Study of solitary wave propagation in micro-structured media

Since the mid-19th century, following John Scott Russell’s observations of a “wave of translation” traveling unaltered along a canal in Edinburgh, solitary waves have intrigued scientists and engineers alike. Initially regarded as a curiosity in fluid dynamics, these localized, non-dispersive waveforms have become central in the study of nonlinear wave propagation due to their remarkable stability and energy-carrying properties.

In recent years, attention has shifted to the behavior of solitary waves in microstructured materials—artificially engineered media with complex internal geometries on a mesoscale. Among the most studied are pantographic structures, lattice-like frameworks made of interconnected beams. These systems exhibit mechanical behaviors that are highly unconventional, often governed by large rotations, internal constraints, and geometric nonlinearities.

Recent research by Barchiesi (2025) and Turco et al. (2022) has shown that pantographic media can support the propagation of rarefaction-type solitary waves, a particularly rare phenomenon in solids, where the wave induces local stretching rather than compression. These findings, largely obtained through advanced numerical simulations, underscore the critical role played by structural architecture in shaping wave dynamics.

From a complementary perspective, Turco and Bilotta (2025) explore discrete nonlinear systems such as the Toda lattice, using tools from the inverse scattering transform—a mathematical method developed in the 1960s to solve integrable nonlinear equations. This framework allows for an exact analytical description of solitary wave solutions in discrete mechanical chains, providing deeper insight into how energy travels in systems with localized interactions.

Together, these contributions paint a picture of wave propagation in which geometry, microstructure, and nonlinearity combine to produce complex and tunable dynamics. The implications extend far beyond academic curiosity: they pave the way for mechanical metamaterials capable of controlling vibrations, isolating specific frequencies, or guiding energy along designed pathways. Such developments hold promise for applications in aerospace engineering, seismic protection, and custom acoustic devices, exemplifying the fusion of theoretical mathematics, physical insight, and technological innovation.

Turco, E., & Bilotta, A. (2025). Inverse scattering transform: an overview and the Toda’s chain as paradigm for discrete systems: E. Turco, A. Bilotta. Continuum Mechanics and Thermodynamics, 37(4), 69.

Barchiesi, E. (2025). Monolayered pantographic waveguides admit elastic rarefaction solitary waves. Meccanica, 1-22.

Turco, E., Barchiesi, E., & dell’Isola, F. (2022). A numerical investigation on impulse-induced nonlinear longitudinal waves in pantographic beams. Mathematics and Mechanics of Solids, 27(1), 22-48.