Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non relativistic limit of the nonlinear Klein-Gordon equation: Uniform in time approximation of KAM solutions

Dario Bambusi, Andrea Belloni, Filippo Giuliani

TL;DR

This work establishes a rigorous, Hamiltonian, nonrelativistic limit for the cubic Klein–Gordon equation on $\mathbb{T}$ by constructing small-amplitude, time-quasi-periodic solutions that, after a Gauge transformation, converge uniformly in time to quasi-periodic solutions of the cubic nonlinear Schrödinger equation. The approach combines a detailed Birkhoff normal form reduction with a KAM scheme, carefully handling $c$-dependent resonances and using translation invariance to control small divisors. The authors prove the existence of a Cantor family of KG tori that share frequencies with NLS tori and quantify the convergence of KG to NLS solutions as $c\to\infty$, including uniform-in-time estimates and measure bounds on the resonant sets. This shows a nonlinear mechanism by which the nonrelativistic limit can be realized uniformly in time on a compact domain, without relying on dispersion, and suggests robust extensions to other semilinear singular limits and almost periodic settings.

Abstract

We study the non relativistic limit of the solutions of the cubic nonlinear Klein--Gordon (KG) equation with periodic boundary conditions on an interval and we construct a family of time quasi periodic solutions which, after a Gauge transformation, converge globally uniformly in time to quasi periodic solutions of the cubic NLS. The proof is based on KAM theory. We emphasize that, regardless of the spatial domain, all the previous results concern approximations valid over compact time intervals.

Non relativistic limit of the nonlinear Klein-Gordon equation: Uniform in time approximation of KAM solutions

TL;DR

This work establishes a rigorous, Hamiltonian, nonrelativistic limit for the cubic Klein–Gordon equation on by constructing small-amplitude, time-quasi-periodic solutions that, after a Gauge transformation, converge uniformly in time to quasi-periodic solutions of the cubic nonlinear Schrödinger equation. The approach combines a detailed Birkhoff normal form reduction with a KAM scheme, carefully handling -dependent resonances and using translation invariance to control small divisors. The authors prove the existence of a Cantor family of KG tori that share frequencies with NLS tori and quantify the convergence of KG to NLS solutions as , including uniform-in-time estimates and measure bounds on the resonant sets. This shows a nonlinear mechanism by which the nonrelativistic limit can be realized uniformly in time on a compact domain, without relying on dispersion, and suggests robust extensions to other semilinear singular limits and almost periodic settings.

Abstract

We study the non relativistic limit of the solutions of the cubic nonlinear Klein--Gordon (KG) equation with periodic boundary conditions on an interval and we construct a family of time quasi periodic solutions which, after a Gauge transformation, converge globally uniformly in time to quasi periodic solutions of the cubic NLS. The proof is based on KAM theory. We emphasize that, regardless of the spatial domain, all the previous results concern approximations valid over compact time intervals.

Paper Structure

This paper contains 34 sections, 47 theorems, 462 equations.

Table of Contents

  1. Main result
  2. Global in time results: a purely nonlinear phenomenon
  3. Discussions
  4. Structure of the Klein-Gordon Hamiltonian
  5. The phase space
  6. Hamiltonian formalism
  7. Symmetries
  8. The Hamiltonians
  9. Decomposition of the KG Hamiltonian
  10. Birkhoff Normal Form
  11. Normal Form step
  12. Generator of the canonical transformations
  13. Proof of Theorem \ref{['Teorema0']}
  14. Expansion on the unperturbed invariant torus
  15. The linearized equation
  16. ...and 19 more sections

Key Result

Theorem 1.1

Fix $p>9/2$, $a \geq 0$ and $\mathcal{J}$ as in insieme, then there exist $C>0$ and ${\tt R}_*>0$ such that for any ${\tt R} \in (0,{\tt R}_*)$ and $c\geq {\tt R}^{{-73/72}}$ the following holds. There exist a set ${\tt \Omega}={\tt \Omega} (c) \subset {\mathbb R}^N$ with positive measure, two Lipeo and for any $\omega \in {\tt \Omega}$, two embeddings such that the functions solve respectively

Theorems & Definitions (98)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 88 more