Impact of the Properties of Microstructured Solids on the Propagation of Hybrid Solitary Waves

Microstructured solids exhibit complex wave propagation dynamics due to the interplay between nonlinearity, dispersion, dissipation, and higher-order spatiotemporal effects induced by their internal architecture. In this work, we investigate how these properties influence the propagation of hybrid solitary waves governed by a generalized strain-wave equation. The main objective is to identify physically admissible hybrid waveforms and to relate each solution family to the characteristic coefficients of the medium. To achieve this, we employ the implicit Bogning (iB) function method, which provides a unified analytical framework for constructing both hyperbolic and trigonometric wave solutions using a hybrid ansatz that combines pulse-like and kink-like components. The analysis reveals that the nonlinear coefficient plays a dominant role in controlling wave amplitude and localization, while the remaining coefficients modulate the waveform structure, including steepness, periodicity, and spatial extent. Several classes of solutions are obtained, including kink-type, antikink, compacton-like, traveling, periodic, and pulse-soliton profiles. For each admissible solution branch, a reduced waveguide equation is derived, allowing a direct physical interpretation of the governing balance between nonlinear and dispersive effects. The novelty of this work lies in the unified analytical treatment of hybrid solitary waves and in the explicit connection established between solution families and coefficient-resolved reduced models. These results provide new insight into wave propagation in engineered microstructured media and offer practical guidelines for designing waveguides with tunable nonlinear responses.