  We calculate magnon dispersions and damping in the Kitaev-Heisenberg model
with an off-diagonal exchange $\Gamma$ and isotropic third-nearest-neighbor
interaction $J_3$ on a honeycomb lattice. This model is relevant to a
description of the magnetic properties of iridium oxides $\alpha$-Li$_2$IrO$_3$
and Na$_2$IrO$_3$, and Ru-based materials such as $\alpha$-RuCl$_3$. We use an
unconventional parametrization of the spin-wave expansion, in which each
Holstein-Primakoff boson is represented by two conjugate hermitian operators.
This approach gives us an advantage over the conventional one in identifying
parameter regimes where calculations can be performed analytically. Focusing on
the parameter regime with the zigzag spin pattern in the ground state that is
consistent with experiments, we demonstrate that one such region is $\Gamma =
K>0$, where $K$ is the Kitaev coupling. Within our approach we are able to
obtain explicit analytical expressions for magnon energies and eigenstates and
go beyond the standard linear spin-wave theory approximation by calculating
magnon damping and demonstrating its role in the dynamical structure factor. We
show that the magnon damping effects in both Born and self-consistent
approximations are very significant, underscoring the importance of non-linear
magnon coupling in interpreting broad features in the neutron-scattering
spectra.
