On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation
Abstract
We consider the Whitham equation $u_t + 2u u_x+Lu_x = 0$, where L is the nonlocal Fourier multiplier operator given by the symbol $m(\xi) = \sqrt{\tanh \xi /\xi}$. 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 several qualitative properties of it, including its optimal $C^{1/2}$-regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol $m(\xi)$, and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol $m(\xi)$ is completely monotone, and provide an explicit representation formula for it.
- Publication:
-
Annales de L'Institut Henri Poincare Section (C) Non Linear Analysis
- Pub Date:
- September 2019
- DOI:
- 10.1016/j.anihpc.2019.02.006
- arXiv:
- arXiv:1602.05384
- Bibcode:
- 2019AIHPC..36.1603E
- Keywords:
-
- Mathematics - Analysis of PDEs;
- 35Q35;
- 42A38;
- 45G05;
- 76B03
- E-Print:
- 40 pages, 3 figures. This version is identical to the one accepted for publication in Annales de l'Institut Henri Poincare, Analyse non lineaire