Shock Waves in Dispersive Eulerian Fluids

Abstract

The long-time behavior of an initial step resulting in a dispersive shock wave (DSW) for the one-dimensional isentropic Euler equations regularized by generic, third-order dispersion is considered by use of Whitham averaging. Under modest assumptions, the jump conditions (DSW locus and speeds) for admissible, weak DSWs are characterized and found to depend only upon the sign of dispersion (convexity or concavity) and a general pressure law. Two mechanisms leading to the breakdown of this simple wave DSW theory for sufficiently large jumps are identified: a change in the sign of dispersion, leading to gradient catastrophe in the modulation equations, and the loss of genuine nonlinearity in the modulation equations. Large amplitude DSWs are constructed for several particular dispersive fluids with differing pressure laws modeled by the generalized nonlinear Schrödinger equation. These include superfluids (Bose–Einstein condensates and ultracold fermions) and “optical fluids.” Estimates of breaking times for smooth initial data and the long-time behavior of the shock tube problem are presented. Detailed numerical simulations compare favorably with the asymptotic results in the weak to moderate amplitude regimes. Deviations in the large amplitude regime are identified with breakdown of the simple wave DSW theory.

Appendix: Numerical Methods

The numerical solution of the gNLS equation (3.1) for the shock tube problem, the initial step in density

$$\begin{aligned} \psi (x,0) = \left\{ \begin{array}{ll} 1 &{}\quad x < x_0 \\ \sqrt{\rho _2} &{}\quad x > x_0 \end{array} \right. , \qquad \rho _2 > 1, \end{aligned}$$

(12.1)

is briefly described here. A pseudospectral, time-splitting method is implemented for the accurate solution of long-time evolution for $x\in (0,L)$. The initial data (11.1) are smoothed by use of the hyperbolic tangent initial condition

$$\begin{aligned} \psi (x,0) = \left\{ \frac{1}{2} [1 + \tanh (x_0-x)](1 - \rho _2) + \rho _2 \right\} ^{1/2} , \end{aligned}$$

where $x_0 = L/2$. Time stepping proceeds by use of second-order Strang splitting Strang (1968) where the linear PDE

$$\begin{aligned} \mathrm{i} \frac{\partial \psi _\mathrm {L}}{\partial t} = -\frac{1}{2} \frac{\partial ^2 \psi _\mathrm {L}}{\partial x^2}, \qquad \psi _\mathrm {L}(x,t) = \psi (x,t), \end{aligned}$$

(12.2)

is advanced half a time step $\Delta t/2$ exactly followed by a full time step of the nonlinear ODE

$$\begin{aligned} \mathrm{i} \frac{\partial \psi _{\mathrm {NL}}}{\partial t} = f(|\psi _{\mathrm {NL}}|^2) \psi _{\mathrm {NL}}, \qquad \psi _{\mathrm {NL}}(x,t) = \psi _\mathrm {L}(x,t+\Delta t/2) . \end{aligned}$$

(12.3)

The linear PDE is then advanced half a time step with the initial data $\psi _\mathrm {L}(x,t+\Delta t/2) = \psi _{\mathrm {NL}}(x,t+\Delta t)$, giving the second-order accurate approximation of $\psi (x,t+\Delta t) \approx \psi _\mathrm {L}(x, \Delta t)$. Equation (11.2) is projected onto a truncated cosine basis of $N$ terms that maintains Neumann ($\psi _x = 0$) boundary conditions, computed efficiently via the fast Fourier transform (FFT), and integrated explicitly in time. The nonlinear ODE (11.3) conserves $|\psi _{\mathrm {NL}}|^2$ so is also integrated explicitly in time. The parameter $\Delta x = L/N$ is the spatial grid spacing of the grid points $x_j = \Delta x (j - 1/2)$, $j = 1,2,\ldots ,N$. The accuracy of the solution is monitored by computing the relative deviation in the conserved $L^2$ norm $E(t) = \int _{\mathbb {R}} | \psi (x,t) |^2 \mathrm{d} x$, $E_{\mathrm {rel}} = |E(t_\mathrm {f})-E(0)|/E(0)$ where $t_\mathrm {f}$ is the final time. All computations presented here exhibit $E_{\mathrm {rel}} < 10^{-8}$. Also, the accurate spatial resolution of the oscillatory structures is supported by the fact that the coefficient of the largest wavenumber in the cosine series is less than $5 \cdot 10^{-10}$ (oftentimes much less). The numerical parameters $L$, $N$, $\Delta t$, and $t_\mathrm {f}$ vary depending upon the nonlinearity strength and jump height; For example, for power-law gNLS with $p = 2$ and $\rho _2 \ge 11$, $N = 2^{16}$, $L = 1,200$, $\Delta t = 0.0002$, and $t_\mathrm {f} = 30$, whereas for $p = 2/3$ with $\rho _2 = 2$, $N = 3 \cdot 2^{14}$, $L = 3,000$, $\Delta t = 0.002$, and $t_\mathrm {f} = 500$.

The extraction of the DSW speeds $v_+$, $s_+$, and minimum density $\rho _\mathrm {min}$ is performed as follows: The precise location of the DSW soliton trailing edge is computed by creating a local cubic spline interpolant through the computed grid points in the neighborhood of the dark soliton minimum. A root finder is applied to the derivative of this interpolant in order to extract the off-grid location of the soliton edge $x_s(t)$ and $\rho _\mathrm {min}\equiv |\psi (x_s(t_\mathrm {f}),t_\mathrm {f})|^2$. The slope of a linear least-squares fit through $x_s(t_j)$ for $j = 1, \ldots , 100$ equispaced $t_j \in [t_\mathrm {f}-1,t_\mathrm {f}]$ determines $s_+$. For the leading, linear wave edge, an envelope function is determined by least-squares fitting two lines, each through about 30 local maxima and minima, respectively, of the DSW density in the vicinity of the trailing edge. The extrema are computed the same as for the soliton minimum. The point of intersection of these two lines is the location of the linear wave edge $x_v(t)$. The same fitting procedure as was used to determine $s_+$ from $x_s(t)$ is used to extract $v_+$ from $x_v(t)$.