Exact Soliton Solutions for a Generalized (2 + 1)‐Dimensional Bogoyavlensky–Konopelchenko Equation With Time‐Dependent Functional Coefficients

This work presents a systematic and significantly applicable methodology for exploring nonlinear evolution equations (NLEE) with variable coefficients. In this analysis, we employ the modified ‐expansion method adapted for variable coefficients to derive soliton wave solutions corresponding to the most general form of the time‐dependent variable coefficient Bogoyavlensky–Konopelchenko (vcBK) equation. This work presents the first application of the variable coefficient analytical method to analyze soliton wave solutions for the vcBK equation. The proposed model characterizes the interplay between a long wave and a Riemann wave within a specific fluid medium and finds broad applicability across diverse areas of fluid dynamics as well as mathematical physics. The transition to a time‐dependent vcBK equation addresses the limitations associated with constant coefficients, thereby enhancing the model’s fidelity to real‐world conditions and providing greater flexibility in the analysis of physical phenomena. Moreover, the work investigates how variable coefficients impact solitons and their interactions in traveling wave solutions and develop innovative soliton wave solutions for the vcBK equation. The acquired results provide a better understanding of related nonlinear models by elucidating the physical behavior of the vcBK model with time‐dependent variable coefficients using contour and three‐dimensional visuals.