Existence of localised normal modes in nonlinear lattices

TL;DR

This paper proves the existence of exponentially localized, time-periodic solutions in nonlinear Hamiltonian lattices, showing how their frequencies relate to the linear spectrum and how their decay can be tuned.

Contribution

It introduces a rigorous proof for localized normal modes in nonlinear lattices using the comparison principle, expanding understanding of localized oscillations.

Findings

Localized solutions exist for soft and hard potentials with frequencies outside the phonon spectrum.

The decay rate of localized states can be continuously tuned by a control parameter.

The proof employs auxiliary linear equations to establish bounds on solutions.

Abstract

We prove the existence of exponentially localised and time-periodic solutions in general nonlinear Hamiltonian lattice systems. Like normal modes, these localised solutions are characterised by collective oscillations at the lattice sites with a uniform time-dependence. The proof of existence uses the comparison principle for differential equations to demonstrate that at each lattice site every half of the fundamental period of oscillations, contributing to localised solutions of the nonlinear lattice, is sandwiched between two oscillatory states of auxiliary linear equations. For soft (hard) on-site potentials the allowed frequencies of the in-phase (out-of-phase) localised periodic solutions lie below (above) the lower (upper) value of the linear spectrum of phonon frequencies. By varying a control parameter the exponential decay of the localised states can be tuned continuously.