  Gamma-ray binaries are composed of a massive star and a rotation-powered
pulsar with a highly relativistic wind. The collision between the winds from
both objects creates a shock structure where particles are accelerated,
resulting in the observed high energy emission. We study the impact of special
relativity on the structure and stability of the colliding wind region and
highlight the differences with colliding winds from massive stars. We focus on
evolution with increasing values of the Lorentz factor of the pulsar wind,
keeping in mind that current simulations are unable to reach the expected
values of the pulsar wind Lorentz factors by orders of magnitude. We use high
resolution numerical simulations with a relativistic extension to the
hydrodynamics code RAMSES we have developed. Using 2D simulations, we focus on
the region close to the binary, neglecting orbital motion. We use different
values of the Lorentz factor of the pulsar wind, up to 16. We find analytic
scaling relations between stellar wind collisions and gamma-ray binaries. They
provide the position of the contact discontinuity. The position of the shocks
strongly depends on the Lorentz factor, the relativistic wind is more
collimated than expected based on non-relativistic simulations. Beyond a
certain distance, the shocked flow is accelerated to its initial velocity and
follows adiabatic expansion. We provide guidance for extrapolation towards more
realistic values of the Lorentz factor of the pulsar wind. We extended the
adaptive mesh refinement code RAMSES to relativistic hydrodynamics. This code
is suited for the study of astrophysical objects such as pulsar wind nebulae,
gamma-ray bursts or relativistic jets and will be part of the next public
release of RAMSES. Using this code we performed simulations of gamma-ray
binaries, highlighting the limits and possibilities of current hydrodynamic
models of such systems.
