Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock-wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, e.g., traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation, hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical one-dimensional (1D) dissipative and dispersive shock problem shows significant differences in shock structure and shock-front speed. Numerical results associated with the three-dimensional experiment show that three- and two-dimensional approximations are in excellent agreement and 1D approximations are in good qualitative agreement. Using 1D DSW theory, it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.

Figure 1

Absorption images of blast-pulse experiments with a BEC. [(a)–(e)] Pulse applied before expansion. (f) Pulse applied during expansion.

Figure 2

Two examples of experiments with blast pulses in slowly rotating BECs.

Figure 3

(Color online) BEC evolved as in the in-trap experiment. On the left is the density ${\mid \Psi (r,0,t)\mid }^2$ and on the right is the radial velocity $\frac{\partial }{\partial r}\arg \Psi (r,0,t)$; both are imaged in the $z=0$ plane. The density scale is not constant throughout the sequence; in each frame, the density is scaled to its largest value so that the dispersing wave is visible. A high-density ring forms accompanied by oscillations on its inner side, a DSW. This DSW expands and propagates outward. The corresponding times are depicted in the middle of the figure. All figures depicting a time evolution of the condensate in this work follow the same form as this figure.

Figure 4

Comparison of the condensate density from the in-trap experiment (top) and numerical simulation (bottom) using the potential $V_{\mathit{it}}$, Eq. (4.7). The image from simulation is a contour plot of the function $\int {\mid \Psi (r,z,t)\mid }^{2}dz$, modeling the experimental imaging process where the photo was taken along the $z$ axis. Both the simulation and experimental pictures were taken at $t=55\mathrm{ms}$. The approximate diameter of the condensate cloud from simulation is $850\mu \mathrm{m}$ and from experiment is $775\mu \mathrm{m}$.

Figure 5

(Color online) BEC evolved as in the out-of-trap experiment. On the left is the density and on the right is the radial velocity, both imaged in the $z=0$ plane. A high-density ring forms with a plateau accompanied by oscillations on its inner and outer sides, two DSWs. These DSWs quickly interact and propagate outward.

Figure 6

Comparison of the condensate density from the out-of-trap experiment (top) and numerical simulation using the potential $V_{\mathit{ot}}$ (3.2) (bottom). The image from simulation is a contour plot of the function $\int {\mid \Psi (r,z,t)\mid }^{2}dz$, modeling the experimental imaging process. The approximate diameter of the condensate cloud from simulation is $116\mu \mathrm{m}$ and from experiment is $363\mu \mathrm{m}$.

Figure 7

(Color online) Dissipative Burgers’ equation shock solution (4.2) with $\nu \to 0$, converging to a traveling discontinuity or classical shock wave.

Figure 8

(Color online) Numerical solutions of Eq. (4.12) for initial data (4.13) and ${\epsilon }^2=0.001$ (dashed), ${\epsilon }^2=0.0001$ (solid). As $\epsilon$ decreases, the wavelength of the oscillations decreases.

Figure 9

(Color online) Elliptic function solutions to the KdV equation for several choices of the parameters $\left\{r_i\right\}$. The solution converges to a constant as $m\to 0$ and to the soliton sech profile as $m\to 1$.

Figure 10

(Color online) Initial data regularization for the KdV dispersive Riemann problem. The dashed line represents the initial data (4.13) for $u$. The solid lines represent the initial data for the Riemann invariants $r_1$, $r_2$, and $r_3$ that regularize the initial data for $u$. This initial data for the Riemann invariants satisfies the three properties of characterization, nondecreasing, and separability (4.19) so a rarefaction-wave solution exists for all time (see Fig. 11).

Figure 11

(Color online) Dissipative (dashed) and dispersive (solid) regularizations of the conservation law (4.4). The dissipative case corresponds to a traveling discontinuity with speed $\frac{1}{2}$ satisfying the jump condition (4.9). The dispersive case is a rarefaction-wave solution to the Whitham equations (4.17) with two associated speeds $v_2^{+}=2∕3$ and $v_2^{-}=-1$. This rarefaction wave modulates the periodic solution (4.14) giving a DSW [see Fig. 12 and Eq. (4.23)].

