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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Gabriele Bruell (Norwegian Institute of Science and Technology)

On a nonlocal shallow water wave equation

The Korteweg-de Vries equation is a famous and well studied model equation in shallow water. The balance between nonlinear and linear (dispersive) terms allows for solitary traveling wave solutions. While the Korteweg-de Vries equation is a valid approximation for long waves in shallow water, it fails to sustain this property for relatively short waves. In addition it can not explain phenomena like wave breaking nor does it admit traveling wave solutions which form cusps. In 1967 G. Whitham introduced a nonlocal alternative to the Korteweg-de Vries equation by keeping the nonlinear part, but replacing the linear term by a nonlocal operator involving the exact dispersion relation of the linearized Euler equations. It turns out that the resulting equation is a better approximation for relatively short waves in shallow water. Moreover, it explains the phenomena of solitary waves as well as wave breaking and traveling periodic waves with cusps.

In this talk we introduce the so-called Whitham equation and focus on its solitary wave solutions. In particular, we show that any solitary wave solution is symmetric with exactly one crest from which the surface decreases exponentially. Moreover, the structure of the Whitham equation allows to conclude that conversely any unique symmetric solution constitutes a traveling wave. In fact, the latter result holds true for a large class of partial differential equations sharing a certain structure.