  We use kinetic simulations of diffusive shock acceleration (DSA) to study the
time-dependent evolution of plane, quasi-parallel, cosmic-ray (CR) modified
shocks. Thermal leakage injection of low energy CRs and finite Alfv\'en wave
propagation and dissipation are included. Bohm diffusion as well as the
diffusion with the power-law momentum dependence are modeled. As long as the
acceleration time scale to relativistic energies is much shorter than the
dynamical evolution time scale of the shocks, the precursor and subshock
transition approach the time-asymptotic state, which depends on the shock sonic
and Alfv\'enic Mach numbers and the CR injection efficiency. For the diffusion
models we employ, the shock precursor structure evolves in an approximately
self-similar fashion, depending only on the similarity variable, x/(u_s t).
During this self-similar stage, the CR distribution at the subshock maintains a
characteristic form as it evolves: the sum of two power-laws with the slopes
determined by the subshock and total compression ratios with an exponential
cutoff at the highest accelerated momentum, p_{max}(t). Based on the results of
the DSA simulations spanning a range of Mach numbers, we suggest functional
forms for the shock structure parameters, from which the aforementioned form of
CR spectrum can be constructed. These analytic forms may represent approximate
solutions to the DSA problem for astrophysical shocks during the self-similar
evolutionary stage as well as during the steady-state stage if p_{max} is
fixed.
