Existence, Stability and Dynamics of Discrete Solitary Waves in a Binary Waveguide Array

TL;DR

This paper investigates the existence, stability, and dynamics of discrete solitary waves in binary waveguide arrays near the anti-continuum limit, revealing how stability varies with the number of excited waveguides and exploring their evolution.

Contribution

It develops a systematic theoretical framework to analyze stability and dynamics of multi-site excitations in binary waveguide arrays near the anti-continuum limit, extending previous work on monoatomic arrays.

Findings

States with one, two, and three excited waveguides are analyzed for stability.

Unstable states often evolve into persistent localized or quasi-periodic states.

Increasing excited sites leads to less regular oscillations in unstable dynamics.

Abstract

Recent work has explored binary waveguide arrays in the long-wavelength, near-continuum limit, here we examine the opposite limit, namely the vicinity of the so-called anti-continuum limit. We provide a systematic discussion of states involving one, two and three excited waveguides, and provide comparisons that illustrate how the stability of these states differ from the monoatomic limit of a single type of waveguide. We do so by developing a general theory which systematically tracks down the key eigenvalues of the linearized system. When we find the states to be unstable, we explore their dynamical evolution through direct numerical simulations. The latter typically illustrate, for the parameter values considered herein, the persistence of localized dynamics and the emergence for the duration of our simulations of robust quasi-periodic states for two excited sites. As the number of excited nodes increase, the unstable dynamics feature less regular oscillations of the solution's amplitude.