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 I propose an equivalent eigenvalue 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 one-dimensional (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 initial excitations and periodic boundary conditions, in which steady-state nonlinear plane wave responses are numerically obtained. Moreover, these nonlinear phenomena are leveraged to achieve tunable frequency splitting and focusing effects. Thus this work opens paradigms for understanding nonlinear wave physics and for achieving tunable wave control capabilities.

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• Acoustic metamaterials
• Lattice dynamics
• Nonlinear acoustics
• Nonlinear waves
• Bloch wave theory