Electronic Research Announcements of the American Mathematical Society, 1997
Quasiperiodic perturbations with two frequencies ( 1 / ε , γ / ε ) (1/\varepsilon ,\gamma /\varep... more Quasiperiodic perturbations with two frequencies ( 1 / ε , γ / ε ) (1/\varepsilon ,\gamma /\varepsilon ) of a pendulum are considered, where γ \gamma is the golden mean number. We study the splitting of the three-dimensional invariant manifolds associated to a two-dimensional invariant torus in a neighbourhood of the saddle point of the pendulum. Provided that some of the Fourier coefficients of the perturbation (the ones associated to Fibonacci numbers) are separated from zero, it is proved that the invariant manifolds split for ε \varepsilon small enough. The value of the splitting, that turns out to be O ( exp ( − c o n s t / ε ) ) \mathrm {O} \left (\exp \left (-\mathrm {const}/\sqrt {\varepsilon }\right )\right ) , is correctly predicted by the Melnikov function.
Breathers are nontrivial time-periodic and spatially localized solutions of nonlinear dispersive ... more 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 consider semilinear Klein-Gordon equations and prove that single bump small amplitude breathers do not exist for generic analytic odd nonlinearities. Breathers with small amplitude can exist only when its temporal frequency is close to be resonant with the Klein-Gordon dispersion relation. For these frequencies, we identify the leading order term in the exponentially small (with respect to the small amplitude) obstruction to the existence of such small breathers in terms of the so-called Stokes constant. We also construct generalized breathers, which are periodic in time and spatially localized solutions up to exponentially small tails. We rely on the spatial dynamics approach...
Electronic Research Announcements of the American Mathematical Society, 1997
Quasiperiodic perturbations with two frequencies ( 1 / ε , γ / ε ) (1/\varepsilon ,\gamma /\varep... more Quasiperiodic perturbations with two frequencies ( 1 / ε , γ / ε ) (1/\varepsilon ,\gamma /\varepsilon ) of a pendulum are considered, where γ \gamma is the golden mean number. We study the splitting of the three-dimensional invariant manifolds associated to a two-dimensional invariant torus in a neighbourhood of the saddle point of the pendulum. Provided that some of the Fourier coefficients of the perturbation (the ones associated to Fibonacci numbers) are separated from zero, it is proved that the invariant manifolds split for ε \varepsilon small enough. The value of the splitting, that turns out to be O ( exp ( − c o n s t / ε ) ) \mathrm {O} \left (\exp \left (-\mathrm {const}/\sqrt {\varepsilon }\right )\right ) , is correctly predicted by the Melnikov function.
Breathers are nontrivial time-periodic and spatially localized solutions of nonlinear dispersive ... more 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 consider semilinear Klein-Gordon equations and prove that single bump small amplitude breathers do not exist for generic analytic odd nonlinearities. Breathers with small amplitude can exist only when its temporal frequency is close to be resonant with the Klein-Gordon dispersion relation. For these frequencies, we identify the leading order term in the exponentially small (with respect to the small amplitude) obstruction to the existence of such small breathers in terms of the so-called Stokes constant. We also construct generalized breathers, which are periodic in time and spatially localized solutions up to exponentially small tails. We rely on the spatial dynamics approach...
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Papers by Tere Seara