Conditions on the Existence of Localized Excitations in Nonlinear Discrete Systems

TL;DR

This paper investigates the conditions under which localized excitations exist in nonlinear Hamiltonian lattices, analyzing their dynamics as multi-frequency excitations on tori, and establishing criteria for their existence and symmetry properties.

Contribution

It introduces a novel mapping approach to analyze localized excitations in nonlinear lattices and derives conditions for their existence, highlighting their generic presence in such systems.

Findings

Conditions for existence of periodic localized excitations

Conditions for multiple frequency localized excitations

Symmetry properties of localized solutions

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 $l \rightarrow \pm \infty$. 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.