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

${?}_{t}u+{?}_x^{3}u?{?}_x^{?1}{?}_y^{2}u+u{?}_{x}u=0$

for functions periodic in y. We prove global well-posedness of this problem for any data in the energy space $\mathbf{E}=\{u\in L^2(\mathbb{R}\times \mathbb{T}),{?}_{x}u\in L^2(\mathbb{R}\times \mathbb{T}),{?}_x^{?1}{?}_{y}u\in L^2(\mathbb{R}\times \mathbb{T})\}$. 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.