On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

TL;DR

This paper proves decay estimates for small perturbations of the soliton in the 1D focusing cubic Klein-Gordon equation, overcoming a small divisor obstacle caused by threshold resonance, using advanced scattering and functional analysis techniques.

Contribution

It establishes decay estimates for codimension one perturbations of the soliton, addressing the small divisor problem due to threshold resonance in the linearized operator.

Findings

Decay estimates up to exponential time scales for small perturbations.

Identification of the small divisor obstacle caused by threshold resonance.

Application of super-symmetric and modified scattering techniques.

Abstract

We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms [76]. The main result of this work establishes decay estimates up to exponential time scales for small "codimension one type" perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with P\"oschl-Teller potentials from [76], and an implementation of a version of an adapted functional framework introduced in [39].