Nonlinear vibrational modes in graphene: group-theoretical results

In-plane nonlinear delocalized vibrations in uniformly stretched single-layer graphene (space group P6mm) are considered with the aid of the group-theoretical methods. These methods were developed by authors earlier in the framework of the theory of the bushes of nonlinear normal modes (NNMs). Each bush represents a set of delocalized NNMs which is conserved in time, and the energy of initial excitation turns out to be trapped in the given bush. The number m of modes entering into the bush defines its dimension. One-dimensional bushes (m = 1) represent individual nonlinear normal modes by Rosenberg which describe periodic dynamical regimes, while bushes of higher dimension (m > 1) describe quasiperiodic motion with m basis frequencies in the Fourier spectrum. Each bush is characterized by a space group, which is a subgroup of the symmetry group of the system equilibrium state. There exist bushes of NNMs of different physical nature. In this paper, we restrict ourselves to study of vibrational bushes. We have found that only 4 symmetry-determined NNMs (one-dimensional bushes), as well as 14 two-dimensional, 1 three-dimensional and 6 four-dimensional vibrational bushes are possible in the single-layer graphene. These dynamical regimes are exact solutions to the motion equations of this two-dimensional crystal. Prospects of further research are discussed. Keywords: lattice dynamics, nonlinear normal modes, anharmonic vibrations, group-theoretical methods, graphene.