Regularity of Structurally Stable Cusp Singularities for Two Families of Quasilinear Wave-type Equations

In this paper, we study two families of quasilinear equations: Hunter-Saxton type and Camassa-Hom type equations, with a paramerter $\lambda \in (0,1)$ whose solutions form cusp singularities. When $\lambda =1$, 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 $\lambda \in (0,1)$. When $\lambda \to 1$, our result indicts the $C^{1\mathrm{/}3}$ regularity at the point where singularity forms, which agrees with the regularity of the generic pre-shock solution.