Focusing modified fractional Korteweg–de Vries dynamics conjecture

Let $u_0\in L^2(\mathbb{R})$ be smooth initial data with a single hump for the focusing modified fractional Korteweg–de Vries equation, with dispersion parameter $\alpha$. Let $Q$ denote the solitary-wave solution at speed $c=1$. Focusing modified fractional Korteweg–de Vries dynamics conjecture. For $\alpha>1$, solutions remain smooth for all $t$ and, as $t\to\infty$, decompose asymptotically into solitary waves and radiation. For $0<\alpha\leq1$, sufficiently small but nonzero mass gives global smooth solutions. For $\alpha=1$, initial data with negative energy and mass larger than the solitary-wave mass blow up at finite time $t^*$, with

$$u(x,t)\sim \frac{1}{\sqrt{L(t)}}Q\left(\frac{x-x_m}{L(t)}\right),\qquad L(t)=c_0(t^*-t).$$

For $0<\alpha<1$, initial data with sufficiently large $L^2$ norm blow up at finite time $t^*$ and finite position $x=x^*$; the blow-up is self-similar, with a profile having non-vanishing $a^\infty$. These claims summarize numerical observations and conjectural dynamics across the focusing regimes; rigorous global behavior and blow-up classification are not established in the source.