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arXiv · 2511.00672

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

Jun Eshima·Luc Deike·Howard A. Stone
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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.

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Jun Eshima, Luc Deike, Howard A. Stone. 2025-11-01. Similarity Solutions of Shock Formation for First-order Strictly Hyperbolic Systems. https://arxiv.org/abs/2511.00672

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