Extended in-band and band-gap solutions of the nonlinear honeycomb lattice

We study the dynamics of extended collective excitations in the pristine honeycomb lattice in the presence of the cubic nonlinearity. We show that not only band-gap excitations but also, stable and quasistable, extended excitations between the two lowest bands of the honeycomb system and labeled as in band exist. We also show that some solutions bifurcate from the saddle points of the Floquet band structure. Among other results, we report the existence of nontrivial stationary solutions even for the Floquet eigenvalue where the Dirac points occur. Numerical findings, in fair agreement with our theoretical predictions, are also reported.

Figure 1

(a) Fraction of a HL showing the indexes of the sublattices and the displacement vectors ${\vec{\delta }}_0$ (black), ${\vec{\delta }}_1$ (blue), and ${\vec{\delta }}_2$ (green), respectively. (b) DPs connecting the two lowest bands of the HL sketched in (a). (c) Dispersion relation along the symmetry points in the HL.

Figure 2

$A$ and $B$ amplitudes vs. $\beta$ at the $\Gamma$ point. Plot symbols indicate different solutions. Bifurcation branches opening to the right (left) correspond to the focusing (defocusing) nonlinearity. Blue stars indicate the position of the fold-bifurcation nodes in the focusing nonlinearity case.

Figure 3

Intensity distribution for a solution with $\beta =0$ in a $64\times 32$ lattice size.

Figure 4

(a) Particular examples of $\beta$ vs. $z$ for amplitude solutions in Fig. (2): solid, dashed, and dotted lines correspond to the $①$, $①$, and $①$  bifurcation branches, respectively. (b) Intensity distribution in the reciprocal lattice space (lines are drawn to show the edges of the first Brillouin zone).

Figure 5

The same as Fig. 2 but for the $M$ point.

Figure 6

The same as Fig 4 but for the $M$ point.