Defocusing modified fractional Korteweg–de Vries dispersion and shock conjecture

Let $u_0\in L^2(\mathbb{R})$ be smooth initial data with a single hump for the defocusing modified fractional Korteweg–de Vries equation, with dispersion parameter $\alpha$. Defocusing modified fractional Korteweg–de Vries conjecture. Initial data with sufficiently small mass disperse and produce global solutions. For $\alpha\geq1/2$, initial data of arbitrary mass produce dispersed global solutions. For $\alpha<1/2$, sufficiently large mass can lead to shock formation in finite time, with a singularity of the form $|x-x^*|^{1/3}$ near $|x|\sim x^*$. The statement summarizes numerical evidence for dispersion in the subcritical and critical regimes and shock formation in the supercritical regime; a general rigorous classification remains open.