Perturbation analysis of nonlinear evanescent waves in a one-dimensional monatomic chain

This paper investigates evanescent waves in one-dimensional nonlinear monatomic chains using a first-order Lindstedt-Poincaré approach. Perturbation approaches applied to traveling waves in similar chains have predicted weakly nonlinear phenomena such as dispersion shifts and amplitude-dependent stability. However, nonlinear evanescent waves have received sparse attention, even though they are expected to serve a critical role in nonlinear interface problems. To aid in their analysis, the nonlinear evanescent waves are categorized herein as either complete or transitional evanescent waves. Complete evanescent waves, including linear evanescent waves, attenuate to zero amplitude in the far field. Transitional evanescent waves, only occurring in softening systems, attenuate to a nontrivial amplitude in the far field, regardless of the initial amplitude, resulting in a saturation effect. For both cases, the presented perturbation approach reveals that the imaginary wave number in the evanescent field is a function of space, rather than a constant value as in its linear counterpart. It also reveals that hardening and softening nonlinearity slow and accelerate the near-field decay, respectively. The predictions obtained from the perturbation approach are verified using numerical simulations with both initial-condition and boundary-continuous excitation, documenting strong agreement.