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.
This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In cite{IsazaLinaresPonce} it was established that if the initial datun $u_0in H^l((b,infty))$ for some $linmathbb Z^+$ and $bin
mathbb R$, then the corresponding solution $u(cdot,t)$ belongs to $H^l((beta,infty))$ for any $beta in mathbb R$ and any $tin (0,T)$. Our goal here is to extend this result to the case where $,lin mathbb R^+$.
In this work we continue our study initiated in cite{GFGP} on the uniqueness properties of real solutions to the IVP associated to the Benjamin-Ono (BO) equation. In particular, we shall show that the uniqueness results established in cite{GFGP} do n
ot extend to any pair of non-vanishing solutions of the BO equation. Also, we shall prove that the uniqueness result established in cite{GFGP} under a hypothesis involving information of the solution at three different times can not be relaxed to two different times.
