  We present a detailed study of soliton compression of ultra-short pulses
based on phase-mismatched second-harmonic generation (\textit{i.e.}, the
cascaded quadratic nonlinearity) in bulk quadratic nonlinear media. The
single-cycle propagation equations in the temporal domain including
higher-order nonlinear terms are presented. The balance between the quadratic
(SHG) and the cubic (Kerr) nonlinearity plays a crucial role: we define an
effective soliton number -- related to the difference between the SHG and the
Kerr soliton numbers -- and show that it has to be larger than unity for
successful pulse compression to take place. This requires that the phase
mismatch be below a critical level, which is high in a material where the
quadratic nonlinearity dominates over the cubic Kerr nonlinearity. Through
extensive numerical simulations we find dimensionless scaling laws, expressed
through the effective soliton number, which control the behaviour of the
compressed pulses. These laws hold in the stationary regime, in which
group-velocity mismatch effects are small, and they are similar to the ones
observed for fiber soliton compressors. The numerical simulations indicate that
clean compressed pulses below two optical cycles can be achieved in a
$\beta$-barium borate crystal at appropriate wavelengths, even for picosecond
input pulses.
