Global well-posedness for the KP-II equation on the background of a non-localized solution

Abstract

Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations: perturbations that are square integrable in $\mathbb{R}\times \mathbb{T}$ and perturbations that are square integrable in ${\mathbb{R}}^2$. In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.