Flexible mechanical metamaterials have been recently shown to support a rich nonlinear dynamic response. In particular, it has been demonstrated that the behavior of rotating-square architected systems in the continuum limit can be described by nonlinear Klein–Gordon equations. Here, we report on a general class of solutions of these nonlinear Klein–Gordon equations, namely cnoidal waves based on the Jacobi elliptic functions sn, cn and dn. By analyzing theoretically and numerically their validity and stability in the design- and wave-parameter space, we show that these cnoidal wave solutions extend from linear waves (or phonons) to solitons, while covering also a wide family of nonlinear periodic waves. The presented results thus reunite under the same framework different concepts of linear and non-linear waves and offer a fertile ground for extending the range of possible control strategies for nonlinear elastic waves and vibrations.

最近已显示出柔性机械超材料可以支持丰富的非线性动态响应。特别是，已经证明，旋转正方形体系在连续极限内的行为可以用非线性Klein-Gordon方程来描述。在这里，我们报告了这些非线性Klein-Gordon方程的一般解，即基于Jacobi椭圆函数sn，cn和dn的正弦波。通过从理论和数值上分析它们在设计和波参数空间中的有效性和稳定性，我们表明这些正弦波解从线性波（或声子）扩展到孤子，同时还涵盖了广泛的非线性周期波族。