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 $s>5/3$. 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.