Figure 12

(Color online) Comparison of a dispersive shock [Eq. (4.23) with ${\epsilon }^2=0.0001$], its average $\overline{\varphi }$ [Eq. (4.15), dashed], and a classical, dissipative shock [Eq. (4.3), Burgers’ solution] plotted at time $t=1$. The DSW front moves faster than its classical (CSW) counterpart (normalized speeds: $v_{\mathit{DSW}}=\frac{2}{3}$ vs $v_{\mathit{CSW}}=\frac{1}{2}$). The average $\overline{\varphi }$ looks very similar to the classical shock, a steep front connected to a constant in the rear, except that the speed of the front is different and the function is continuous.

Figure 13

(Color online) Numerical solution of the KdV equation with an initial step (top) and the asymptotic DSW solution (4.23) (bottom) for ${\epsilon }^2=0.001$.

Figure 14

(Color online) Numerical solution of the KdV equation (4.12) with the step initial data (4.24) for different values of $\epsilon$ and the zero-dissipation (dispersion) limit $\epsilon =0$, the rarefaction wave (4.8). The plot corresponds to $t=1$.

Figure 15

(Color online) Regularized initial data (solid) for the dispersive Riemann problem (4.31). The variables $r_{+}$ and $r_{-}$ (dashed) are the Riemann invariants for the Euler equations satisfying Eqs. (4.30). The $\left\{r_i\right\}$ are the Riemann invariants for the Whitham equations (4.43) describing the modulation of the periodic wave (4.37). The $\left\{r_i\right\}$ satisfy the properties of being nondecreasing and separable (4.45), so a continuous rarefaction solution exists for all time (see Fig. 16).

Figure 16

(Color online) Solution to classical (dashed) and dispersive (solid) Riemann problems for the Euler equations with initial data shown in Fig. 15. The classical solution consists of three constant states connected by a rarefaction wave (with speeds $v_{-}$ and $v_{-}^{+}$) and a dissipative shock wave (with speed $v_{+}$). The dispersive regularization involves a pure rarefaction solution of the Whitham equations (4.43) where only $r_3$ varies according to Eqs. (4.48). The DSW speeds $v_3^{-}=\sqrt{{\rho }_0}$ and $v_3^{+}=\frac{8{\rho }_0-8\sqrt{{\rho }_0}+1}{2\sqrt{{\rho }_0}-1}$ are different from the classical shock speed $v_{+}$. This solution modulates the periodic solution (4.37) giving a DSW (see Fig. 17).

Figure 17

(Color online) DSW (GP, oscillatory) and classical shock (Euler, discontinuity). The dashed curve is the averaged DSW. The upper plot is the density $\rho$, the middle plot is the velocity $u$, and the lower plot is the momentum $\rho u$ for the parameters ${\rho }_0=2.5$, $t=1$, and ${\epsilon }^2=0.002$. A DSW in BEC is an expanding region with the constant trailing- and leading-edge speeds $v_3^{-}=\sqrt{{\rho }_0}$ and $v_3^{+}=\frac{8{\rho }_0-8\sqrt{{\rho }_0}+1}{2\sqrt{{\rho }_0}-1}$, respectively. When $1<{\rho }_0<4$, the DSW looks similar to the one pictured here. For ${\rho }_0⩾4$, the situation is different (see Fig. 21). The trailing dip is locally a gray soliton, verified by comparing the maximums and minimums of the density and velocity with the NLS gray-soliton solution (4.50).

Figure 18

(Color online) The density and velocity of the gray-soliton solution (4.50) with labeled maximum and minimum values for comparison with the trailing dip in the DSW Fig. 17.

