Singularity formation as a wetting mechanism in a dispersionless water wave model
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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.
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