  We present here a semi-analytical solution of the problem of particle
acceleration at non-linear shock waves with a free escape boundary at some
location upstream. This solution, besides allowing us to determine the spectrum
of particles accelerated at the shock front, including the shape of the cutoff
at some maximum momentum, also allows us to determine the spectrum of particles
escaping the system from upstream. This latter aspect of the problem is crucial
for establishing a connection between the accelerated particles in
astrophysical sources, such as supernova remnants, and the cosmic rays observed
at the Earth. An excellent approximate solution, which leads to a
computationally fast calculation of the structure of shocks with an arbitrary
level of cosmic ray modification, is also obtained.
