Energy Dynamics of Long-Wave Low-Amplitude Disturbances in an Anharmonic One-Dimensional Lattice

We present analytical investigations of evolution of localized disturbances during their propagation in an infinite monoatomic nonlinear one-dimensional lattice, specifically the α-Fermi-Pasta-Ulam (FPU) chain. We focus on two key disturbance characteristics: the position of the energy center and the energy radius. Restricting our analysis to long-wave low-amplitude disturbances, we investigate the dynamics in the α-FPU chain and its two continuous versions described by the Boussinesq and Korteweg-de Vries (KdV) equations. Utilizing the energy dynamics approach and leveraging the known property of the KdV equation that any localized disturbance eventually decomposes into a set of non-interacting solitons and a dispersive oscillatory tail, we establish a similarity between the behavior of the disturbance in the linear chain and the nonlinear chain under consideration. Namely, at large time scales, the disturbance energy center propagates and the energy radius increases linearly in time, meaning dispersion also occurs at a constant velocity, analogous to the linear case. It was also found that, prior to its decomposition into non-interacting components, a disturbance in the KdV equation generally evolves as if subjected to an effective force from the medium. Furthermore, for two reduced versions of the KdV equation-one lacking the dispersive term and the other lacking the nonlinear term-the energy center of any disturbance moves with constant velocity. These results generalize the behavior observed in harmonic chains to weakly nonlinear systems and provide a unified framework for understanding energy transport.