Conditions on the existence of localized excitations in nonlinear discrete systems

Abstract

We use recent results that localized excitations in nonlinear Hamiltonian lattices can be viewed and described as multiple-frequency excitations. Their dynamics in phase space takes place on tori of corresponding dimension. For a one-dimensional Hamiltonian lattice with nearest neighbor interaction we transform the problem of solving the coupled differential equations of motion into a certain mapping ${\mathit{M}}_{\mathit{l}+1}$=F(${\mathit{M}}_{\mathit{l}}$,${\mathit{M}}_{\mathit{l}\mathrm{\ensuremath{-}}1}$), where ${\mathit{M}}_{\mathit{l}}$ for every l (lattice site) is a function defined on an infinite discrete space of the same dimension as the torus. We consider this mapping in the ``tails'' of the localized excitation, i.e., for l\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}. For a generic Hamiltonian lattice the thus-linearized mapping is analyzed. We find conditions of existence for periodic (one-frequency) localized excitations as well as of multiple-frequency excitations. The symmetries of the solutions are obtained. As a result we find that the existence of localized excitations can be a generic property of nonlinear Hamiltonian lattices in contrast to nonlinear Hamiltonian fields.