Discrete Breathers in a Honeycomb Lattice Near a Semi-Dirac Point

TL;DR

This paper investigates the behavior and stability of discrete breathers in a nonlinear honeycomb lattice near a semi-Dirac point, revealing hybrid structures and stability regimes through analytical and numerical methods.

Contribution

It introduces a detailed analysis of discrete breathers near a semi-Dirac point, combining asymptotic regimes and stability analysis in a nonlinear honeycomb lattice.

Findings

Breathers are stable over a wide parameter range.

Identified an instability transition for breathers.

Analyzed the stability of nonlinear plane waves bifurcating from zero.

Abstract

We study the dynamics of discrete breathers -- spatially localized and time-periodic solutions -- inside the bandgap of a nonlinear honeycomb lattice where the dispersion landscape approaches a so-called semi-Dirac point in which the bands cross linearly in one direction and quadratically in the orthogonal direction. By studying breather dynamics in two opposing asymptotic regimes, near the continuum and anti-continuum limits, we capture the features of hybrid coherent structures on the lattice that are highly discrete at the breather's central peak and have tails well approximated by exact separable solutions to an effective long-wave PDE theory at spatial infinity. We find that breathers are dynamically stable over a wide range of parameters and locate an instability transition. Finally, we analyze the Floquet stability of spatially extended nonlinear plane waves bifurcating from the zero solution at the edges of the gap and how they shape breather profiles inside the gap.