  One dimensional nonlinear lattices with harmonic long range interaction
potentials (LRIP) having an inverse power kernel type, are studied. For the
nearest neighbour nonlinear interaction we consider the anharmonic potential of
the Fermi-Pasta-Ulam problem and the \phi^3+\phi^4 potential as well. The
continuum limit is obtained following the method used by Ishimori and several
Boussinesq and KdV type equations with supplementary Hilbert transform terms
are found. These nonlocal terms are introduced by the LRIP. For the
\phi^3+\phi^4 nearest neighbour interactions the continuum approximation turns
out to admit exact bilinearization in Hirota formalism. Exact rational
nonsingular solutions are found. The integrability of these nonlocal equations
and the connection with perturbed KdV are also discussed.
