Existence of localized excitations in nonlinear Hamiltonian lattices

Abstract

We consider time-periodic nonlinear localized excitations (NLE's) on one-dimensional translationally invariant Hamiltonian lattices with an arbitrary finite interaction range and an arbitrary finite number of degrees of freedom per unit cell. We analyze a mapping of the Fourier coefficients of the NLE solution. NLE's correspond to homoclinic points in the phase space of this map. Using dimensionality properties of separatrix manifolds of mapping we show the persistence of NLE solutions under perturbations of the system, provided that the NLE's exist for the given system. For a class of nonintegrable Fermi-Pasta-Ulam chains, we rigorously prove the existence of NLE solutions.