Asymptotic Analysis of discrete nonlinear localised modes in a Kagome lattice

TL;DR

This paper analyzes nonlinear localized modes in a Kagome lattice, deriving new coupled NLS equations near flat band intersections and exploring their solitary wave solutions through asymptotic and numerical methods.

Contribution

It introduces a novel coupled NLS system derived from asymptotic analysis of a nonlinear Kagome lattice at flat band intersections.

Findings

Existence of a flat band and Dirac points in the dispersion relation.

Derivation of a new coupled NLS system for small amplitude waves.

Numerical simulations confirming solitary wave solutions.

Abstract

We describe a nonlinear kagome lattice with nonlinear dynamics described by Klein-Gordon interactions with a scalar unknown at each node, such as might occur in a nonlinear electrical lattice. We show that the dispersion relation has three bands - a flat band and two other surfaces which may meet in Dirac points or be separated by a gap. By using multiple scales asymptotic methods, we find a variety of reductions to nonlinear Schrodinger (NLS) systems, some of which are similar to those obtained previously, and have the Townes soliton as a solution. We find a novel system of coupled NLS equations, by forming an asymptotic expansion for small amplitude weakly nonlinear waves around the point where the flat band meets the upper surface of the dispersion relation. We analyse this 2+1 dimensional system using Lie symmetries, and find further reductions to more complicated solitary wave solutions. Numerical simulations of the wave are also presented.