Pulse solutions of the fractional effective models of the Fermi-Pasta-Ulam lattice with long-range interactions

We study the analytical solutions of the fractional Boussinesq equation (FBE), which is an effective model for the Fermi-Pasta-Ulam 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 the FBE be reduced 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.