PreprintarXiv licencearXiv
Regularity of Structurally Stable Cusp Singularities for Two Families of Quasilinear Wave-type Equations
Abstract
In this paper, we study two families of quasilinear equations: Hunter-Saxton type and Camassa-Hom type equations, with a paramerter λ∈(0,1) whose solutions form cusp singularities. When λ=1, the first system becomes the scalar conservation law. The main result of this paper is to give regularity of two types of structurally stable singularities: Type I on the singular curve, Type II at the point where cusp singularity forms, for some λ∈(0,1). When λ→ 1, our result indicts the C1/3 regularity at the point where singularity forms, which agrees with the regularity of the generic pre-shock solution.
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BibTeX
@misc{armstrong2026regularity,
title = {{Regularity of Structurally Stable Cusp Singularities for Two Families of Quasilinear Wave-type Equations}},
author = {Samuel J. Armstrong and Geng Chen and Tao Huang and Yannan Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12926},
eprint = {2609.12926},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12926},
}
RIS
TY - UNPB
TI - Regularity of Structurally Stable Cusp Singularities for Two Families of Quasilinear Wave-type Equations
AU - Samuel J. Armstrong
AU - Geng Chen
AU - Tao Huang
AU - Yannan Shen
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12926
UR - https://arxiv.org/abs/2609.12926
AB - In this paper, we study two families of quasilinear equations: Hunter-Saxton type and Camassa-Hom type equations, with a paramerter λ∈(0,1) whose solutions form cusp singularities. When λ=1, the first system becomes the scalar conservation law. The main result of this paper is to give regularity of two types of structurally stable singularities: Type I on the singular curve, Type II at the point where cusp singularity forms, for some λ∈(0,1). When λ→ 1, our result indicts the C^1/3 regularity at the point where singularity forms, which agrees with the regularity of the generic pre-shock solution.
ER -