Mathematics Colloquia and Seminars

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Surface waves are waves that propagate along a boundary or interface, with energy that is localized near the surface. Physical examples include water waves on the free surface of a fluid, Rayleigh waves on an elastic half-space, waves on a vorticity discontinuity in the flow of an ideal fluid, and surface plasmons on a metal-dielectric interface. We will describe some of the history of these, and related surface waves, discuss how they are affected by nonlinearity, and describe a general Hamiltonian framework for their analysis. The weakly nonlinear evolution of dispersive surface waves, such as water waves, is described by well-known PDEs, such as the KdV equation or the nonlinear Schr\"odinger equation. The nonlinear evolution of nondispersive surface waves, such as Rayleigh waves or waves on a vorticity discontinuity, is described by novel nonlocal, quasi-linear, singular integrodifferential equations, and we will discuss some of their properties.