Breather modes of fully nonlinear mass-in-mass chains

TL;DR

This paper models a nonlinear mass-in-mass chain, deriving asymptotic solutions that reveal different governing equations (NLS or Ginzburg-Landau) depending on the nonlinearities involved.

Contribution

It introduces a new model for fully nonlinear mass-in-mass chains and systematically derives asymptotic solutions using multiple timescales analysis.

Findings

Dynamics governed by NLS for certain nonlinearities

Ginzburg-Landau equation arises with quadratic nonlinearities

Asymptotic solutions constructed systematically

Abstract

We propose a model for a chain of particles coupled by nonlinear springs in which each mass has an internal mass and all interactions are assumed to be nonlinear. We show how to construct an asymptotic solution of this system using multiple timescales, the systematic solution of coupled equations by repeated application of a consistency condition. Our results show that for some combinations of nonlinearity the dynamics are governed by the NLS as in the more usual mass-in-mass chains with linear interactions between inner and outer masses. However, when both nonlinearities have quadratic components, we show that the asymptotic reduction results in a Ginzburg-Landau equation instead of NLS.