Unstable dynamics of solitary traveling waves in a lattice with long-range interactions

In this work we revisit the existence, stability and dynamics of unstable traveling solitary waves in the context of lattice dynamical systems. We consider a nonlinear lattice of an $\alpha$-Fermi-Pasta-Ulam type with the additional feature of all-to-all harmonic long-range interactions whose strength decays exponentially with distance. The competition between the nonlinear nearest-neighbor terms and the longer-range linear ones yields two parameter regimes where the dependence of the energy $H$ of the traveling waves on their velocity $c$ is non-monotonic and multivalued, respectively. We examine both cases, and identify the exact (up to a prescribed numerical tolerance) traveling waves. To investigate the stability of the obtained solutions, we compute their Floquet multipliers, thinking of the traveling wave problem as a periodic one modulo shifts. We show that in the general case when the relationship between $H$ and $c$ is not single-valued, the sufficient but not necessary criterion for stability change is $H'(s)=0$, where $s$ is the parameter along the energy-velocity curve. Perturbing the unstable solutions along the corresponding eigenvectors, we identify two different scenarios of the dynamics of their transition to stable branches. In the first one, the perturbed wave slows down after expelling a dispersive wave. The second scenario involves an increase in the velocity of the perturbed wave accompanied by the formation of a slower small-amplitude traveling solitary wave.