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

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 singularities for nonzero time intervals, whereas the latter is shown to develop nonstandard velocity shocks instantaneously. Polynomial bottom topographies simplify the hierarchy, as they contribute only a finite number of inhomogeneous forcing terms to the equations in the recursion relations. However, we show that truncation to finite-dimensional systems and polynomial solutions is in general only possible for the case of a quadratic bottom profile. In this case, the system's evolution can reduce to, and is completely described by, a low-dimensional dynamical system for the time-dependent coefficients. This system encapsulates all the nonlinear properties of the solution for general power series initial data, and, in particular, governs the loss of regularity in finite times at the dry point. For the special case of parabolic bottom topographies, an exact, self-similar solution class is introduced and studied to illustrate via closed-form expressions the general results.