  Context. We have presented a numerical model for the non-thermal emission of
gamma-ray binaries in a pulsar-wind driven scenario. Aims. We apply this model
to one of the best-observed gamma-ray binaries, the LS 5039 system. Methods.
The model involves a joint simulation of the pulsar- and stellar-wind
interaction and the transport of electronic pairs from the pulsar wind
accelerated at the emerging shock structure. We compute the synchrotron and
inverse Compton emission in a post-processing step, while consistently
accounting for relativistic beaming and $\gamma \gamma$-absorption in the
stellar radiation field. Results. The stellar- and pulsar-wind interaction
leads to the formation of an extended, asymmetric wind collision region
developing strong shocks, turbulent mixing, and secondary shocks in the
turbulent flow. Both the structure of the collision region and the resulting
particle distributions show significant orbital variation. Next to the
acceleration of particles at the bow-like pulsar wind and Coriolis shock the
model naturally accounts for the reacceleration of particles at secondary
shocks contributing to the emission at very-high-energy (VHE) gamma-rays. The
model successfully reproduces the main spectral features of LS 5039. While the
predicted lightcurves in the high-energy and VHE gamma-ray band are in good
agreement with observations, our model still does not reproduce the X-ray to
low-energy gamma-ray modulation, which we attribute to the employed magnetic
field model. Conclusions. We successfully model the main spectral features of
the observed multiband, non-thermal emission of LS 5039 and thus further
substantiates a wind-driven interpretation of gamma-ray binaries. Open issues
relate to the synchrotron modulation, which might be addressed through a
magnetohydrodynamic extension of our model.
