Global Well-Posedness and Blow-Up for the fifth order $L^2-$critical KP-I equation

In the current paper, we investigate the fifth order modified KP-I eqaution, namely \begin{equation*} \partial_t u-\partial_{x}^{5}u-\partial_{x}^{-1}\partial_{y}u+\partial_{x}(u^3)=0. \end{equation*} This equation is $L^2$ critical and we prove on $\mathbb{R}\times\mathbb{R}$ that it is globally well posed in the natural energy space if the $L^2$ norm of the initial data is less the $L^2$ norm of the ground state associated to this equation. We also find a subspace of the natural energy space associated to this equation where we have local well-posedness, nevertheless if the initial data is sufficiently localized we obtain blow-up. On $\mathbb{R}\times \mathbb{T},$ we prove global well-posedness in the energy space for small data.