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The Existence of Full-Dimensional KAM tori for one-dimensional nonlinear Klein-Gordon equation

Hongzi Cong, Siming Li, Xiaoqing Wu

TL;DR

This work proves the existence and linear stability of full-dimensional KAM tori for the one-dimensional nonlinear Klein-Gordon equation in the non-relativistic limit under periodic boundary conditions. It extends Bourgain's one-dimensional NLS techniques to NLKG by constructing an abstract KAM framework that remains effective as the speed of light $c$ grows, and by enforcing a momentum-conserving transformation to avoid double degeneracy. The main result shows the invariant tori carry amplitudes with subexponential decay $|I_n| \in [\tfrac14 e^{-2r(\ln^{\sigma}\lfloor n\rfloor)}, 4 e^{-2r(\ln^{\sigma}\lfloor n\rfloor)}]$ and are linearly stable, valid for a large-measure subset of the frequency space and uniformly in $c$. This advances the understanding of long-time stability and almost-periodic dynamics in HPDEs, bridging the NLKG with NLS and NLW regimes in the non-relativistic limit.

Abstract

In this paper, we investigate the almost-periodic solutions for the one-dimensional nonlinear Klein-Gordon equation within the non-relativistic limit under periodic boundary conditions. Specifically, by employing the method introduced in \cite{Bourgain2005JFA}, we establish the existence and linear stability of full-dimensional tori with subexponential decay for the equation.

The Existence of Full-Dimensional KAM tori for one-dimensional nonlinear Klein-Gordon equation

TL;DR

This work proves the existence and linear stability of full-dimensional KAM tori for the one-dimensional nonlinear Klein-Gordon equation in the non-relativistic limit under periodic boundary conditions. It extends Bourgain's one-dimensional NLS techniques to NLKG by constructing an abstract KAM framework that remains effective as the speed of light grows, and by enforcing a momentum-conserving transformation to avoid double degeneracy. The main result shows the invariant tori carry amplitudes with subexponential decay and are linearly stable, valid for a large-measure subset of the frequency space and uniformly in . This advances the understanding of long-time stability and almost-periodic dynamics in HPDEs, bridging the NLKG with NLS and NLW regimes in the non-relativistic limit.

Abstract

In this paper, we investigate the almost-periodic solutions for the one-dimensional nonlinear Klein-Gordon equation within the non-relativistic limit under periodic boundary conditions. Specifically, by employing the method introduced in \cite{Bourgain2005JFA}, we establish the existence and linear stability of full-dimensional tori with subexponential decay for the equation.
Paper Structure (15 sections, 11 theorems, 187 equations)

This paper contains 15 sections, 11 theorems, 187 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Comments and Outline of the proof.
  5. The Norm of the Hamiltonian
  6. Representation and norm of the perturbation
  7. Some properties of the norm
  8. Another norm of the Hamiltonian and its properties.
  9. KAM iteration
  10. KAM iteration lemma
  11. Convergence
  12. Proof of the main theorem
  13. Application to the one-dimensional nonlinear Klein-Gordon equation
  14. Proof of Theorem \ref{['th1']}
  15. Measure Estimate

Key Result

Theorem 1.4

Given any $2<\sigma\le 3$, $r>1$, $\gamma'>0$ and $c\in [1,\infty)$. There exists a subset $\mathcal{R}\subset\Pi_c$ satisfying $\hbox{meas}\ {\mathcal{R}} = O( \gamma')$, such that for any $\omega\in\Pi_c\setminus {\mathcal{R}}$, there exists a small $\epsilon_*:=\epsilon_*(\sigma,r ,\gamma)>0$ d (2) the frequencies on $\mathcal{E}$ are prescribed to be $( \omega_{n} )_{n\in\mathbb{Z}}$; (3) th

Theorems & Definitions (34)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 24 more