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

TL;DR

This paper proves the global well-posedness of the KP-II equation for perturbations of non-decaying solutions, such as the KdV-line soliton, in both $\R\times \T$ and $\R^2$ settings, addressing transverse stability.

Contribution

It establishes the global well-posedness of the KP-II equation for non-localized perturbations of non-decaying solutions, extending stability analysis.

Findings

Global well-posedness for $\R\times \T$ perturbations

Global well-posedness for $\R^2$ perturbations

Addresses transverse stability of non-decaying solutions

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 $ \R\times \T $ and perturbations that are square integrable in $ \R^2 $. In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.