Small-energy stability and self-similar blow-up conjecture for the Novikov–Veselov equation

Let $E$ satisfy $|E|\ll 10$. A small-energy blow-up conjecture asserts that the KdV soliton for small $a$ is stable under NV dynamics, while the KdV soliton for large $a$ is unstable against an $L^{\infty}$ blow-up in finite time. NV solutions corresponding to localized initial data of sufficiently small $L^{2}$ norm are global in time, whereas localized initial data of sufficiently large $L^{2}$ norm blow up in finite time. Moreover, a blow-up at time $t^{*}$ is self-similar for $t\sim t^{*}$ according to

$$v \sim \frac{1}{L^{2}}Q\left(\frac{z-z^{*}}{L}\right),\qquad L=\sqrt{t^{*}-t},$$

where $z^{*}$ is the apparently finite blow-up location and $Q$ is the lump. These claims summarize numerical observations for small $|E|$; the authors caution that numerical blow-up studies are challenging, and the conjectured self-similar profile and threshold behavior remain analytically unresolved.