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 neighbour interaction we transform the problem of solving the coupled differential equations of motion into a certain mapping
M
l+1
=F(
M
l
,
M
l−1
)
, where
M
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→±∞
. For a generic Hamiltonian lattice the thus linearized mapping is analyzed. We find conditions of existence of 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.