Figure 1

Density profile in vicinity of the $\delta$ potential represented by a (red) vertical straight line. The flow is stationary, with a velocity directed toward positive $x$ as indicated by the arrow. The density in the region $x<0$ is a portion of a dark soliton (see Sec. 3b) with asymptotic upstream density ($n_1$) and subsonic velocity. The flow in the downstream region $x>0$ has a constant density and a supersonic velocity.Reuse & Permissions

Figure 2

Sketch of the density profile $n(x,t)$ at a given time $t>0$. The flow is directed toward positive $x$. The regions denoted as $A$, $B$, $C$, and $D$ are presented in the text. Points $X_{-}\left(t\right)$ and $X_{+}\left(t\right)$ are the small amplitude edge and soliton edge of the DSW (region $B$). The characteristic densities $n_0$, $n_1$, and $n_2$ are defined in the text [Eqs. (10), (24), and (26)].Reuse & Permissions

Figure 3

The Riemann invariants plotted as functions of the self-similar variable $x/t$ in regions $A$, $B$, and $C$. In region $C$ the (red) dashed line (not considered in Sec. 3) corresponds to the soliton train with ${\lambda }_2={\lambda }_3=x/t-u_1/2$ discussed in Sec. 5 [Eq. (86)]. The figure is drawn for the situation studied in Sec. 5, with $n_0=1$, $u_0=1$, and $\kappa =5.2$. The corresponding velocities of the edges of the DSW (region $B$) are $V_{-}=-2.4$ and $V_{+}=-0.48$.Reuse & Permissions

Figure 4

Boundaries of the region of parameters for which the DSW is expelled upstream from the obstacle. The vertical axis is the speed of the incident beam in units of the speed of sound $c_0=\sqrt{n_0}$. The horizontal axis is the dimensionless strength $\kappa /\sqrt{n_0}$ of the $\delta$ potential. The DSW is expelled from the region of the obstacle when the point representative of the system lies between the two solid curves which correspond to Eqs. (66) and (67). The dashed curves are the approximate boundaries (68), (69) (valid when $\kappa /\sqrt{n_0}\ll 1$), (70), and (71) (valid when $\kappa /\sqrt{n_0}\gg 1$).Reuse & Permissions

Figure 5

Boundaries of the region (78) for which the hydraulic solution exists and the upstream DSW is detached from a thick and smooth obstacle (here we use units such that $n_0=1$). The dashed lines correspond to the approximations (79) which are valid when $U_{\mathrm{m}}\ll 1$.Reuse & Permissions

Figure 6

Density profiles at $t=500$. The three upper panels present the results of numerical simulations performed in the case where the obstacle is represented by the Gaussian potential (81) and the incident beam by (80) with $\Delta =20$, 10, and 5. The lower panel is the analytical result corresponding to a point-like obstacle (9) and a step-like incident beam (10) and (11), that is, $\Delta =0$.Reuse & Permissions

Figure 7

Density profiles at $t=500$. These plots are enlargements of specific parts of the two lower plots of Fig. 6. The left plot concerns a part of the DSW ($-590<x<-560$), and the right one concentrates on the region of the obstacle. In both the right and left plots the black solid line corresponds to the analytical density profile. In the right plot the red solid line is the numerical density computed in the case $\Delta =5$. The numerical profile is represented by red points in the left plot, because it would be indistinguishable from the analytical result if represented with a solid line.Reuse & Permissions

Figure 8

The thick solid line represents the analytical prediction (87) for the positions of the points of minimal density of the soliton train at time $t=500$. The value $u_1=0.039$ was used in the plot of Eq. (87) [see discussion below Eq. (85)]. The dots show the same quantity obtained from the density profile $n(x,t=500)$ computed numerically in the case $\Delta =20$ (see Fig. 6, upper panel).Reuse & Permissions