On the periodic Zakharov-Kuznetsov equation

We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on ${\mathbb{T}}^2$. We prove the local well-posedness for given data in $H^s({\mathbb{T}}^2)$ whenever $s>5\mathrm{/}3$. More importantly, we prove that this equation is of quasi-linear type for initial data in any Sobolev space on the torus, in sharp contrast with its semi-linear character in the ${\mathbb{R}}^2$ and $\mathbb{R}\times \mathbb{T}$ settings.