  We consider the two dimensional generalization of the Korteweg-de Vries
equation, the generalized Zakharov-Kuznetsov (ZK) equation, $u_t +
\partial_{x_1}(\Delta u + u^p) = 0, (x_1,x_2) \in \mathbb{R}^2$. It is known
that solitons are stable for nonlinearities $p < 3$ and unstable for $p > 3$,
which was established by Anne de Bouard in [5] generalizing the arguments of
Bona-Souganidis-Strauss in [1] for the gKdV equation. The $L^2$-critical case
with $p=3$ has been open and in this paper we prove that solitons are unstable
in the cubic ZK equation. This matches the situation with the critical gKdV
equation, proved in 2001 by Martel and Merle in [22]. While the general
strategy follows [22], the two dimensional case creates several difficulties
and to deal with them, we design a new virial-type quantity, revisit
monotonicity properties and, most importantly, develop new pointwise decay
estimates, which can be useful in other contexts.
