Nonlinear Wave Propagation in an Extended (3+1)-Dimensional Calogero-Bogoyavlenskii-Schiff equation

The (3+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff fluid equation with variable coefficients serves as a mathematical model for nonlinear wave propagation in non-homogeneous media characterized by spatially and temporally varying physical parameters. This formulation addresses limitations inherent in constant-coefficient models by incorporating realistic variations encountered in complex fluid environments. The present investigation applies the Variable Coefficient Generalized Abel Equation Method, a specialized analytical technique designed for handling variable-coefficient nonlinear partial differential equations of high dimension. We construct exact traveling-wave solutions including kink, lump soliton, breather and periodic waves. Each solution class manifests characteristic physical behaviors typical of nonlinear wave interactions in inhomogeneous media. Also the qualitative analysis examines phase transitions between different solution types, stability properties under parameter perturbations and the spectrum of accessible dynamical regimes. This analysis delineates critical parameter thresholds governing qualitative changes in wave propagation characteristics. Corresponding three-dimensional surface plots and phase portraits provide comprehensive visualization of spatio-temporal evolution patterns, amplitude profiles, localization features and structural diversity exhibited by these analytical solutions. The graphical representations underscore practical relevance to applications including turbulent flow modeling, oceanic surface wave dynamics and plasma wave instabilities occurring within non-uniform physical environments. This systematic study extends the catalog of known analytical solutions for variable-coefficient CBS models while establishing foundational analytical tools essential for investigating nonlinear wave phenomena in realistic, spatially-varying physical systems.