Figure 1

Location of the three poles $q_1$, $q_2$, and $q_3$ of $\chi (q,-Mq)$ in the complex $q$ plane. For positive (negative) $X=x+Mt$ the integral in Eq. (16) is evaluated by closing the contour from above (below). As a result, for $M>M_{\mathrm{crit}}$ (damped) density oscillations are observed upstream from the obstacle (i.e., for $X<0$).Reuse & Permissions

Figure 2

$M_{\mathrm{crit}}=V_{\mathrm{crit}}/c_s$ as a function of the dimensionless damping parameter $\eta$, such as given by Eq. (26). The dashed lines correspond to the asymptotic expressions $M_{\mathrm{crit}}\simeq 1-\frac{3}{2}{(\eta /2)}^{2/3}$ (for low $\eta$) and $M_{\mathrm{crit}}\simeq 2/\left(3\sqrt{3}\eta \right)$ (for large $\eta$). The insets represent typical density profiles in the presence of a repulsive $\delta$-peak impurity for $M<M_{\mathrm{crit}}$ (lower left inset) and $M>M_{\mathrm{crit}}$ (upper right inset).Reuse & Permissions

Figure 3

$\delta \rho (X=x+Mt)$ for a $\delta$-impurity potential (left panels) and a Gaussian potential (30) of width $\sigma =0.5$ (right panels) as given by perturbation theory [Eq. (15)]. The plots are drawn for a system in which $\eta =0.5$ and in this case $M_{\mathrm{crit}}=0.5$. The two upper panels correspond to a velocity below $M_{\mathrm{crit}}$ ($M=0.4$) and the two lower ones to a velocity above $M_{\mathrm{crit}}$ ($M=1.75$). In the lower right panel the dashed gray line corresponds to the approximation (29).Reuse & Permissions

Figure 4

$F_d/{\varkappa }^2$ as a function of $M=V/c_s$ for different values of the dimensionless damping parameter $\eta$. The curves are drawn for the $\delta$-impurity potential (19): $U_{\mathrm{ext}}\left(X\right)=\varkappa \delta \left(X\right)$. For each curve the abscissa of the white dot spots the value of $M_{\mathrm{crit}}\left(\eta \right)$. For each $\eta$ this critical velocity is reached exactly when $F_d/{\varkappa }^2=\frac{2}{9}$.Reuse & Permissions

Figure 5

$F_d/{\varkappa }^2$ as a function of $M=V/c_s$ for different values of the dimensionless damping parameter $\eta$. The solid curves are drawn for a Gaussian-impurity potential of width $\sigma =0.5$. The black dashed line is the corresponding asymptotic result (45). The gray dashed line is the result for a $\delta$-impurity potential, shown for comparison.Reuse & Permissions

Figure 6

$\overline{\rho }=\rho \left(0\right)$ as a function of $\eta$ for $M=3$ and different values of $\varkappa$ (left panel) and as a function of $\varkappa$ for $\eta =0.05$ and several values of $M$ (right panel).Reuse & Permissions

Figure 7

Comparison of the Whitham theory with the numerical solution of Eq. (46). The plot is drawn in the case $\eta =1$, $M=3$, and $\varkappa =4$. The numerics corresponds to the dashed black line. Whitham envelopes are shown by thin red solid lines, and the upstream dispersive shock wave oscillatory structure obtained by substitution of the solution of the Whitham Eqs. (83) into Eqs. (75) is shown by a red solid line (for $x⩽0$). The downstream ($x⩾0$) hydraulic approximation is shown by a green solid line.Reuse & Permissions

Figure 8

Different profiles $\rho \left(x\right)$ for flows past a $\delta$-impurity potential of type (19). For all the profiles the damping parameter is $\eta =0.05$. For the upper row $M=3$, then $M=1.2$ for the row below, $M=1$ for the following one, and finally $M=0.5$ for the lower row. The value of $\varkappa$ is indicated in each plot. In each plot the black solid line corresponds to the numerical solution of Eq. (46), the (red online) thin line to the perturbative result and the (green online) dashed line to the result of the hydraulic approximation which is only relevant for $x⩾0$ (see Sec. 4a).Reuse & Permissions

Figure 9

Different regimes of flow past a $\delta$-impurity potential of type (19) in the $(\varkappa ,M=V/c_s)$ plane. The main plot corresponds to the dissipative system (with $\eta =0.05$). The inset is drawn for $\eta =0$. In both plots the shaded regions correspond to stationary flows, the white ones to time-dependent and dissipative flows and the horizontal dashed line indicated the transition from a localized wake to a regime of Cherenkov emission as predicted by perturbation theory. In the inset, the subsonic ($M<1$) shaded region is superfluid and the supersonic ($M>1$) one is dissipative; the exact equation of the boundaries between the different domains is given in Ref. 31. In the main plot the dots with error bars represent the numerically determined boundary of the stationary domain. They are connected by a dashed line to guide the eye. The other dashed line in this plot represents the $\eta =0$ result (shown for comparison).Reuse & Permissions

Figure 10

Position of $q_1$, $q_2$, and $q_3$ in the complex $q$ plane. The figure is drawn in the case $\eta =0.1$. The arrows indicate the direction of motion of the poles when $M$ increases from 0 to $\infty$[53].Reuse & Permissions