Envelope solitons in a piezoelectric metamaterial beam obeying the nonlinear Schrödinger equation

Nonlinear metamaterials exhibit rich dynamics, including amplitude-dependent behavior, scale disparities, and bifurcations. These unique characteristics provide additional tunability for nonlinear responses, inspiring the development of functional metamaterials with capabilities such as energy focusing/redirection, mechanical logic, and non-reciprocal acoustics. Piezoelectric metamaterials consisting of arrays of piezoelectric patches bonded on an elastic substrate are well-known for linear concepts such as tunable bandgaps. While these linear metamaterials enable the manipulation of acoustic waves with electric signals, the exploration on nonlinear piezoelectric metamaterials remains limited. In this work, a nonlinear piezoelectric metamaterial is proposed using Duffing-type shunt circuits. The cubic nonlinear inductance in the shunt circuit can be realized through a synthetic impedance circuit with digital control. Homogenization of the governing equations yields a pair of coupled partial differential equations suitable for perturbation analysis. Subsequent analysis in the weakly nonlinear regime reveals that the evolution of a wave packet in the metamaterial is governed by the Nonlinear Schrödinger Equation (NLSE), which is well-known for supporting envelope solitary waves. In addition, NLSE-based solitons can be achieved with either hardening or softening nonlinear shunt inductance, depending on the frequency and wavenumber of the wave. The single envelope soliton solutions of NLSE predicted analytically are validated through nonlinear finite element simulations. These results pave the way for novel nonlinear piezoelectric metamaterials capable of electrically tunable nonlinear wave propagation, with potential applications such as physical reservoir computing leveraging soliton collisions and wave-based mechanical logic.