Center stable manifolds around line solitary waves of the Zakharov--Kuznetsov equation

TL;DR

This paper constructs center stable manifolds around unstable line solitary waves for the Zakharov--Kuznetsov equation and demonstrates their orbital stability, contributing to understanding the asymptotic behavior of solutions near these waves.

Contribution

It introduces a novel construction of center stable manifolds for unstable solitary waves of the Zakharov--Kuznetsov equation using a graph transform approach.

Findings

Orbital stability of unstable line solitary waves on the center stable manifolds.

Construction of the manifolds using a contraction map on the graph space.

Application of bilinear estimates and modified mobile distance in the analysis.

Abstract

In this paper, we construct center stable manifolds of unstable line solitary waves for the Zakharov--Kuznetsov equation on $\mathbb{R}\times \mathbb{T}_L$ and show the orbital stability of the unstable line solitary waves on the center stable manifolds, which yields the asymptotic stability of unstable solitary waves on the center stable manifolds near by stable line solitary waves. The construction is based on the graph transform approach by Nakanishi and Schlag. Applying the bilinear estimate on Fourier restriction spaces by Molinet and Pilod and modifying the mobile distance by Nakanishi and Schlag, we construct a contraction map on the graph space.