Solitons in one-dimensional mechanical linkage

TL;DR

This paper investigates soliton solutions in one-dimensional classical chains, revealing their topological and nonlinear properties, and explores their dynamics at boundaries between inhomogeneous chains.

Contribution

It introduces a method to define and analyze solitons in discrete nonlinear chains without continuum approximation, including their boundary interactions.

Findings

Identification of quasi-periodic solutions as solitons in discrete chains

Demonstration of soliton transmission and reflection at chain boundaries

Linking topological zero-energy modes to nonlinear soliton dynamics

Abstract

It has been observed that certain classical chains admit topologically protected zero-energy modes that are localized on the boundaries. The static features of such localized modes are captured by linearized equations of motion, but the dynamical features are governed by its nonlinearity. We study quasi-periodic solutions of nonlinear equations of motion of one-dimensional classical chains. Such quasi-periodic solutions correspond to periodic trajectories in the configuration space of the discrete systems, which allows us to define solitons without relying on a continuum theory. Furthermore, we study the dynamics of solitons in inhomogeneous systems by connecting two chains with distinct parameter sets, where transmission or reflection of solitons occurs at the boundary of the two chains.