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Dispersive estimate for quasi-periodic Klein-Gordon equation on 1-d lattices

Hongyu Cheng

TL;DR

This work analyzes dispersive behavior for the discrete Klein-Gordon equation on a 1D lattice with a quasi-periodic potential. The authors implement a quantitative KAM reducibility scheme for the associated Schrödinger cocycle to obtain a robust spectral representation via a spectral transform, enabling precise oscillatory-integral estimates on the spectrum. They prove a linear dispersive bound of order $\langle t\rangle^{-1/3}$ with logarithmic corrections and an energy bound under Diophantine frequencies and small analytic perturbations, and extend the analysis to a nonlinear defocusing/focusing KG equation with $\kappa>5$ to obtain small-data global well-posedness and decay, using a Duhamel contraction framework. The results illuminate long-time dynamics and stability in quasi-periodic lattice models by connecting spectral reducibility, Bloch-like transforms, and sharp time-decay estimates.

Abstract

The dispersive estimate plays a pivotal role in establishing the long-term behavior of solutions to the nonlinear equation, thereby being crucial for investigating the well-posedness of the equation.In this work we prove that the solutions to Klein-Gordon equation on 1-d lattices follow the dispersive estimate provided that potential is quasi-periodic with Diophantine frequencies and closed to positive constants.

Dispersive estimate for quasi-periodic Klein-Gordon equation on 1-d lattices

TL;DR

This work analyzes dispersive behavior for the discrete Klein-Gordon equation on a 1D lattice with a quasi-periodic potential. The authors implement a quantitative KAM reducibility scheme for the associated Schrödinger cocycle to obtain a robust spectral representation via a spectral transform, enabling precise oscillatory-integral estimates on the spectrum. They prove a linear dispersive bound of order with logarithmic corrections and an energy bound under Diophantine frequencies and small analytic perturbations, and extend the analysis to a nonlinear defocusing/focusing KG equation with to obtain small-data global well-posedness and decay, using a Duhamel contraction framework. The results illuminate long-time dynamics and stability in quasi-periodic lattice models by connecting spectral reducibility, Bloch-like transforms, and sharp time-decay estimates.

Abstract

The dispersive estimate plays a pivotal role in establishing the long-term behavior of solutions to the nonlinear equation, thereby being crucial for investigating the well-posedness of the equation.In this work we prove that the solutions to Klein-Gordon equation on 1-d lattices follow the dispersive estimate provided that potential is quasi-periodic with Diophantine frequencies and closed to positive constants.
Paper Structure (14 sections, 14 theorems, 160 equations)

This paper contains 14 sections, 14 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. The result of a KAM procedure
  3. Schrödinger cocycle
  4. A quantitative KAM reducibility
  5. Spectral transform
  6. Oscillatory integral on the spectrum
  7. Auxiliary lemmas
  8. The estimate of integral with general M
  9. Proof of Lemma 3.1.
  10. Proof of main results
  11. Proof of Theorem \ref{['theorem1']}
  12. Proof of Theorem \ref{['theorem2']}
  13. Appendix A. Van der Corput lemma
  14. Appendix B. Cocycles, rotation number

Key Result

Theorem 1.1

Consider the linear Klein-Gordon equation mainequation+ with the potential $V$ defined by 202310230 and assume $\omega\in D C_{d}(\gamma, \tau), P\in C_{r}^{\omega}(\mathbb{T}^{d},\mathbb{R}),\ \theta\in\mathbb{T}^{d}.$ Then there exist $\varepsilon_{*}=\varepsilon_{*}(r,\gamma,\tau,d)>0,$ and an ab

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 8 more