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

TL;DR

This paper derives explicit analytical pulse solutions for the fractional Boussinesq equation modeling long-range interactions in the Fermi-Pasta-Ulam lattice, revealing how solutions evolve and depend on the decay exponent s.

Contribution

It provides the first explicit analytical solutions to the fractional Boussinesq equation for long-range FPU models, showing how pulse behavior varies with the decay exponent s.

Findings

Solutions are localized pulses or anti-pulses depending on s

Solutions evolve from localized to delocalized over time

Explicit solutions are obtained by reducing the fractional PDE to an ODE

Abstract

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.