  When the pressure of particles accelerated at shock waves is no longer
negligible compared to the kinetic pressure of the gas, the linear theory of
diffusive shock acceleration breaks down. This is expected in particular when
the shock sweeps up preexisting cosmic rays, or when multiple shocks
reaccelerate successively the same particles. To describe these systems, one
has to account for the nonlinear backreaction of the particles on the
magnetohydrodynamic flow. Using an up-to-date semi-analytical model of particle
reacceleration at nonlinear shocks, we show that the presence of prexisting
energetic particles strongly affects the shock profile, in such a way that the
reacceleration of non thermal particles or the acceleration of particles from
the thermal bath becomes less efficient. We further describe the evolution of
the distribution of particles after several shocks and study the properties of
the asymptotic solution. We detail the case of identical shocks as well as more
realistic scenarios, including the heating of the medium or superbubble
environments. When the particles are efficiently confined in the acceleration
region, it is generally found that the spectrum converges toward a concave
solution after a few tens of shocks, with a spectral index around 3.5 at the
highest energy. The postshock cosmic ray pressure reaches an asymptotic value
of about 4-5% of the ram pressure of one shock. Most of the shock pressure is
transferred to escaping particles.
