Numerical global-regularity and blow-up conjecture for fractional KdV equations

Let $u_0 in L_2(\mathbb{R})$ be smooth initial data with a single hump. The fKdV equation is conjectured to have the following behavior. For $\alpha>0.5$, solutions remain smooth for all $t$ and, for large $t$, decompose asymptotically into solitons and radiation. For $0<\alpha\leq 0.5$, solutions with sufficiently small but non-zero mass remain smooth for all $t$. For $\alpha=0.5$, solutions with negative energy and mass larger than the soliton mass blow up at finite time $t^*$ and infinite $x^*$, with

$$u(x,t)\sim \frac{1}{\sqrt{L(t)}}Q_1\left(\frac{x-x_m}{L(t)}\right),\qquad L=c_0(t^*-t),$$

where $c_0$ is a constant and $Q_1$ is the solitary-wave solution for $c=1$, and

$$\lVert u_x\rVert_2\sim \frac{1}{L^2(t)}.$$

For $1/3<\alpha<0.5$, solutions with sufficiently large $L_2$ norm blow up at finite time $t^*$ and finite $x=x^*$; a soliton-type hump separates from the initial hump and eventually blows up, with

$$u(x,t)\sim \frac{1}{L^\alpha(t)}U\left(\frac{x-x_m}{L(t)}\right),\qquad L=c_1(t^*-t)^{\frac{1}{1+\alpha}},$$

where $c_1$ is a constant and $U$ is a solution vanishing for $|y|\to\infty$, if such a solution exists, and

$$\lVert u_x\rVert_2\sim \frac{1}{L^{2\alpha+1}(t)}.$$

For $0<\alpha<1/3$, solutions with sufficiently large $L_2$ norm blow up at finite time $t^*$ and finite $x=x^*$; the maximum of the initial hump evolves directly into a blow-up, whose profile appears to be given by the stated profile equation. Numerical fKdV blow-up conjecture. The preceding global-regularity, asymptotic-decomposition, blow-up, and profile assertions hold for the indicated ranges of $\alpha$ and initial data. These claims summarize numerical findings about fractional dispersive perturbations of Burgers-type dynamics; they are not established as general theorems in the supplied text.