Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic stability of the sine-Gordon kink

Gong Chen, Jonas Luhrmann

Abstract

We establish the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. Our proof consists of a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects combined with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major challenge is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the moving kink. Our analysis crucially relies on two remarkable null structures in the quadratic nonlinearities of the evolution equation for the radiation term and of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is not specific to the sine-Gordon model and extends to many other asymptotic stability problems for moving solitons in relativistic scalar field theories on the line.

Asymptotic stability of the sine-Gordon kink

Abstract

We establish the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. Our proof consists of a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects combined with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major challenge is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the moving kink. Our analysis crucially relies on two remarkable null structures in the quadratic nonlinearities of the evolution equation for the radiation term and of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is not specific to the sine-Gordon model and extends to many other asymptotic stability problems for moving solitons in relativistic scalar field theories on the line.

Paper Structure

This paper contains 61 sections, 71 theorems, 1108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The sine-Gordon model
  3. Motivation: Asymptotic stability of moving kinks
  4. Main result
  5. References
  6. Organization of the paper
  7. Acknowledgements
  8. Overview of the Proof.
  9. Linearized operator
  10. Distorted Fourier theory and linear decay estimates
  11. Modulation and evolution equation for the effective profile
  12. Slow local decay and the null structures
  13. Bootstrap setup
  14. Further comments
  15. Preliminaries
  16. ...and 46 more sections

Key Result

Theorem 1.1

For any $\ell_0 \in (-1,1)$ there exist constants $0 < \varepsilon_0 \ll 1$, $0 < \delta \ll 1$, and $C \geq 1$ such that for any $x_0 \in \mathbb R$, and any $\bm{u}_0 = (u_{0,1}, u_{0,2}) \in H^3_x(\mathbb R) \times H^2_x(\mathbb R)$ with the solution to the sine-Gordon equation equ:intro_sG_1st_order with initial data exists globally in time and there exist continuously differentiable paths $

Figures (3)

  • Figure 2.1: Spectral features of the operator ${\bf L}_\ell$ defined in \ref{['equ:overview_definition_bfLell']}: The orange bands indicate the essential spectrum, the blue dot corresponds to the zero eigenvalue, and the red dots represent the threshold resonances.
  • Figure 4.1: Contour: $\Gamma_{R,\delta,\epsilon}^+$.
  • Figure 11.1: Summary of the decomposition of $\langle \xi\rangle \partial_\xi \mathcal{I}_{2,\mathrm{schem}}^{\delta_0}(s,\xi)$.

Theorems & Definitions (145)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 135 more