Frequency-momentum representation of moving breathers in a two dimensional hexagonal lattice

We study nonlinear excitations propagating in a hexagonal layer which is a model for the cation layer of silicates. We consider their properties in the frequency-momentum or $\omega-k$ representation, extending the theory on pterobreathers in their moving frame for the first time to two dimensions. It can also be easily extended to three dimensions. Exact traveling waves in the $\omega-k$ representation are within {\em resonant} planes, each plane corresponding in the moving frame to a single frequency. These frequencies are integer multiples of a frequency called the fundamental frequency. A breather is within a resonant plane called the breather plane and has a single frequency in the moving frame. The intersection of the resonant planes with the phonon surfaces produce co-traveling wings with a small set of frequencies. The traveling waves obtained by perturbing the system consist of a breather and a soliton traveling together and are quasi-exact. These traveling waves can be used as seeds to obtain exact traveling waves, also formed by a breather and a soliton. The wings do exist but they are usually very small.