  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 $s$-H\"older regularity, attained
in the cusp.
