Klein–Roudenko–Stoilov blow-up rate conjecture for the critical generalized Zakharov–Kuznetsov equation

Let $Q$ denote the unique positive radial solution on $\mathbb{R}^2$ of

$$-\Delta Q+Q-Q^3=0.$$

Consider the critical two-dimensional Zakharov–Kuznetsov equation

$$u_t+\partial_{x_1}(\Delta u+u^3)=0,$$

with initial data $u(0,x_1,x_2)=u_0(x_1,x_2)\in H^1(\mathbb{R}^2)$. For sufficiently localized $u_0\in\mathcal{S}(\mathbb{R}^2)$ satisfying $\|u_0\|_{L^2}>\|Q\|_{L^2}$, let $\lambda(t)$ and $(x(t),y(t))$ be the blow-up scale and modulation parameters.

Klein–Roudenko–Stoilov conjecture. The solution blows up at a finite time $T$, and, as $t\to T$,

$$u(x,y,t)-\frac{1}{\lambda(t)}Q\left(\frac{x-x(t)}{\lambda(t)},\frac{y-y(t)}{\lambda(t)}\right)\to\widetilde{u}\in L^2,$$

with

$$\|\nabla u(t)\|_{L^2}\sim\frac{1}{(T-t)^{1/2}},\qquad \lambda(t)\sim(T-t)^{1/2},$$

and

$$x(t)\sim\frac{1}{T-t},\qquad y(t)\to y^*\in\mathbb{R}.$$

The paper states that its findings instead give an exponent approximately equal to $3/4$, so the conjectured exponent $1/2$ is contradicted by the paper's claimed result.