article · Mathematics in Applied Sciences and Engineering
This research examines exact travelling-wave solutions for a non-dissipative, double-dispersive strain-wave equation that models how nonlinear waves move through microstructured solids. By transforming the governing partial differential equation into an ordinary differential equation, the extended Bogning, Djeumen, Tchaho, and Kofane method using implicit Bogning functions is applied to identify and classify valid analytical wave profiles. This approach uses an algebraic substitution process to establish coefficient-range equations and parameter constraints, keeping only branches where all collected coefficients cancel out completely. The verified solutions encompass hyperbolic forms, such as localized, singular localized, regular pulse-type, and dark-soliton-like profiles, alongside trigonometric periodic and singular periodic regimes. These analytical profiles establish exact mathematical descriptions that characterize how internal microstructure and dispersion parameters influence nonlinear wave shape, localization, and propagation behaviour.
Understanding how strain waves travel through complex materials with microscopic structures is fundamental for materials science and mechanical modelling. By deriving exact mathematical solutions rather than approximations, this study provides clear baselines to verify simulations of wave dynamics. This helps researchers predict how mechanical pulses and acoustic energy disperse or concentrate in advanced materials.
This work represents early-stage fundamental mathematical research with no direct commercial product described. The derived exact solutions can be used by computational mechanics researchers and software developers as analytical benchmarks to validate simulation tools for wave propagation in microstructured solids. Because the abstract focuses strictly on mathematical derivation and classification, any practical engineering or commercial application remains distant.
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This work investigates exact travelling-wave solutions of a non-dissipative double-dispersive strain-wave equation describing nonlinear wave propagation in microstructured solids. The Bogning--Djeumen--Tchaho--Kofane method extended to implicit Bogning functions is applied to construct and classify admissible analytical wave profiles of the model. By using a travelling-wave transformation, the governing nonlinear partial differential equation is reduced to an ordinary differential equation, which is then treated through an implicit Bogning-function ansatz. The substitution procedure leads to a coefficient-range equation whose admissible index pairs and parameter constraints determine the possible solution branches. A key aspect of the analysis is the direct verification of the obtained amplitudes: only branches for which all collected coefficients vanish identically are retained as exact solutions. The resulting verified solutions include hyperbolic and trigonometric forms. The hyperbolic branches describe non-periodic travelling structures such as localized, singular localized, regular pulse-type, and dark-soliton-like profiles, whereas the trigonometric branches represent periodic or singular periodic propagation regimes. Graphical representations are provided to illustrate the influence of the admissible parameter constraints on the shape, localization, and propagation of the waves. The results show that the extended BDKm framework offers a systematic coefficient-based approach for deriving, verifying, and classifying exact wave solutions in double-dispersive microstructured media. The obtained profiles may serve as benchmark solutions for future numerical simulations, stability studies, and parameter-sensitivity analyses of nonlinear strain-wave dynamics.
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DOI: 10.5206/mase/25108
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