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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.00672 |
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| _version_ | 1866908953135284224 |
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| author | Eshima, Jun Deike, Luc Stone, Howard A. |
| author_facet | Eshima, Jun Deike, Luc Stone, Howard A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00672 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Similarity Solutions of Shock Formation for First-order Strictly Hyperbolic Systems Eshima, Jun Deike, Luc Stone, Howard A. Analysis of PDEs Mathematical Physics 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. |
| title | Similarity Solutions of Shock Formation for First-order Strictly Hyperbolic Systems |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2511.00672 |