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

Abstract

In this article, we address the Cauchy problem for the KP-I equation

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