in Appendix \[app:1dGPE\], while in Appendix \[app:3dGPE\] we discuss and evaluate the criteria for applicability of the reduction to a 1D model, and compare with a simulation using the 3D GPE. We find that the qualitative features in the 1D simulation are consistent with the results of 3D simulation. This indicates that the results of our detailed analysis of the 1D simulation should apply as well to the 3D system.

The GP wavefunction, $\Psi(x,t)$, in our simulation involves a condensate component and the excitation modes generated during the sweep: $$\begin{aligned}
\label{eq:psi0}
\Psi(x,t)=\Psi_{\rm bf}(x,t)+\sum_{j}\psi_{j}(x,t),\end{aligned}$$ where $\Psi_{\rm bf}$ indicates the background condensate flow, and $\psi_{j}$ denotes its excitation mode satisfying the Bogoliubov–de Gennes (BdG) equation [@pethick]. In our simulation, there are three modes that we observed: the BCR mode, the HR mode, and the HR partner (labeled by $j=$ BCR, HR, and p, respectively). The role of each mode will be explained in the later sections. In regions where the flow is slowly varying, each component in Eq. \[eq:psi0\] behaves locally as a Wentzel-Kramers-Brillouin (WKB) plane wave with a characteristic frequency $\omega$ and wavevector $k$. Here we introduce two techniques to resolve individual components and their spectral properties.

Windowed Fourier transform {#sec:WFT0}
