Gradient catastrophe and Peregrine soliton in nonlinear flexible mechanical metamaterials

We explore the generation of extreme wave events in mechanical metamaterials using the regularization of the gradient catastrophe theory developed by A. Tovbis and M. Bertola for the nonlinear Schr\"odinger equation. According to this theory, Peregrine solitons can locally emerge in the semiclassical limit of the nonlinear Schr\"odinger equation. Our objective is to determine whether the phenomenon of gradient catastrophe can occur in a class of architected structures designated as flexible mechanical metamaterials, both with and without losses. We demonstrate theoretically and numerically that this phenomenon can occur in a canonical example of such flexible mechanical metamaterial, a chain of rotating units, studied earlier for its ability to support robust nonlinear waves such as elastic vector solitons. We find that in the presence of weak losses, the gradient catastrophe persists although the amplitude of extreme generated events is smaller and their onset is delayed compared to the lossless configuration.