Waves of maximal height for a class of nonlocal equations with homogeneous symbols
Abstract
We discuss the existence and regularity of periodic travelingwave solutions of a class of nonlocal equations with homogeneous symbol of order $r$, where $r>1$. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic travelingwave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol $m(k)=k^{2}$. Thereby we recover its unique highest $2\pi$periodic, peaked travelingwave solution, having the property of being exactly Lipschitz at the crest.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 DOI:
 10.48550/arXiv.1810.00248
 arXiv:
 arXiv:1810.00248
 Bibcode:
 2018arXiv181000248B
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 25 pages