Equivalent eigenvalue approach to nonlinear plane waves revealing nonlinearity-induced bandgap opening and closing phenomena

Recent studies have shown some unusual nonlinear dispersion behaviors that are disconnected from the linear regime. However, existing analytical techniques, such as perturbation methods, fail to correctly capture these behaviors. Here we propose a general theoretical approach that converts the nonlinear wave equation to an equivalent linear eigenvalue problem, which directly gives the nonlinear dispersion relation and modal vectors. The theoretical approach is employed to 1D phononic chains and 2D hexagonal lattices with alternating softening and hardening nonlinearities, revealing amplitude-induced bandgap opening and closing phenomena. The theoretical results are validated via full-scale simulations with periodic boundary conditions, in which steady-state nonlinear plane wave responses are numerically obtained. Moreover, we leverage these nonlinear phenomena to achieve tunable frequency splitting and focusing effects. Thus, our work opens new paradigms for understanding nonlinear wave physics and for achieving novel wave control capabilities.