Critical-mass blow-up and radiation dichotomy for quartic gKdV

Let $Q$ be the soliton of the quartic generalized Korteweg–de Vries equation, and let $u(x,t)$ solve that equation for $n=4$ with smooth initial data $u_0\in L_2(\mathbb{R})$ of sufficiently large mass and with a single maximum. Write $E[u_0]$ for the energy and $\lVert u_0\rVert_2$ for the $L_2$ norm. Critical quartic gKdV conjecture. If $E[u_0]>E[Q]$ and $\lVert u_0\rVert_2<\lVert Q\rVert_2$, then the solution is radiated away to infinity; if $E[u_0]<E[Q]$ and $\lVert u_0\rVert_2>\lVert Q\rVert_2$, then the solution blows up at a finite time $t^*$, even for initial data outside the class $\mathcal A$. This conjecture concerns the two proposed dynamical regimes near the quartic gKdV soliton. The paper reports numerical compatibility with it, while the cited blow-up theory gives an asymptotic description in a more restricted setting.