# Mats Ehrnstrom: On Whitham’s conjecture of a highest cusped wave for a nonlocal shallow water wave equation

Submitted by Andreas L. Freund on October 15, 2015 - 11:00am

Speaker: Mats Ehrnstrom, Norwegian Technical University

Date: October 15, 2015

Title: On Whitham’s conjecture of a highest cusped wave for a nonlocal shallow water wave equation

Abstract: We consider the Whitham equation $$u_t + 2u u_x+Lu_x = 0$$, where L is the non-local (Fourier multiplier) operator given by the symbol $$m(\xi) = ({\frac{\tanh(\xi)}{\xi}})^{1/2}$$. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of $$P$$-periodic solutions, and give qualitative properties of it. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol $$m$$. This is joint work with Erik Wahlén.

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