Global well-posedness of partially periodic KP-I equation in the energy space and application

In this article, we address the Cauchy problem for the KP-I equation [partial_t u + partial_x^3 u -partial_x^{-1}partial_y^2u + upartial_x u = 0] for functions periodic in $y$. We prove global well-posedness of this problem for any data in the energy space $mathbb{E} = left{uin L^2left(mathbb{R}timesmathbb{T}right),~partial_x u in L^2left(mathbb{R}timesmathbb{T}right),~partial_x^{-1}partial_y u in L^2left(mathbb{R}timesmathbb{T}right)right}$. We then prove that the KdV line soliton, seen as a special solution of KP-I equation, is orbitally stable under this flow, as long as its speed is small enough.