  We study analytical solutions of the Fractional Boussinesq Equation (FBE),
which is an effective model for the Fermi-Pasta-Ulam (FPU) one-dimensional
lattice with long-range couplings. The couplings decay as a power-law with
exponent s, with 1 < s < 3, so that the energy density is finite, but s is
small enough to observe genuine long-range effects. The analytic solutions are
obtained by introducing an ansatz for the dependence of the field on space and
time. This allows to reduce the FBE to an ordinary differential equation, which
can be explicitly solved. The solutions are initially localized and they
delocalize progressively as time evolves. Depending on the value of s the
solution is either a pulse (meaning a bump) or an anti-pulse (i.e., a hole) on
a constant field for 1 < s < 2 and 2 < s < 3, respectively.
