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

Jonas Luhrmann, Wilhelm Schlag

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 Pöschl-Teller potentials from [76], and an implementation of a version of an adapted functional framework introduced in [39].

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

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 Pöschl-Teller potentials from [76], and an implementation of a version of an adapted functional framework introduced in [39].
Paper Structure (40 sections, 35 theorems, 698 equations)

This paper contains 40 sections, 35 theorems, 698 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. References
  4. Main difficulties
  5. Vector field method for 1D Klein-Gordon equations with a potential
  6. Long-range nature of the constant coefficient cubic nonlinearity
  7. Localized quadratic nonlinearity and the threshold resonance of the linearized operator
  8. Overview of the proof
  9. Spectral decomposition
  10. Iterated Darboux transformations
  11. Structure of the transformed equation
  12. Adapted functional framework and weighted energy estimates
  13. Shooting argument and conclusion of the proof of Theorem \ref{['thm:main']}
  14. Further remarks
  15. Preliminaries
  16. ...and 25 more sections

Key Result

Theorem 1.1

There exist constants $C, \delta_0 > 0$ and a Lipschitz function $h \colon \mathcal{A}_0 \to \mathbb R$ with and such that denoting the following holds:

Theorems & Definitions (69)

  • Theorem 1.1: Kowalczyk-Martel-Muñoz KMM19
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 1.5
  • proof
  • Remark 1.6
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['prop:dispersive_decay_estimate']}
  • Lemma 2.2
  • ...and 59 more