Linear dynamics of the GP equation

We will now develop a WKB theory for small-scale wave-packets, described by a linearized GP equation, with and without the presence of a background condensate. As is traditional with any WKB-type method we assume the existence of a scale separation $\varepsilon \ll 1$, as explained in section . In this analysis we will take $l\sim 1$ so that any spatial derivatives of a given large-scale quantity (e.g. the potential $U$ or the condensate) are of order $\varepsilon$. The transition to WKB phase-space is achieved through the application of the Gabor transform [23],

$$\hat{g}(\vec{x},\vec{k},t) = \int f(\varepsilon^{*}\vert\vec{x} ... ..., \E^{i\vec{k}\cdot(\vec{x} - \vec{x}_{0})} \, g(\vec{x}_{0},t)\,d\vec{x}_{0},$$ (2)

where $f$ is an arbitrary function fastly decaying at infinity. For our purposes it will be sufficient to consider a Gaussian of the form

$$f(\vec{x}) = \frac{1}{(2\pi)^{d}}\,e^{- x^2},$$

where $d$ is the number of space dimensions. The parameter $\varepsilon^{*}$ is small and such that $\varepsilon \ll \varepsilon^{*} \ll 1$. Hence, our kernel $f$ varies at the intermediate-scale. A Gabor transform can therefore be thought of as a localized Fourier transform, and in the limit $\epsilon^*\to 0$ becomes an exact Fourier transform. Physically, one can view a Gabor transform as a wavepacket distribution function over positions $\vec{x}$ and wavevectors $\vec{k}$.