Similarity Solutions of Shock Formation for First-order Strictly Hyperbolic Systems

TL;DR

The paper addresses how shocks form in general one-dimensional first-order strictly hyperbolic systems by extending the Burgers' equation self-similarity framework to the full system. It a priori identifies a universal, Burgers-type self-similar structure that governs the local behavior near a finite-time shock, with the solution taking the form $\boldsymbol{f}(x,t) = \boldsymbol{f}_* + (t_*-t)^{1/2} F(\xi) \boldsymbol{e}$ along an eigen-direction, where $\xi = (x - x_* - \lambda (t_*-t))/(c (t_*-t)^{3/2})$ and $F$ satisfies $-\xi = F + K F^3$. The constants $K>0$ and $c$ are the only arbitrary parameters (up to scaling) and can be computed from the Jacobian $\boldsymbol{M}$ and its derivatives; higher-order corrections are $O(t_*-t)$. The approach is verified numerically on the shallow water equations, demonstrating correct blow-up rates for first and second derivatives and a consistent self-similar collapse, thereby furnishing a universal local description of shock formation applicable to a broad class of hyperbolic PDEs. This offers a practical, analytically tractable framework for diagnosing and understanding singularity formation in nonlinear wave systems.

Abstract

Shocks due to hyperbolic partial differential equations (PDEs) appear throughout mathematics and science. The canonical example is shock formation in the inviscid Burgers' equation $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$. Previous studies have shown that when shocks form for the inviscid Burgers' equation, for positions and times close to the shock singularity, the dynamics are locally self-similar and universal, i.e., dynamics are equivalent regardless of the initial conditions. In this paper, we show that, in fact, shock formation is self-similar and universal for general first-order strictly hyperbolic PDEs in one spatial dimension, and the self-similarity is like that of the inviscid Burgers' equation. An analytical formula is derived and verified for the self-similar universal solution.

Figures (3)

• Figure 1: Example evolution of the shallow water equations (\ref{['eq:swe_nondim']}), taking periodic boundary conditions for the domain $x \in [0,1]$ with initial condition $(u(x,0),\eta(x,0))=(\sin(2\pi x), 1)$. (a) Velocity $u(x,t)$, where the evolution from gray to black shows evolution in time of ($t = 0,0.05, 0.1, 0.15, 0.195$). The shock singularity occurs at $t_* \approx 0.196$ (thus, the final timestep shown is just before the shock forms). Inset shows magnified views of the shock region at $t = 0.195$. (b) Analogous to (a) for the water height $h(x,t)$.
• Figure 2: Verification of the similarity solution exponents and prefactors. The shallow water equations (\ref{['eq:swe_nondim']}) are solved numerically, taking periodic boundary conditions for the domain $x \in [0,1]$ with initial condition $(u(x,0),\eta(x,0))=(\sin(2\pi x), 1)$. The numerical solution of the shallow water equations (SWE) (solid curves) is compared against the similarity solution (S) (dashed lines) predicted by the similarity solution (\ref{['eq:first_deriv_example']}, \ref{['eq:second_deriv_example']}). (a,b) Log-log plot of $\max |\partial u/\partial x|$, $\max |\partial \eta/\partial x|$ (green, purple) as the shock forms $t \rightarrow t_*^-$, showing a $(t_*-t)^{-1}$ blowup. There are no fitting parameters for the prediction (\ref{['eq:first_deriv_example']}). (b) Log-log plot of $\max |\partial^2 u/\partial x^2|$, $\max |\partial^2 \eta/\partial x^2|$ (green, purple) as the shock forms $t \rightarrow t_*^-$, showing a $(t_*-t)^{-5/2}$ blowup, for the best fit $K \approx 0.14$, the only unknown in the prediction (\ref{['eq:second_deriv_example']}). Insets show a magnified view for $t_*-t \in [10^{-3}, 1.2 \times 10^{-3}]$.
• Figure 3: Verification of the similarity solution profiles. The shallow water equations (\ref{['eq:swe_nondim']}) are solved numerically, taking periodic boundary conditions for the domain $x \in [0,1]$ with initial condition $(u(x,0),\eta(x,0))=(\sin(2\pi x), 1)$. The numerical solutions (solid curves) are colored gray to black as $t\rightarrow t_*^-$. The reference frame $\overline{x} = x-x_*-\lambda (t_*-t)$ of the shock is considered. The orange dashed curves show the similarity solution prediction $-X = Y+KY^3$ where $X$ is the horizontal axis and $Y$ is the vertical axis for $K \approx 0.14$ as predicted in \ref{['fig:verif']}(b). (a) $u$ against $\overline{x}$. (b) $u$ against $\overline{x}$, suitably rescaled according to the similarity solution (\ref{['eq:simil_soln_example']}). (c,d) Analogous to (a,b) for the water height $\eta$.