Novel soliton solutions via the Nucci reduction and the extended direct algebraic methods; Chaotic dynamics and bifurcation analysis

Mathematicians, engineers, and related scientists continue to investigate soliton solutions of nonlinear differential equations and their potential applications. The present work focuses on the (3 + 1)-dimensional Vakhnenko–Parkes equation, a nonlinear evolution model arising in various physical contexts. While many analytical approaches have been proposed for lower-dimensional or reduced forms of this equation, exact analytical solutions and detailed dynamical characteristics of the full (3 + 1)-dimensional model remain limited in the literature.
In this study, the Nucci reduction method and an extended direct algebraic approach are utilized to construct exact solutions of the governing equation. Several of the obtained wave structures, including specific kink-type, kink–bell-shaped, and kink–singular soliton solutions, are newly derived for the (3 + 1)-dimensional Vakhnenko–Parkes equation and have not been reported previously. In addition to these novel solutions, hyperbolic- and trigonometric-type wave solutions are recovered as limiting cases of the general analytical forms.
Beyond the analytical framework, the dynamical characteristics of the model are examined by means of bifurcation structures and phase-plane representations, allowing the identification of parameter intervals associated with qualitative changes in solution behavior. The sensitivity analysis clarifies how variations in system parameters influence the stability and persistence of the obtained wave patterns. Two- and three-dimensional numerical representations are provided to illustrate the temporal and spatial evolution of these solutions and to support the analytical results.
Taken together, the analytical constructions and the accompanying dynamical examination indicate that the (3+1)-dimensional Vakhnenko–Parkes equation supports a wide range of nonlinear behaviors, including regimes associated with chaotic dynamics for specific parameter configurations. These findings extend the current understanding of higher-dimensional nonlinear wave models and their soliton-related structures.