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Orbital stability of plane waves in the Klein-Gordon equation against localized perturbations

Emile Bukieda, Louis Garénaux, Björn de Rijk

Abstract

We investigate the stability and long-term behavior of spatially periodic plane waves in the complex Klein-Gordon equation under localized perturbations. Such perturbations render the wave neither localized nor periodic, placing its stability analysis outside the scope of the classical orbital stability theory for Hamiltonian systems developed by Grillakis, Shatah, and Strauss. Inspired by Zhidkov's work on the stability of time-periodic, spatially homogeneous states in the nonlinear Schrödinger equation, we develop an alternative method that relies on an amplitude-phase decomposition and leverages conserved quantities tailored to the perturbation equation. We establish an orbital stability result of plane waves that is locally uniform in space, accommodating $L^2$-localized perturbations as well as unbounded phase modulations. Our result is sharp in the sense that it holds up to the spectral stability boundary.

Orbital stability of plane waves in the Klein-Gordon equation against localized perturbations

Abstract

We investigate the stability and long-term behavior of spatially periodic plane waves in the complex Klein-Gordon equation under localized perturbations. Such perturbations render the wave neither localized nor periodic, placing its stability analysis outside the scope of the classical orbital stability theory for Hamiltonian systems developed by Grillakis, Shatah, and Strauss. Inspired by Zhidkov's work on the stability of time-periodic, spatially homogeneous states in the nonlinear Schrödinger equation, we develop an alternative method that relies on an amplitude-phase decomposition and leverages conserved quantities tailored to the perturbation equation. We establish an orbital stability result of plane waves that is locally uniform in space, accommodating -localized perturbations as well as unbounded phase modulations. Our result is sharp in the sense that it holds up to the spectral stability boundary.

Paper Structure

This paper contains 13 sections, 7 theorems, 114 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Goal, challenges, and strategy
  3. Main result
  4. Discussion and outlook
  5. Organization
  6. Data availability statement
  7. Acknowledgments
  8. Coordinate transformation and energy
  9. Local existence analysis
  10. Energy estimates
  11. Proof of the main result
  12. Numerics
  13. Spectral stability analysis

Key Result

Theorem 3

Let $(a, k, \omega) \in (0,\infty) \times {\mathbb R}^2$ with $(k,\omega) \neq (0,0)$. Assume that the existence condition e:disp-rel and the spectral condition e:condition are satisfied. Then, there exist constants $C,\delta > 0$ such that, whenever $\theta_\infty \in {\mathcal{C}}^1({\mathbb R},{\ there exists a unique global classical solution to the Klein-Gordon equation e:KG with initial con

Figures (3)

  • Figure 1: Plots of the $L^2$-norms of the polar coordinates $\varrho(t)$ and $\vartheta(t)$ of the perturbed plane-wave solution \ref{['e:planewavepert']} with parameters \ref{['e5.1']} and initial condition \ref{['e:IC']}, along with those of their spatial derivatives. The norms of $\varrho(t)$, $\varrho_x(t)$ and $\vartheta_x(t)$ remain bounded in time. In contrast, the top-right panel illustrates that the $L^2$-norm of $\vartheta(t)$ grows over time.
  • Figure 2: Log-log plot of the $L^2$-norm of the phase $\vartheta(t)$ of the perturbed plane-wave solution \ref{['e:planewavepert']} with parameters \ref{['e5.1']} and initial condition \ref{['e:IC']}. The best linear fit, shown in red, is computed using the last quarter of data points. Its slope is approximately $0.5051$, suggesting that $\lVert\vartheta(t)\rVert_{L^2(\mathbb{R})}$ grows with rate $\sqrt{t}$.
  • Figure 3: Plots of the real part of the perturbed plane-wave solution \ref{['e:planewavepert']} (blue), with parameters \ref{['e5.1']} and initial condition \ref{['e:IC']}, along with its phase $\vartheta(t_0)$ (orange) at different times $t_0$. The initial perturbations are given by \ref{['e5.1']} scaled by ${\frac{7}{4}}$ for improved visibility. The perturbation triggers a phase defect that travels outwards in both spatial directions.

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • proof
  • ...and 8 more