Programmable Solitons in an Electromechanical Metamaterial Obeying the Nonlinear Schrödinger Equation via Duffing Shunts

The study of solitary waves in nonlinear acoustic/elastic metamaterials has garnered significant attention due to their shape-invariant properties, where dispersion and nonlinearity are perfectly balanced. Various research groups have made considerable progress in designing such metamaterials, leveraging geometric or material nonlinearities to support solitary wave propagation. Recent advances in our lab in the domain of piezoelectric structures have introduced a novel approach to incorporating nonlinearity with precise tunability and programmability: integrating piezoelectric patches connected to synthetic impedance circuits on a linear substrate or waveguide. These circuits can emulate basic linear and nonlinear electronic components —resistive, capacitive, or inductive— via digital signal processing, offering tunable and programmable characteristics. In this work, for the first time, we explore the use of synthetic impedance circuits in nonlinear metamaterials for soliton generation. We demonstrate that the dynamics of a piezoelectric metamaterial beam coupled with intentionally designed Duffing-type nonlinear shunt circuits can be described by the Nonlinear Schrödinger Equation (NLSE), a well-known model supporting envelope solitons. By adjusting the linear and nonlinear inductance values in the Duffing shunts, we achieve tunable control over both the dispersion relation and the system's nonlinearity. This allows for the initiation of solitary waves around the locally resonant bandgap and the generation of programmable solitons with either hardening or softening nonlinearity on demand. These findings may open new avenues for the development of novel programmable acoustic/elastic metamaterials with potential applications in targeted energy transfer, enhanced sensing, and physical intelligence.