Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations

We study traveling waves for a class of fractional Korteweg–De Vries and fractional Degasperis–Procesi equations with a parametrized Fourier multiplier operator of order $s \in (-1, 0)$. For both equations there exist local analytic bifurcation branches emanating from a curve of constant solutions, consisting of smooth, even and periodic traveling waves. The local branches extend to global solution curves. In the limit we find a highest, cusped traveling-wave solution and prove its optimal $s$-Hölder regularity, attained in the cusp.