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 \[\partial_t u + \partial_x^3 u -\partial_x^{-1}\partial_y^2u + u\partial_x u = 0\] for functions periodic in $y$. We prove global well-posedness of this problem for any data in the energy space $\mathbb{E} = \left\{u\in L^2\left(\mathbb{R}\times\mathbb{T}\right),~\partial_x u \in L^2\left(\mathbb{R}\times\mathbb{T}\right),~\partial_x^{-1}\partial_y u \in L^2\left(\mathbb{R}\times\mathbb{T}\right)\right\}$. 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.