Abstract
AbstractWave 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.
Highlights
In general, sloped boundaries add a layer of difficulty to the study of the hydrodynamics of water waves, and as such have been a classical subject of the literature, stemming from the seminal papers by Carrier and Greenspan [7, 12], and Gurtin [13], among others
Our work follows in these footsteps, and focusses on the dynamics of singular points in the initial value problem for hydrodynamic systems, including that of a contact line viewed as a “vacuum” point, according to the terminology of gas-dynamics often adopted for hyperbolic systems
We have discussed several aspects of non-smooth wave front propagation in the presence of bottom topography, including the extreme case of vacuum/dry contact points, within the models afforded by long wave asymptotics and their hyperbolic mathematical structure
Summary
In general, sloped boundaries add a layer of difficulty to the study of the hydrodynamics of water waves, and as such have been a classical subject of the literature, stemming from the seminal papers by Carrier and Greenspan [7, 12], and Gurtin [13], among others. Sloshing solutions for water layers in parabolic bottoms are shown to exist at all times depending on the value of the initial free surface curvature with respect to that of the bottom, while global, full interval gradient catastrophes can occur when the relative values are in the opposite relation The analysis of these solutions builds upon our previous work in this area [5, 4, 3], extending it to non-flat horizontal bottoms, and highlights the effects of curvature in the interactions with topography. While this solution class is special, it can govern the evolution of more general, analytic initial data setups, at least up to a gradient catastrophe time [4]. Online supplementary material complements the exposition by showing animations of representatives of the exact sloshing solutions we derived
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