Local wellposedness of the modified KP-I equations in periodic setting with small initial data

We prove local well-posedness of partially periodic and periodic modified KP-I equations, 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 \(H^{s,s}(\mathbb{R}\times \mathbb{T})\) 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 three results require the initial data to be small.