Direction-dependent invariant waveforms and stability in two-dimensional, weakly nonlinear lattices

This study presents higher-order multiple scales analysis aimed at revealing angle- and amplitude-dependent invariant waveforms, and plane-wave stability, in two-dimensional periodic media. Multi-harmonic invariant waves arise from successive orders of particular solutions appearing in the multiple scales analysis. Simulations of nonlinear shear lattices confirm that inclusion of higher-order terms in the injected waveforms significantly reduce the ensuing growth of higher harmonics. These simulations also confirm the predicted directional-dependence of harmonic coefficients. In addition, the study assesses plane-wave stability using a local fixed-point analysis applied to the evolution equations, revealing angle-dependence in the stability characteristics. Based on the directional dependence uncovered, the study concludes with implications for encryption strategies and damage detection using weakly nonlinear lattices. (C) 2019 Elsevier Ltd. All rights reserved.