Fractional Benjamin–Bona–Mahony global behavior and cusp conjecture

Let $u_0\in L^2(\mathbb{R})$ be smooth initial data with a single hump for the fractional Benjamin–Bona–Mahony equation, with parameter $0<\alpha$. Fractional Benjamin–Bona–Mahony conjecture. Initial data of sufficiently small mass disperse and give global solutions for $0<\alpha<1$. For $\alpha\geq1/3$, solutions are global in time. For $\alpha>1/2$, the long-time behavior of sufficiently large-mass data is characterized by solitary waves and radiation. For $\alpha<1/3$, sufficiently large mass can lead to cusp formation in finite time. These assertions summarize numerical observations about dispersive shocks and cusp formation; the global dynamics and finite-time singularity mechanism remain open.