Localization in Coupled Finite Vibro-Impact Chains: Discrete Breathers and Multibreathers

TL;DR

This paper studies localized periodic solutions called discrete breathers and multibreathers in a 2D array of coupled vibro-impact chains, highlighting how impact nonlinearities enable explicit solutions and stability analysis, with coupling causing symmetry breaking.

Contribution

It introduces a model with impact nonlinearities allowing explicit computation and stability analysis of breathers and multibreathers in coupled chains, revealing symmetry-breaking effects.

Findings

Explicit solutions for breathers and multibreathers are derived.

Stability can be analyzed using simple linear algebra methods.

Coupling induces symmetry-breaking in breather solutions.

Abstract

We examine the dynamics of strongly localized periodic solutions (discrete breathers) in two-dimensional array of coupled finite one-dimensional chains of oscillators. Localization patterns with both single and multiple localization sites (multibreathers) are considered. The model is scalar, i.e. each particle can move only parallel to the axis of the chain it belongs to. The model involves symmetric parabolic on-site potential with rigid constraints (the displacement domain of each particle is finite) and a linear nearest-neighbor coupling in the chain, and also between the neighbors in adjacent chains. When the particle approaches the constraint, it undergoes an elastic Newtonian impact. The rigid impact constraints are the only source of nonlinearity in the system. The model allows easy computation of highly accurate approximate solutions for the breathers and multibreathers with an arbitrary set of localization sites in conservative setting. The vibro-impact nonlinearity permits explicit derivation of a monodromy matrix for the breather and multi-breather solutions. Consequently, the stability of the derived breather and multibreather solutions can be studied in the framework of simple methods of linear algebra, without additional approximations. It is shown that due to the coupling of the chains, the breather solutions can undergo the symmetry breaking.