  We examine the role of long--range interactions on the dynamical and
statistical properties of two 1D lattices with on--site potentials that are
known to support discrete breathers: the Klein--Gordon (KG) lattice which
includes linear dispersion and the Gorbach--Flach (GF) lattice, which shares
the same on--site potential but its dispersion is purely nonlinear. In both
models under the implementation of long--range interactions (LRI) we find that
single--site excitations lead to special low--dimensional solutions, which are
well described by the undamped Duffing oscillator. For random initial
conditions we observe that the maximal Lyapunov exponent $\lambda $ %: (a)
tends to a positive value for KG and grows like $\varepsilon^(.25)$ for GF as
the energy density $\varepsilon=E/N$ increases; (b) saturates to a positive
value as the number of particles $N$ increase, scales as $N^{-0.12}$ in the KG
model and as $N^{-0.27}$ in the GF with LRI, suggesting in that case an
approach to integrable behavior towards the thermodynamic limit. Furthermore,
under LRI, their non-Gaussian momentum distributions are distinctly different
from those of the FPU model.