Figure 19

(Color online) Numerical solution of the dispersive Riemann problem (left) and its asymptotic solution (right) at $t=1.5$ for ${\rho }_0=2.5$ and $\epsilon =0.03$. The numerically determined trailing-edge speed, averaged from $t=1.2$ to $t=1.5$, is 1.592, approximately equal to $\sqrt{{\rho }_0}=1.581$, the theoretical result (4.49). The trailing-edge minimum density is 0.183, approximately the theoretical value ${(2-\sqrt{{\rho }_0})}^2=0.175$.

Figure 20

(Color online) Numerical solution of the dispersive Riemann problem for rarefaction initial data. As $\epsilon \to 0$, the dispersive regularization converges strongly to the dissipative regularization, the rarefaction wave Eqs. (4.29). The parameters are ${\rho }_0=0.2$, $u_0=2\sqrt{{\rho }_0}-2$, and $t=0.5$.

Figure 21

(Color online) DSW shock profile (solid) and its average (dashed) plotted for $t=1$ when ${\rho }_0⩾4$ (here ${\rho }_0=10$) giving rise to a vacuum point at $x∕t={\xi }_v\approx 7.4$ [see Eqs. (4.55)] of zero density and infinite velocity. The average velocity $\overline{\nu }$ exhibits a jump at the vacuum point but the average momentum $\overline{\varphi \nu }$ does not. This indicates that the variables $\rho$ and $\rho u$ are the most natural variables for the problem.

Figure 22

(Color online) Average momentum $\overline{\psi \nu }$ [Eqs. (4.40)] (dashed) and the modulated DSW momentum $\psi \nu$ [Eqs. (4.37)] (solid) with the rarefaction solution (4.48) and the incorrect choice of constant sign $\sigma \equiv +1$ and the correct choice $\sigma =\mathrm{sgn}(\xi -{\xi }_v)$ (dash dotted) for ${\rho }_0=10$. The boundary condition (4.57) is not satisfied and there is a kink at the vacuum point for the incorrect $\sigma$.

Figure 23

(Color online) Development and propagation of DSW in the 3D in trap experiment with (dashed) and without (solid) the expansion potential. On the left is the BEC density ${\mid \Psi \mid }^2$ and on the right is the BEC radial velocity $\epsilon {\left(\arg \Psi \right)}_x$. The expansion potential does not alter the structure of the DSW, just the speeds. The 3D and 2D densities are in excellent agreement with a maximum relative error of less than 1%. The difference cannot be seen in this graphic.

Figure 24

(Color online) Comparison of the in-trap experiment in 3D with an expansion potential $\Psi (r,z=0,t)$ (solid) and the 1D modified in-trap experiment without the expansion potential ${\Psi }_{1D}(x,t)$ (dashed). The basic DSW structure appears in both the 3D and 1D simulations but the amplitudes and speeds are different. Therefore, the agreement is only qualitative.

Figure 25

(Color online) Comparison of 2D simulations for BEC (solid) and gas dynamics (dashed). The dissipative regularization used in the gas dynamics simulation was calculated using Clawpack [20]. The densities agree in smooth regions not affected by breaking (see the circled region).

Figure 26

(Color online) 3D shock development and interaction for the modified out-of-trap experiment. The left plot shows the evolution of the density ${\mid \Psi (r,0,t)\mid }^2$ and the right plot shows the evolution of the radial velocity ${\left[\arg \Psi (r,0,t)\right]}_r$, both in the $z=0$ plane. As with the in-trap experiment, the maximum absolute difference in density from the 2D evolution has a relative error of less than 1%. The two straight lines represent the position of the rightmost DSW front. The slope is inversely proportional to the speed. As predicted by DSW interaction theory [38], once the two DSWs interact, the rightmost DSW front’s speed increases.

Figure 27

(Color online) Comparison of 3D (solid) and 1D (dashed) shock development. The left plot shows the evolution of density; the right plot shows the evolution of the velocity. There is good qualitative agreement over this short time scale.

Figure 28

(Color online) Shock evolution in 2D BEC (solid) and the dissipative regularization of the 2D Euler equations (dashed). The left plot is density and the right plot is radial velocity. The dissipative regularization was calculated using Clawpack [20].