  In this article, we consider the Cauchy problem for the cubic (mass-critical)
Zakharov-Kuznetsov equations in dimension two: $$\partial_t
u+\partial_{x_1}(\Delta u+u^3)=0,\quad (t,x)\in [0,\infty)\times
\mathbb{R}^{2}.$$ For initial data in $H^1$ close to the soliton with a
suitable space-decay property, we fully describe the asymptotic behavior of the
corresponding solution. More precisely, for such initial data, we show that
only three possible behaviors can occur: 1) The solution leaves a tube near
soliton in finite time; 2) the solution blows up in finite time; 3) the
solution is global and locally converges to a soliton. In addition, we show
that for initial data near a soliton with non-positive energy and above the
threshold mass, the corresponding solution will blow up as described in Case 2.
  Our proof is inspired by the techniques developed for mass-critical
generalized Korteweg-de Vries equation (gKdV) equation in a similar context by
Martel-Merle-Rapha\"el. More precisely, our proof relies on refined modulation
estimates and a modified energy-virial Lyapunov functional. The primary
challenge in our problem is the lack of coercivity of the Schr\"odinger
operator which appears in the virial-type estimate. To overcome the difficulty,
we apply a transform, which was first introduced in Kenig-Martel [13], to
perform the virial computations after converting the original problem to the
adjoint one. Th coercivity of the Schr\"odinger operator in the adjoint problem
has been numerically verified by Farah-Holmer-Roudenko-Yang [9].
