Evolution of interface singularities in shallow water equations with variable bottom topography

Abstract

Wave front propagation with nontrivial bottom topography is studied within the formalism of hyperbolic long wave models. Evolution of nonsmooth initial data is examined, and, in particular, the splitting of singular points and their short time behavior is described. In the opposite limit of longer times, the local analysis of wave fronts is used to estimate the gradient catastrophe formation and how this is influenced by the topography. The limiting cases when the free surface intersects the bottom boundary, belonging to the so-called "physical" and "nonphysical" vacuum classes, are examined.Solutions expressed by power series in the spatial variable lead to a hierarchy of ordinary differential equations for the time-dependent series coefficients, which are shown to reveal basic differences between the two vacuum cases: for nonphysical vacuums, the equations of the hierarchy are recursive and linear past the first two pairs, whereas for physical vacuums, the hierarchy is nonrecursive, fully coupled, and nonlinear. The former case may admit solutions that are free of singularitiesThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.