Department of Mathematics

The best known examples of hyperbolic surface waves are the Rayleigh waves that occur in elasticity. These waves are localized near the boundary of a half-space, and are generated by earthquakes and used in signal processing devices. Similar surface waves can also propagate along some shocks or interfaces, such as tangential discontinuities in magnetohydrodynamics and propagating phase boundaries in the compressible Euler equations. The existing general theory of initial boundary value problems for hyperbolic partial differential equations in a half-space (due to Kreiss and Sakamoto) or with a shock (due to Majda) requires that the boundary conditions or jump conditions satisfy a uniform Lopatinski condition, and this uniform condition fails when surface waves are present. It is therefore interesting to study the nonlinear development of such surface waves, both because of their physical interest and in order to understand the well-posedness of hyperbolic initial boundary-value problems that are not uniformly stable. We will describe an unusual nonlocal evolution equations that provides an asymptotic description of weakly nonlinear hyperbolic surface waves. Numerical solutions of this equation show the formation of singularities in finite time, together with some surprising effects of nonlocality.