[Paper Review] On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting

Otávio M. L. Gomide, Marcel Guàrdia+2 more|arXiv (Cornell University)|Jul 30, 2021

Nonlinear Waves and Solitons63 references4 citations

TL;DR

This paper rigorously establishes the existence of exponentially small homoclinic splitting in semilinear Klein-Gordon equations with analytic odd nonlinearities, identifying the Stokes constant as the leading-order obstruction to small breather existence. It proves that for generic analytic odd nonlinearities, no small breathers exist—confirming a long-standing heuristic from Kruskal and Segur via singular perturbation and complex analysis techniques in spatial dynamics.

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}.

Motivation & Objective

• To resolve the long-standing conjecture that small breathers are generically absent in non-integrable nonlinear Klein-Gordon equations.
• To identify the exponentially small obstruction to homoclinic orbit formation in a singular perturbation framework.
• To rigorously connect the formal asymptotic analysis of Kruskal and Segur to a dynamical systems framework using spatial dynamics.
• To characterize the leading-order term in the splitting of stable and unstable manifolds via the Stokes constant, independent of frequency.
• To prove generic non-vanishing of the Stokes constant, implying non-existence of small breathers for generic analytic odd nonlinearities.

Proposed method

• Formulates breathers as homoclinic orbits in a spatial dynamics setting, reducing the problem to invariant manifold splitting.
• Applies a singular perturbation approach where a Bogdanov-Takens bifurcation is coupled with infinitely many rapidly oscillatory directions.
• Analyzes the analytic continuation of invariant manifold parameterizations to compute the exponentially small splitting distance.
• Introduces and solves the 'inner equation' in the complexified evolution variable to capture the exponentially small terms.
• Uses complex matching estimates between outer and inner solutions to control the splitting distance across different domains.
• Employs a fixed-point argument in Banach spaces to rigorously estimate the invariant manifolds and their separation.

Experimental results

Research questions

• RQ1 What is the leading-order term in the exponentially small splitting of stable and unstable manifolds for small homoclinic orbits in semilinear Klein-Gordon equations?
• RQ2 How does the Stokes constant—dependent only on the nonlinearity—determine the obstruction to breather existence?
• RQ3 For which classes of analytic odd nonlinearities do small breathers fail to exist, and why?
• RQ4 Can the formal asymptotic prediction of Kruskal and Segur regarding exponentially small splitting be rigorously justified?
• RQ5 Is the Stokes constant generically non-zero for analytic odd nonlinearities, implying non-existence of small breathers?

Key findings

• The leading-order term in the exponentially small splitting of invariant manifolds is determined by the Stokes constant, which depends analytically on the nonlinearity but not on the temporal frequency.
• For generic analytic odd nonlinearities, the Stokes constant is non-vanishing, implying that no small breathers exist, even when the frequency is close to resonance.
• The paper proves that the formal asymptotic prediction of Kruskal and Segur on exponentially small splitting is rigorously valid in this context.
• The inner equation is solved via a fixed-point argument in a Banach space of analytic functions, enabling precise control of the exponentially small terms.
• Complex matching estimates between outer and inner solutions yield a precise asymptotic expression for the splitting distance, with exponential decay governed by the inverse of the amplitude.
• The non-vanishing of the Stokes constant is established generically via a perturbative argument in the nonlinearity parameter, confirming the generic absence of small breathers.