  We interpret the purely spectral forward Maxwell equation with up to 3${^{\rm
rd}}$ order induced polarizations for pulse propagation and interactions in
quadratic nonlinear crystals. The interpreted equation, also named nonlinear
wave equation in frequency domain, includes both quadratic and cubic
nonlinearities, delayed Raman effects and anisotropic nonlinearities. The full
potential of this wave equation is demonstrated by investigating simulations of
solitons generated in the process of ultrafast cascaded second-harmonic
generation. We show that a balance in the soliton delay can be achieved due to
competition between self-steepening, Raman effects and self-steepening-like
effects from cascading originating in the group-velocity mismatch between the
pump and second harmonic. We analyze the first-order contributions, and show
that this balance can be broken to create fast or slow pulses. Through further
simulations we demonstrate few-cycle compressed solitons in extremely short
crystals, where spectral phenomena such as blue/red shifting, non-stationary
radiation in accordance with the non-local phase matching condition and
dispersive-wave generation are observed and marked, which help improving the
experimental knowledge of cascading nonlinear soliton pulse compression.
