  Ground state properties and phonon dispersion curves of a classical linear
chain model describing a crystal with an incommensurate phase are studied. This
model is the DIFFOUR (discrete frustrated phi4) model with an extra
fourth-order term added to it. The incommensurability in these models may arise
if there is frustration between nearest-neighbor and next-nearest-neighbor
interactions. We discuss the effect of the additional term on the phonon
branches and phase diagram of the DIFFOUR model. We find some features not
present in the DIFFOUR model such as the renormalization of the
nearest-neighbor coupling. Furthermore the ratio between the slopes of the soft
phonon mode in the ferroelectric and paraelectric phase can take on values
different from -2. Temperature dependences of the parameters in the model are
different above and below the paraelectric transition, in contrast with the
assumptions made in Landau theory. In the continuum limit this model reduces to
the Landau free energy expansion for type II incommensurate crystals and it can
be seen as the lowest-order generalization of the simplest Lifshitz-point
model. Part of the numerical calculations have been done by an adaption of the
Effective Potential Method, orginally used for models with nearest-neighbor
interaction, to models with also next-nearest-neighbor interactions.
