Studying bifurcations and chaotic dynamics in the generalized hyperelastic-rod wave equation through Hamiltonian mechanics

This article introduces a novel modified <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfenced open="(" close=")"> <m:mrow> <m:mfrac> <m:mrow> <m:mi>G</m:mi> <m:mo accent="false">′</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:math> ({[-0.75em]{}{0ex}},{G^{} }{{G}²}) expansion method, combined with Maple software, for solving the exact solutions of the generalized hyperelastic-rod wave equation (GHRWE). The GHRWE has extensive applications in various fields, including the study of dark soliton molecules in nonlinear optics, the propagation of longitudinal waves in fractional derivative viscoelastic materials, and the investigation of the local well-posedness and dispersive limit behavior of the generalized hyperelastic rod wave equation. Through this method, we have obtained a variety of solutions, including U-shaped dark solitons, inverted U-shaped solitons, single W-shaped solitons, and bright-dark alternating solitons. These solutions not only enrich the solution set of GHRWE but also provide a theoretical basis for understanding and predicting behaviors in nonlinear dynamical systems. Our research delves into the impact of fractional order variations on soliton solution characteristics, such as shape and amplitude, and analyzes the effects of different fractional derivative selections on soliton properties. In addition, a significant contribution of this study is the construction of the Hamiltonian system for GHRWE using the trial equation method, revealing complex bifurcations and chaotic dynamics that have eluded previous research. Ultimately, we employed the Chebyshev method, renowned for its exceptional accuracy and numerical stability, to rigorously validate our analytical solutions, ensuring minimal error margins. Through these efforts, we have not only advanced the understanding of GHRWE but also provided new perspectives and tools for research in related fields.