Mathematics Colloquia and Seminars

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Examples of hyperbolic surface waves are Rayleigh waves on an elastic half-space, and a variety of waves that propagate along shock waves and contact discontinuities. Nonlinear Rayleigh waves have been of some recent technological interest because of the use of ultrasonic surface acoustic wave devices in electronics, and the behavior of surface waves on shocks and contacts is important for the nonlinear stability or instability of shocks and contacts. Such surface waves have an interesting nonlocal, nonlinear self-interaction. We will describe the general Hamiltonian structure of the nonlocal equations that describe them, and show that surface waves on a tangential discontinuity in incompressible magnetohydrodynamics satisfy the simplest equation with this structure. We will also mention some open questions related to the formation of singularities in nonlinear hyperbolic surface waves, and the global existence of weak solutions.