Dispersive-shock and exponential blow-up-time conjecture for small-dispersion gKdV

Let $u_0\in L_2(\mathbb R)$ be smooth initial data with a single maximum, let $\epsilon\ll1$ be the dispersion parameter, and let $t_c$ be the break-up time of the corresponding generalized Hopf solution. Small-dispersion gKdV conjecture. For $n\geq4$, the solution of the gKdV equation has a dispersive shock for times between $t\sim t_c$ and the blow-up time $t^*$. Moreover,

$$t^*(\epsilon)=t_0^*\exp(\alpha\epsilon),$$

where $t_0^*>t_c$ and $\alpha$ are constant with respect to $\epsilon$. The claim summarizes the paper’s numerical findings for the dispersive shock and the dependence of blow-up time on small dispersion; it is presented as a conjectural analytic description rather than a theorem.