  We study solitons in one-dimensional quadratic nonlinear photonic crystals
with modulation of both the linear and nonlinear susceptibilities. We derive
averaged equations that include induced cubic nonlinearities and numerically
find previously unknown soliton families. The inclusion of the induced cubic
terms enables us to show that solitons still exist even when the effective
quadratic nonlinearity vanishes and conventional theory predicts that there can
be no soliton. We demonstrate that both bright and dark forms of these solitons
are stable under propagation.
