PHYSICAL REVIEW A (2014)
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.