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

• 成果类型: Article
• 署名作者: Gomide, Otavio M. L.; Guardia, Marcel; Seara, Tere M.; Zeng, Chongchun
• 署名单位: Universidade Federal de Goias; University of Barcelona; Centre de Recerca Matematica (CRM); Universitat Politecnica de Catalunya; Universitat Politecnica de Catalunya; University System of Georgia; Georgia Institute of Technology
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01327-y
• 发表日期: 2025
• 页码: 661-777
• 关键词: modulating pulse solutions; analytic unfoldings; reversible-systems; inner equation; normal forms; wave; separatrices; Orbits; MANIFOLDS; breakdown

摘要

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 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 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 (Phys. Rev. Lett. 58(8):747, 1987) 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 inner equation.