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

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.

Figures (5)

• Figure 1: (a) schematic of honeycomb lattice with A (red) and B (blue) sites and staggered coupling strengths 1 (bold) and $\lambda$ (dashed); (b) the first Brillouin zone, $\mathcal{B},$ and reciprocal lattice vectors; (c) the linear dispersion relation, $\omega_{\pm}^2(\mathbf{k};\lambda),$ in \ref{['eq2']} over $\mathcal{B}$ for the case $\lambda=1$ and $\omega_0=2$.
• Figure 1: (a) truncated honeycomb lattice with hexagonal Dirichlet boundaries used in computations; (b) view of the converged spatial profile at $t=0$ of the midgap discrete breather when $\lambda=0.48\lesssim\lambda_*=1/2$, seeded from the $(+)$-state in \ref{['cn']} ($g=1$) with $a_*\sim 0.82025$, $N=16,$ and $M=4800$; (c) the same breather shown in (b), viewed at a perpendicular angle to the one in (b); (d) the profile of the 1D midgap soliton to the continuum system in section \ref{['sec:pde']} in the $Z$-direction; (e) profile of the transverse 1D NLS soliton in section \ref{['sec:pde']} in the $H$-direction; (f) midgap breather like in (b) but seeded from the $(-)$-state in \ref{['cn']} ($g=-1$).
• Figure 1: Magnitude of the Floquet spectra of the hardening nonlinear plane wave in \ref{['plane1']} over the Brillouin zone, depicted by black lines, with $S=250$. Regions where the magnitude equals one are stable and those greater than one are unstable. (a) shows when the gap is small and so $a^2\ll 1$, the band-edge plane wave undergoes a tangent bifurcation near the wavevector $\mathbf{M}$. (b)-(f) show that as the gap widens and $a^2$ increases there are growing regions of instability and a persistent neighborhood of stability around $\mathbf{M}.$ (e) shows when $a^2=a_{c,\Gamma}^2=4\lambda/3\approx 0.445$, where \ref{['plane1']} undergoes another tangent bifurcation near the $\Gamma$-point. (f) shows just after the tangent bifurcation at $a\gtrsim a_{c,\Gamma}$.
• Figure 2: Top panels: dispersion relation, \ref{['eq2']}, plotted over $\mathcal{B}$ when $\lambda=\lambda_*$, showing the semi-Dirac point along the (a) $k_y-$ and (b) $k_x-$directions. Bottom panels: zoomed-in view of the dispersion relation, \ref{['eq2']}, centered at $\mathbf{M},$ with an open bandgap (blue curves) for a value $\lambda<\lambda_*$ again along the (a) $k_y-$ and (b) $k_x-$directions.
• Figure 2: Floquet multipliers, $\mu,$ at different values of $\lambda$ on the same continuation curve as the breather shown in Figure \ref{['fig3']}(b) and (c).

Theorems & Definitions (2)

• Theorem 2.1
• Proof 1