Discrete Breathers and Multi-Breathers in Finite Vibro-Impact Chain

TL;DR

This paper derives exact analytical solutions for discrete breathers and multi-breathers in a finite vibro-impact chain, analyzing their stability and the effects of chain finiteness and localization site proximity.

Contribution

It introduces a novel model with rigid impact constraints allowing explicit solutions and stability analysis of localized oscillations in finite chains.

Findings

Exact solutions for breathers and multi-breathers are derived.

Finiteness of the chain significantly influences solution stability.

Proximity of localization sites affects existence and stability patterns.

Abstract

We explore dynamics of discrete breathers and multi-breathers in finite one-dimensional chain. The model involves parabolic on-site potential with rigid constraints and linear nearest-neighbor coupling. The rigid non-ideal impact constraints are the only source of nonlinearity and damping in the model. The model allows derivation of exact analytic solutions for the breathers and multi-breathers with arbitrary set of localization sites, both in conservative and forced-damped settings. We choose periodic boundary conditions; exact solutions for other types of the boundary conditions are also possible. Local character of the nonlinearity allows explicit derivation of a monodromy matrix for the breather solutions. Consequently, a stability of the derived breather and multi-breather solutions can be efficiently studied in the framework of simple methods of linear algebra, and with rather moderate computational efforts. We demonstrate that finitness of the chain fragment and proximity of the localization sites strongly effect existence and stability patterns of these localized solutions.