Numerical cusp-formation conjecture for Whitham and fractional KdV equations

Let $u_0\in L_2(\mathbb{R})$ be smooth initial data with a single negative hump. For $-1<\alpha<0$, solutions to the Whitham equation and to the fKdV equations with sufficiently small mass remain smooth for all $t$ and are radiated away. For $\alpha=-1/2$, solutions with negative initial data and sufficiently large mass develop, at $t^*>t_c$, a cusp of the form $|x-x^*|^{1/3}$, while the supremum norm remains bounded at the blow-up point. Solutions with positive initial data and sufficiently large norm mass develop, at $t^*<t_c$, a cusp of the form $|x-x^*|^{1/2}$. Numerical Whitham–fKdV cusp conjecture. The stated global-radiation and cusp-formation behavior holds for the indicated equations, parameters, and initial data. The claims are based on numerical observations of singularity formation and remain open in the supplied text.