  The output pulses of a commercial high-power femtosecond fiber laser or
amplifier are typically around 300-500 fs with a wavelength around 1030 nm and
10s of $\mu$J pulse energy. Here we present a numerical study of cascaded
quadratic soliton compression of such pulses in LiNbO$_3$ using a type I phase
matching configuration. We find that because of competing cubic material
nonlinearities compression can only occur in the nonstationary regime, where
group-velocity mismatch induced Raman-like nonlocal effects prevent compression
to below 100 fs. However, the strong group velocity dispersion implies that the
pulses can achieve moderate compression to sub-130 fs duration in available
crystal lengths. Most of the pulse energy is conserved because the compression
is moderate. The effects of diffraction and spatial walk-off is addressed, and
in particular the latter could become an issue when compressing in such long
crystals (around 10 cm long). We finally show that the second harmonic contains
a short pulse locked to the pump and a long multi-ps red-shifted detrimental
component. The latter is caused by the nonlocal effects in the nonstationary
regime, but because it is strongly red-shifted to a position that can be
predicted, we show that it can be removed using a bandpass filter, leaving a
sub-100 fs visible component at $\lambda=515$ nm with excellent pulse quality.
