  The behavior of a class of solutions of the shallow water Airy system
originating from initial data with discontinuous derivatives is considered.
Initial data are obtained by splicing together self-similar parabolae with a
constant background state. These solutions are shown to develop velocity and
surface gradient catastrophes in finite time and the inherent persistence of
dry spots is shown to be terminated by the collapse of the parabolic core. All
details of the evolution can be obtained in closed form until the collapse
time, thanks to formation of simple waves that sandwich the evolving
self-similar core. The continuation of solutions asymptotically for short times
beyond the collapse is then investigated analytically, in its weak form, with
an approach using stretched coordinates inspired by singular perturbation
theory. This approach allows to follow the evolution after collapse by
implementing a spectrally accurate numerical code, which is developed alongside
a classical shock-capturing scheme for accuracy comparison. The codes are
validated on special classes of initial data, in increasing order of
complexity, to illustrate the evolution of the dry spot initial conditions on
longer time scales past collapse.
