2)} \,d\vec{q}d\vec{p}$$ (2.213)

To proceed we recall that $\vec{x}$ and $\nabla_{\vec{x}}$ do not commute (and likewise $\nabla_{\vec{\xi}}$ and $\vec{\xi}$) and their commutation relation can be expressed as $[{\rm i}\vec{q}\cdot\vec{x},\vec{p}\cdot\nabla_{\vec{x}}/2] = -\frac{1}{2}{\rm i}\vec{q}\cdot\vec{p}$ (and $[\vec{p}\cdot\nabla_{\vec{\xi}},{\rm i}\vec{q}\cdot\vec{\xi}/2] = \frac{1}{2}{\rm i}\vec{p}\cdot\vec{q}$). With the aid of the Baker-Campbell-Hausdorff (BCH) identity^2.9we can simplify the Weyl operator as follows

$$\Omega(\vec{x}+\vec{\xi}/2,-{\rm i}\nabla_{\vec{\xi}}-{\rm i}\nabla_{\vec{x}}/2) = \int \int {\hat \omega}(\vec{q},\vec{p})\, e^{{\rm i}\vec{q}\cdot... ...}\vec{q}\cdot\vec{\xi}/2}\,e^{-{\rm i}\vec{p}\cdot\vec{q}/4} \,d\vec{q}d\vec{p}$$

$$= \int \int {\hat \omega}(\vec{q},\vec{p})\,e^{{\rm i}\vec{q}\cdot\... ...}\cdot\nabla_{\vec{x}}/2}\,e^{{\rm i}\vec{q}\cdot\vec{\xi}/2}\,d\vec{q}d\vec{p}$$

Now, the corresponding Weyl symbol is written as

$$\omega(\vec{x}+{\rm i}{\nabla_{\vec{k}}}/2,\vec{k}-{\rm i}\nabla_{\vec{x}}/2) = \int \int {\hat \omega}(\vec{q},\vec{p})\,e^{{\rm i}\vec{q}\cdot\... ...\cdot\nabla_{\vec{x}}/2}\,e^{-\vec{q}\cdot\nabla_{\vec{k}}/2}\,d\vec{q}d\vec{p}$$

$$= \int \int \underbrace{{\hat \omega}(\vec{q},\vec{p})\,e^{{\rm i}\... ...vec{x}}/2}\,e^{-\vec{q}\cdot\nabla_{\vec{k}}/2}}_{({\rm II})}\,d\vec{q}d\vec{p}$$

The first part (I) is simply the Fourier transform of ${\hat \omega}(\vec{q},\vec{p})$, that is, Eq. (2.206), resulting in $\omega(\vec{x},\vec{k})$. With respect to part (II) we recall that the exponential function can be expressed as its Taylor series, such as

  $$e^{\vec{p}\cdot\nabla_{\vec{x}}/2} = \sum_{n=0}^{\infty} \frac{1}{n!} \left [ \frac{1}{2}\vec{p}\cdot\nabla_{\vec{x}} \right ]^n$$ (2.214)

Then keeping in mind that multiplication by $\vec{q}$ or $\vec{p}$ will lead to a derivative in the corresponding Fourier variable $\vec{x}$ or $\vec{k}$, respectively, we obtain the following

  $$\omega(\vec{x}+{\rm i}{\nabla_{\vec{k}}}/2,\vec{k}-{\rm i}\nabla_... ...{x}}- \overleftarrow{\nabla}_{\vec{x}}\cdot\nabla_{\vec{k}} \right ) \right )$$ (2.215)

with the left-pointing arrow above the derivative indicating that it acts on the symbol standing to the left (here $\omega$). Inserting in Eq. (2.213) yields

  $$\frac{\partial W}{\partial t}(\vec{x},\vec{k}) = -{\rm i} \omega(... ...ot\overrightarrow{\nabla}_{\vec{x}} \right )\, W(\vec{x},\vec{k}) + \rm {c.c.}$$ (2.216)

where the right-pointing arrow implies that the derivative operates on the symbol to the right (here $W$). The first term of the right hand side of Eq. (2.217) represents the product of the two symbols and is known as the Moyal product (Cohen, 2012). Clearly, this product is not commutative because the product of the associated operators in physical space is a non-commutative operator. However, by virtue of the Weyl rule of association, the proper ordering of the arguments $\vec{x}$ and $-{\rm i}\nabla_{\vec{x}}$ of operator $\Omega$ is obtained such that Eq. (2.203) recasts to the correct transport equation for $W(\vec{x},\vec{k})$, assuming the usual WKB ansatz (further details and discussion on this topic can be found in Smit and Janssen, 2013 and in Akrish et al., 2020).


Eq. (2.217) is the most general form of the phase-space equation, suitable for the spectral description of the statistically inhomogeneous wave field. However, this equation is given in the form of an infinite series (through the expansion of the exponential function), which is not readily numerically tractable even for direct evaluation. Thus, an approximation is introduced by truncating the series expansion, which is the objective of the next section.