Highest Cusped Waves for the Fractional KdV Equations

In this paper we prove the existence of highest, cusped, traveling wave solutions for the fractional KdV equations $f_t+f{f}_x=\mathrm{\mid }D{\mathrm{\mid }}^{\alpha }{f}_x$ for all $\alpha \in (-1,0)$ and give their exact leading asymptotic behavior at zero. The proof combines careful asymptotic analysis and a computer-assisted approach.