Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting

Otávio M. L. Gomide, Marcel Guardia, Tere M. Seara, Chongchun Zeng

TL;DR

This paper proves that small amplitude breathers are generically absent for semilinear Klein-Gordon equations with analytic odd nonlinearities, except when the temporal frequency is close to resonance with the linear spectrum. The authors develop a spatial-dynamics framework, reducing breathers to homoclinic orbits and performing a meticulous outer/inner analysis combined with complex matching to capture exponentially small separations of invariant manifolds. The main quantitative contribution is the identification of a leading-order obstruction governed by a Stokes constant $C_{\mathrm{in}}$, analytic in the nonlinearity but independent of $\omega$, which explains the Kruskal–Segur-type nonexistence phenomenon; they also prove the existence of generalized breathers with tails and establish near-resonant bifurcation structure for higher-mode indices $k$. The results illuminate the delicate interplay between fast oscillations and hyperbolic dynamics in nonintegrable Klein-Gordon equations, providing rigorous justification for exponential-small splitting and offering a precise mechanism for breather formation and breakdown in this class of PDEs.

Abstract

Breathers are nontrivial time-periodic and spatially localized solutions of nonlinear dispersive partial differential equations (PDEs). Families of breathers have been found for certain integrable PDEs but are believed to be rare in non-integrable ones such as nonlinear Klein-Gordon equations. In this paper we show that small amplitude breathers of \emph{any} temporal frequency do not exist for semilinear Klein-Gordon equations with generic analytic odd nonlinearities. A breather with small amplitude exists only when its temporal frequency is close to be resonant with the linear Klein-Gordon dispersion relation. Our main result is that, for such frequencies, we rigorously identify the leading order term in the exponentially small (with respect to the small amplitude) obstruction to the existence of small breathers in terms of the so-called \emph{Stokes constant}, which depends on the nonlinearity analytically, but is independent of the frequency. This gives a rigorous justification of a formal asymptotic argument by Kruskal and Segur \cite{KS87} in the analysis of small breathers. We rely on the spatial dynamics approach where breathers can be seen as homoclinic orbits. The birth of such small homoclinics is analyzed via a singular perturbation setting where a Bogdanov-Takens type bifurcation is coupled to infinitely many rapidly oscillatory directions. The leading order term of the exponentially small splitting between the stable/unstable invariant manifolds is obtained through a careful analysis of the analytic continuation of their parameterizations. This requires the study of another limit equation in the complexified evolution variable, the so-called \emph{inner equation}.

On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting

TL;DR

This paper proves that small amplitude breathers are generically absent for semilinear Klein-Gordon equations with analytic odd nonlinearities, except when the temporal frequency is close to resonance with the linear spectrum. The authors develop a spatial-dynamics framework, reducing breathers to homoclinic orbits and performing a meticulous outer/inner analysis combined with complex matching to capture exponentially small separations of invariant manifolds. The main quantitative contribution is the identification of a leading-order obstruction governed by a Stokes constant , analytic in the nonlinearity but independent of , which explains the Kruskal–Segur-type nonexistence phenomenon; they also prove the existence of generalized breathers with tails and establish near-resonant bifurcation structure for higher-mode indices . The results illuminate the delicate interplay between fast oscillations and hyperbolic dynamics in nonintegrable Klein-Gordon equations, providing rigorous justification for exponential-small splitting and offering a precise mechanism for breather formation and breakdown in this class of PDEs.

Abstract

Breathers are nontrivial time-periodic and spatially localized solutions of nonlinear dispersive partial differential equations (PDEs). Families of breathers have been found for certain integrable PDEs but are believed to be rare in non-integrable ones such as nonlinear Klein-Gordon equations. In this paper we show that small amplitude breathers of \emph{any} temporal frequency do not exist for semilinear Klein-Gordon equations with generic analytic odd nonlinearities. A breather with small amplitude exists only when its temporal frequency is close to be resonant with the linear Klein-Gordon dispersion relation. Our main result is that, for such frequencies, we rigorously identify the leading order term in the exponentially small (with respect to the small amplitude) obstruction to the existence of small breathers in terms of the so-called \emph{Stokes constant}, which depends on the nonlinearity analytically, but is independent of the frequency. This gives a rigorous justification of a formal asymptotic argument by Kruskal and Segur \cite{KS87} in the analysis of small breathers. We rely on the spatial dynamics approach where breathers can be seen as homoclinic orbits. The birth of such small homoclinics is analyzed via a singular perturbation setting where a Bogdanov-Takens type bifurcation is coupled to infinitely many rapidly oscillatory directions. The leading order term of the exponentially small splitting between the stable/unstable invariant manifolds is obtained through a careful analysis of the analytic continuation of their parameterizations. This requires the study of another limit equation in the complexified evolution variable, the so-called \emph{inner equation}.

Paper Structure

This paper contains 32 sections, 38 theorems, 474 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Non-existence of small amplitude breathers
  3. Main quantitative results: leading order of the exponentially small obstruction
  4. Birth of small homoclinics via "eigenvalue collision": exponentially small splitting of separatrices
  5. The spatial dynamics approach for the Klein-Gordon equation
  6. Analysis of the first bifurcation ($k=1$) with oddness assumption in $t$
  7. Heuristics of the proof of Theorem \ref{['maintheorem']}; exponentially small bounds
  8. Strategy of the proof of Theorem \ref{['maintheorem']}; exponentially small asymptotics
  9. Description of the Proof of Theorem \ref{['maintheorem']}
  10. Estimates of the invariant manifolds in complex domains
  11. Analysis close to the singularities
  12. The distance between the invariant manifolds
  13. Estimates of the invariant manifolds: Proof of Theorem \ref{['outerthm']}
  14. Banach Spaces and Linear Operators
  15. Fixed Point Argument
  16. ...and 17 more sections

Key Result

Theorem 1.2

Fix $r>0$. Then there exists an open and dense set $\mathcal{U}\subset\mathcal{F}_r$ such that for any $f\in\mathcal{U}$ the following holds. For any $\sigma \in (0, 1)$, there exists $\rho^*>0$ such that there does not exist any solution $u(x, t)$ to kleingordonrev which:

Figures (12)

  • Figure 1: Multi-bump (left) and single-bump (right) functions according to Definition \ref{['D:bump']}.
  • Figure 2: Multi-bump (left) and single-bump (right) solutions in the spatial dynamics framework.
  • Figure 3: Real positive homoclinic \ref{['homoclinic']} to $0$ of the Duffing equation \ref{['singularlimit']} in the critical manifold $\mathcal{M}$.
  • Figure 4: The (infinite dimensional) transverse section $\Sigma$ (see \ref{['section']}) where we measure the distance between the perturbed manifolds $W^u(0)$ and $W^s(0)$.
  • Figure 5: Outer domains $D^{\mathrm{out},u}_{\kappa}$ and $D^{\mathrm{out},s}_{\kappa}$.
  • ...and 7 more figures

Theorems & Definitions (72)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 2.1
  • Proposition 2.2
  • Remark 2.3
  • Theorem 3.1: Outer
  • Remark 3.2
  • ...and 62 more