In this paper we construct and approximate breathers in the DNLS model starting from the continuous limit: such periodic solutions are obtained as perturbations of the ground state of the NLS model in H 1 (R n ), with n = 1, 2. In both the dimensions we recover the Sievers-Takeno (ST) and the Page (P) modes; furthermore, in R 2 also the two hybrid (H) modes are constructed. The proof is based on the interpolation of the lattice using the Finite Element Method (FEM).

Upon initial excitation of a few normal modes the energy distribution among all modes of a nonlinear atomic chain (the Fermi-Pasta-Ulam model) exhibits exponential localization on large time scales. At the same time, resonant anomalies (peaks) are observed in its weakly excited tail for long times preceding equipartition. We observe a similar resonant tail structure also for exact time-periodic Lyapunov orbits, coined q-breathers due to their exponential localization in modal space. We give a simple explanation for this structure in terms of superharmonic resonances. The resonance analysis agrees very well with numerical results and has predictive power. We extend a previously developed perturbation method, based essentially on a Poincare-Lindstedt scheme, in order to account for these resonances, and in order to treat more general model cases, including truncated Toda potentials. Our results give a qualitative and semiquantitative account for the superharmonic resonances of q-breathers and natural packets.

Abstract. We construct small amplitude breathers in 1D and 2D Klein-Gordon infinite lattices. We also show that the breathers are well approximated by the ground state of the nonlinear Schrödinger equation. The result is obtained by exploiting the relation between the Klein Gordon lattice and the discrete Non Linear Schrödinger lattice. The proof is based on a Lyapunov-Schmidt decomposition and continuum approximation techniques introduced in [9], actually using its main result as an important lemma.

We consider a finite but arbitrarily large Klein-Gordon chain, with periodic boundary conditions. In the limit of small couplings in the nearest neighbor interaction, and small (total or specific) energy, a high order resonant normal form is constructed with estimates uniform in the number of degrees of freedom. In particular, the first order normal form is a generalized discrete nonlinear Schrödinger model, characterized by all-to-all sites coupling with exponentially decaying strength.

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