# Waves of maximal height for a class of nonlocal equations with homogeneous symbols

Bruell, Gabriele; Dhara, Raj Narayan

##### Abstract:
We discuss the existence and regularity of periodic traveling-wave 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 traveling-wave 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 2π-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht/Preprint Publikationsjahr 2018 Sprache Englisch Identifikator ISSN: 2365-662X urn:nbn:de:swb:90-869143 KITopen-ID: 1000086914 Verlag Karlsruher Institut für Technologie (KIT) Umfang 25 S. Serie CRC 1173 ; 2018/26 Schlagwörter highest wave, singular solution, fractional KdV equation, nonlocal equation with homogeneous symbol
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