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

TL;DR

This paper demonstrates that shock formation in general first-order strictly hyperbolic PDEs in one dimension exhibits self-similar and universal behavior, extending known results from the Burgers' equation.

Contribution

It establishes the universality of shock formation in a broad class of hyperbolic PDEs and derives an analytical formula for the self-similar solution.

Findings

Shock formation is self-similar for general hyperbolic PDEs.

The universal self-similar solution is analytically derived.

Shock dynamics near singularity are independent of initial conditions.

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 for the self-similar universal solution.