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Wave front propagation with non-trivial bottom topography is studied withinthe formalism of hyperbolic long wave models. Evolution of non-smooth initialdata is examined, and in particular the splitting of singular points and theirshort time behaviour is described. In the opposite limit of longer times, thelocal analysis of wavefronts is used to estimate the gradient catastropheformation and how this is influenced by the topography. The limiting cases whenthe free surface intersects the bottom boundary, belonging to the so-called"physical" and "non-physical" vacuum classes, are examined. Solutions expressedby power series in the spatial variable lead to a hierarchy of ordinarydifferential equations for the time-dependent series coefficients, which areshown to reveal basic differences between the two vacuum cases: fornon-physical vacuums, the equations of the hierarchy are recursive and linearpast the first two pairs, while for physical vacuums the hierarchy isnon-recursive, fully coupled and nonlinear. The former case may admit solutionsthat are free of singularities for nonzero time intervals, while the latter isshown to develop non-standard velocity shocks instantaneously. We show thattruncation to finite dimensional systems and polynomial solutions is in generalonly possible for the case of a quadratic bottom profile. In this case thesystem's evolution can reduce to, and is completely described by, a lowdimensional dynamical system for the time-dependent coefficients. This systemencapsulates all the nonlinear properties of the solution for general powerseries initial data, and in particular governs the loss of regularity in finitetimes at the dry point. For the special case of parabolic bottom topographies,an exact, self-similar solution class is introduced and studied to illustratevia closed form expressions the general results.

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