Breaking in water wave models

The surface of an ocean wave, after some time, may become vertical and accelerate infinitely rapidly; thereafter a portion of the surface overturns, projects forward and forms a jet of water. Think of the stunning Hokusai wave. The complexity of the governing equations of the water wave problem, however, prevents a detailed account of "breaking." Whitham in the 1970s conjectured that a model combining the water wave dispersion and a nonlinearity of the shallow water equations would capture the phenomenon. I will present its proof and use Whitham's model to illustrate the Benjamin-Feir instability of Stokes' periodic waves in water. I will discuss breaking, instabilities and ill-posedness for related, nonlinear dispersive equations.

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