On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting by Otávio M. L. Gomide & Marcel Guardia & Tere M. Seara & Chongchun Zeng ISBN 101007/S0022202501327Y instant download

On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting by Otávio M. L. Gomide & Marcel Guardia & Tere M. Seara & Chongchun Zeng ISBN 101007/S0022202501327Y instant download

AbstractBreathers are nontrivial time-periodic and spatially localized solutions of nonlinear dispersive partial differential equations (PDEs). Families of breathers have beenfound for certain integrable PDEs but are believed to be rare in non-integrable onessuch as nonlinear Klein-Gordon equations. In this paper we show that small amplitude breathers of any temporal frequency do not exist for semilinear Klein-Gordonequations with generic analytic odd nonlinearities. A breather with small amplitudeexists only when its temporal frequency is close to be resonant with the linear KleinGordon dispersion relation. Our main result is that, for such frequencies, we rigorously identify the leading order term in the exponentially small (with respect tothe small amplitude) obstruction to the existence of small breathers in terms of theso-called Stokes constant, which depends on the nonlinearity analytically, but is independent of the frequency. This gives a rigorous justification of a formal asymptoticargument by Kruskal and Segur (Phys. Rev. Lett. 58(8):747, 1987) in the analysisof small breathers. We rely on the spatial dynamics approach where breathers canbe seen as homoclinic orbits. The birth of such small homoclinics is analyzed via asingular perturbation setting where a Bogdanov-Takens type bifurcation is coupledto infinitely many rapidly oscillatory directions. The leading order term of the exponentially small splitting between the stable/unstable invariant manifolds is obtainedthrough a careful analysis of the analytic continuation of their parameterizations. Thisrequires the study of another limit equation in the complexified evolution variable, theso-called inner equation.