bf}\sim |\Psi_{\rm bf}(x)|e^{ik_{\rm bf}(x)x}$, its spatial WFT exhibits the local wavevector $k_{\rm bf}(x)$ for each $x$, which determines the local flow velocity in the laboratory frame, $\hbar k_{\rm bf}(x)/m$. This is shown as the main streak in Fig. \[fig:exp\](b). Similarly, the spatial WFT of the density, $n(x)=|\Psi(x)|^2$, separates the background condensate density $n_{\rm bf}(x)$ ($k=0$) from the superimposed spatial oscillations (with nonzero $k$). An example is shown in Fig. \[fig:GP\_growth\](b), which is the spatial WFT of density profile in Fig. \[fig:GP1\](g), evaluated at the center of the oscillatory region. The spectrum has a central peak $n_{\rm bf}$ as the background density, and two side peaks $n_k$, indicating the oscillatory component.

Moving average of the GP wavefunction {#sec:smoothing0}
-------------------------------------

To separate fast-oscillating components in Eq. \[eq:psi0\] from the slowly-varying parts, we implement a smoothing procedure on the GP wavefunction. The procedure is equivalent to calculating the moving average of a discrete data set, which smooths out short-range fluctuations. Here the moving average of wavefunction $\Psi(x)$ is defined as $$\begin{aligned}
\label{eq:smoothing0}
\bar{\Psi}(x)=\frac{1}{2D_s} \int^{x+D_s}_{x-D_s} dy\, \Psi(y),\end{aligned}$$ where the integral serves as a square window of width $2D_s$ centered at $x$, over which $\Psi(x)$ is being averaged. For components in $\Psi(x)$ with wavelength shorter 