Local Well-Posedness of the Modified Kadomtsev-Petviashvili I Equations

We define the anisotropic Sobolev spaces as $H^{s_1,s_2}(M\times N)=\{g\in L^2(M\times N):\|g\|_{H^{s_1,s_2}}=\|\widehat{g}(\xi,\eta)[(1+\xi^2)^{\frac{s_1}{2}}+(1+\eta^2)^{\frac{s_2}{2}}]\|_{L^2(M^{\ast}\times N^{\ast})}2$, in $H^{s,s}(\mathbb{R}\times \mathbb{T})$ if $l=3$ and $s>2$, in $H^{s,s}(\mathbb{T}\times \mathbb{T})$ if $l=3$ and $s>\frac{19}{8}$, and in $H^{s,s}(\mathbb{R}\times \mathbb{T})$ if $l=5$ and $s>\frac{5}{2}$. All four results require the initial data to be small