Traveling Solitons in Long-Range Oscillator Chains

TL;DR

This paper explores the existence, properties, and stability of traveling solitons in a long-range anharmonic oscillator chain, deriving analytical solutions and validating them through numerical simulations.

Contribution

It introduces a perturbative analytical approach for traveling solitons in a long-range FPU chain using a discrete NLS equation, highlighting differences from short-range models.

Findings

Traveling envelope solitons are analytically derived for the long-range FPU chain.

Kink-solitons can exist but are unstable due to non-analytic dispersion relations.

Numerical simulations confirm the analytical predictions.

Abstract

We investigate the existence and propagation of solitons in a long-range extension of the quartic Fermi-Pasta-Ulam (FPU) chain of anharmonic oscillators. The coupling in the linear term decays as a power-law with an exponent greater than 1 and less than 3. We obtain an analytic perturbative expression of traveling envelope solitons by introducing a Non Linear Schrodinger (NLS) equation for the slowly varying amplitude of short wavelength modes. Due to the non analytic properties of the dispersion relation, it is crucial to develop the theory using discrete difference operators. Those properties are also the ultimate reason why kink-solitons may exist but are unstable, at variance with the short-range FPU model. We successfully compare these approximate analytic results with numerical simulations.