On the Cauchy problem for the periodic fifth-order KP-I equation

The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation $${\mathrm{\partial }}_{t}u-{\mathrm{\partial }}_x^{5}u-{\mathrm{\partial }}_x^{-1}{\mathrm{\partial }}_y^{2}u+u{\mathrm{\partial }}_{x}u=0,(t,x,y)\in \mathbb{R}\times {\mathbb{T}}^2$$ We prove global well-posedness for constant $x$ mean value initial data in the space $$\mathbb{E}$ = {u\in L^2,~\partial_x^2 u \in L^2,~\partial_x^{-1}\partial_y u \in L^2}$ which is the natural energy space associated with this equation.