Global well-posedness of periodic KP-I initial value problem in the energy space

The periodic KP-I initial value problem $\partial_t u+\partial_x^3 u-\partial_x^{-1}\partial_y^2 u+\partial_x (u^2/2)=0$ on $T_{x,y}^2\times R_t,$u(0)=\phi$is globally well-posed in the energy space$E^1 = E^1 (T^2)=\phi: T^2\to R:\hat\phi(0,n)=0$for all$n\in Z \ 0$and$||\phi||_{E^1 (T^2)}=||\hat{\phi}(m,n)(1+|m|+|n/m|)||_{l^2(Z^2)}<\infty$.