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Local wellposedness of the modified KP-I equations in periodic setting with small initial data (1911.09767v2)
Published 20 Nov 2019 in math.AP
Abstract: We prove local well-posedness of partially periodic and periodic modified KP-I equations, namely for $\partial_t u+(-1){\frac{l+1}{2}}\partiall_x u-\partial_x{-1}\partial_y2 u+u2\partial_x u=0$ 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.