An enormous diversity of fractional-soliton solutions with sensitive prodigy to the \(Tzitz\acute{e}ica\)–Dodd–Bullough equation

The central objective of this study is to explore the dynamic response of fractional-soliton solutions within a nonlinear \(Tzitz\acute{e}ica\)–Dodd–Bullough (TDB) equation enhancing the long-distance optical communication, developing advanced materials with unique electromagnetic properties, and contributing to a deeper understanding of complex phenomena. This fractional version integrates fractional derivatives to facilitate the modeling of anomalous diffusion and various other non-local phenomena. We approach the governing model using the extended direct algebraic method, leading to the derivation of fractional-soliton solutions. These solutions are not only exhibited but also have their physical implications elucidated, with two fractional derivative definitions serving as the interpretive tools: the \(\beta\)-derivative and a novel local derivative. The aforementioned integration approach enables the derivation of numerous modern optical soliton solutions, encompassing dark, semi-bright, as well as solutions involving trigonometric, mixed hyperbolic, rational functions, and dark singular solitons. This method effectively highlights the fractional impact of the derived physical phenomena on the fTBD equation. Additionally, the fractional dynamical system undergoes a thorough sensitivity analysis, with the results being graphically represented. To facilitate this, the model undergoes transformation into a planar dynamical system via the Galilean transformation, allowing for an evaluation of the sensitivity performance.