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 -SE (-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 & ouml;lder regularity, attained in the cusp.