Discrete breathers in honeycomb Fermi-Pasta-Ulam lattices

TL;DR

This paper analyzes discrete breathers in a two-dimensional honeycomb Fermi-Pasta-Ulam lattice, deriving conditions for their existence, energy thresholds, and symmetry properties, and compares them across different lattice geometries.

Contribution

It introduces a multiple-scale analysis approach to derive NLS equations for breathers in honeycomb lattices and characterizes their energy thresholds and symmetry features.

Findings

Derived moving and stationary breather solutions.

Identified energy thresholds depending on wavenumber.

Compared breather properties across lattice types.

Abstract

We consider the two-dimensional Fermi-Pasta-Ulam lattice with hexagonal honeycomb symmetry, which is a Hamiltonian system describing the evolution of a scalar-valued quantity subject to nearest neighbour interactions. Using multiple-scale analysis we reduce the governing lattice equations to a nonlinear Schrodinger (NLS) equation coupled to a second equation for an accompanying slow mode. Two cases in which the latter equation can be solved and so the system decoupled are considered in more detail: firstly, in the case of a symmetric potential, we derive the form of moving breathers. We find an ellipticity criterion for the wavenumbers of the carrier wave, together with asymptotic estimates for the breather energy. The minimum energy threshold depends on the wavenumber of the breather. We find that this threshold is locally maximised by stationary breathers. Secondly, for an asymmetric potential we find stationary breathers, which, even with a quadratic nonlinearity generate no second harmonic component in the breather. Plots of all our findings show clear hexagonal symmetry as we would expect from our lattice structure. Finally, we compare the properties of stationary breathers in the square, triangular and honeycomb lattices.