Blow-Up Dynamics for the L² critical case of the $2$D Zakharov-Kuznetsov equation

We investigate the blow-up dynamics for the L² critical two-dimensional Zakharov-Kuznetsov equation {equation*} {cases} ∂_t u+∂x_1 (Δ u+u^3)=0, x=(x_1,x_2)∈ R^2, t ∈ R\\ u(0,x_1,x_2)=u_0(x_1,x_2)∈ H^1(R^2), {cases} {equation*} with initial data u₀ slightly exceeding the mass of the ground state Q. Employing methodologies analogous to the Martel-Merle-Raphael blow-up theory for L² critical equations, more precisely for the critical NLS equation and the quintic generalized Korteweg-de Vries equation, we categorize the solution behaviors into three outcomes: asymptotic stability, finite-time blow-up, or divergence from the soliton's vicinity. The construction of the blow-up solution involves the bubbling of the solitary wave which ensures the universal behavior and stability of the blow-up.