Nonlinear skin breathing modes in one-dimensional nonreciprocal mechanical lattices

TL;DR

This paper explores how nonreciprocity and nonlinearity in a one-dimensional mechanical lattice lead to the emergence of nonlinear skin breathing modes, which are boundary-localized, time-periodic oscillations with tunable properties.

Contribution

It introduces a theoretical framework for nonlinear skin breathing modes in nonreciprocal lattices, deriving explicit frequency shifts and predicting boundary breathers, extending the nonlinear non-Hermitian skin effect to dynamic modes.

Findings

Nonlinear skin breathing modes emerge from linear counterparts at any nonreciprocal strength.

Explicit nonlinear frequency shifts depend on amplitude, nonlinearity, lattice size, and nonreciprocity.

Modes can be stable or unstable, with skin localization inherited from asymmetric couplings.

Abstract

We investigate the interplay of nonreciprocity and nonlinearity in a one-dimensional nonlinear Klein-Gordon chain of classical oscillators coupled by asymmetric springs, akin to a mechanical analogue of the Hatano-Nelson model with onsite nonlinearity. Using multiple-scale analysis, we show that families of nonlinear skin breathing modes -- time-periodic, boundary-localized oscillations -- emerge from their linear counterparts at any nonreciprocal strength. We derive an explicit nonlinear frequency shift for these families of nonlinear breathing modes, showing its dependence on amplitude, nonlinearity type, lattice size, and nonreciprocity, and we predict the emergence of genuine skin end breathers at the boundary once their oscillation frequency and higher harmonics enter the spectral gaps of the linear spectrum. Numerical pseudo-arclength continuation confirms full families of solutions for both hardening and softening nonlinearities. Furthermore, the Floquet analysis shows that these modes can be either linearly stable or unstable, with Floquet eigenvectors exhibiting skin localization inherited from the asymmetric couplings. Our results extend the nonlinear non-Hermitian skin effect from stationary modes to intrinsically time-periodic excitations, providing a pathway to engineer and control breathing modes in nonreciprocal mechanical metamaterials.