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

We define the anisotropic Sobolev spaces as $H^{s_1,s_2}(M\times N)=\{g\in L^2(M\times N):\|g\|_{H^{s_1,s_2}}=\|\widehat{g}(\xi,\eta)[(1+\xi^2)^{\frac{s_1}{2}}+(1+\eta^2)^{\frac{s_2}{2}}]\|_{L^2(M^{\ast}\times N^{\ast})}<\infty\}$, where $M$ or $N$ can be either the real line $\mathbb{R}$ or the torus $\mathbb{T}$. We prove local well-posedness of modified KP-I equations in the KP hierarchy, namely for $\partial_t u+(-1)^{\frac{l+1}{2}}\partial^l_x u-\partial_x^{-1}\partial_y^2 u+u^2\partial_x u=0$ in the anisotropic Sobolev space $H^{s,0}(\mathbb{R}\times \mathbb{R})$ if $l=3$ and $s>2$, 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.