Periodic and quasiperiodic traveling waves in nonlinear lattices with odd elasticity

Abstract: Discrete nonlinear systems support a rich variety of localized and extended wave phenomena, with their dynamics sensitively dependent on the symmetries of the underlying interaction forces within the lattice. Odd elasticity, emerging in effective models of active materials, breaks the action-reaction symmetry of the local interactions and gives rise to new wave behavior. We investigate the existence and stability of traveling waves in a nonlinear lattice with odd elasticity, where the coupling force between adjacent units depends asymmetrically on the deformations of the coupled units (nonreciprocal elastic coupling). We demonstrate the existence of periodic and quasiperiodic traveling waves and analyze their spectral stability using the master stability framework. In particular, we identify the onset of Eckhaus instability based on the curvature of the associated master stability curve. This approach enables a quantitative analysis of size effects, specifically the bounds on lattice sizes for which a given traveling wave is stable. The stability analysis for quasiperiodic waves is based on an effective description of the envelope of the response through a rotating wave approximation, which agrees well with direct numerical simulations. Our findings establish a unified framework for understanding wave propagation characteristics in nonlinear lattices, for both periodic and quasiperiodic wave profiles. We highlight, qualitatively and quantitatively, the role of nonreciprocal stiffness on the existence and stability of nonlinear traveling waves in dissipative systems, and discuss how localization and stability depend on the interplay between nonlinearity, dissipation and odd elasticity.