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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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