ABSTRACT
Continuous, piecewise differentiable exact solutions with discontinuous derivatives of the shallow water (Airy’s) hyperbolic system are constructed by splicing together self-similar parabolae with the evolution from a constant background state. These new solutions are used to illustrate the mechanism by which “vacuum states,” mathematically corresponding to the hyperbolic-parabolic transition points of the governing equations, can be filled by the evolution of the hyperbolic system, and in particular how dry spots persist until a non-generic gradient catastrophe develops at the dry point(s). The continuation of solutions asymptotically for short times beyond the catastrophe is then investigated analytically, in its weak form, with an approach inspired by the stretched coordinates used in singular perturbation theory.
