Exact wave structures and bifurcation for the (2+1)-Dimensional bogoyavlensky-Konopelchenko equation using modified extended direct algebraic technique.

This study investigates exact analytical solutions and bifurcation dynamics of the (2+1)-dimensional Bogoyavlensky-Konopchenko (BK) equation, a nonlinear model that arises in fluid mechanics and nonlinear wave propagation. The model's coefficients represent physical parameters such as dispersion, nonlinearity, and external forcing in inhomogeneous media, which complicate the derivation of closed-form solutions. To address this, we employ the Modified Extended Direct Algebraic (MEDAM) technique, a powerful analytical tool for solving nonlinear partial differential equations (NLPDEs). The method systematically generates a diverse range of exact solutions, namely dark solitons, singular solitons, singular periodic solutions, exponential solutions, and Jacobi elliptic functions, each corresponding to distinct physical wave structures, by leveraging symbolic computation and compatibility conditions. A bifurcation analysis of the associated dynamical system is conducted, identifying critical parameter thresholds that govern transitions between periodic orbits, stable solitons, and unbounded trajectories. Phase portraits illustrate saddle points, centers, and stability characteristics, providing qualitative insights into nonlinear wave stability. Key results demonstrate the efficacy of MEDAM in handling the equation's coefficients, revealing new families of solutions with physical relevance. The dark soliton solutions correspond to localized intensity dips on a continuous background, characteristic of phenomena such as optical pulses in nonlinear fibers or density holes in plasmas. Singular soliton solutions exhibit waveform coherence with unbounded amplitudes at isolated points, relevant to wave focusing events in fluid dynamics. The Jacobi elliptic solutions represent periodic wave trains that arise in modulated wave propagation, while the bifurcation analysis identifies critical parameter thresholds governing transitions between stable periodic oscillations and unstable soliton regimes-insights that are essential for predicting wave stability in geophysical flows, plasma confinement, and optical communication systems. Previous work employing specialized techniques to obtain exact solutions for the BK equation was restricted to soliton solutions under constrained conditions. In contrast, our study introduces the Modified Extended Direct Algebraic Method (MEDAM), which achieves systematic generation of diverse solution classes not previously reported for this equation, including dark solitons, singular solitons, singular periodic waves, Jacobi elliptic functions, and exponential solutions; the relaxation of prior constraints on coefficient parameters, enabling a broader range of physically relevant solutions; and integration of bifurcation analysis that identifies critical parameter thresholds where transitions occur between periodic orbits, stable solitons, and unbounded trajectories-an aspect entirely absent from earlier investigations. The derived solutions are presented in terms of hyperbolic, trigonometric, and exponential functions, their dynamics visualized through graphical representations to elucidate wave interactions and propagation patterns. Additionally, constraints on the coefficients for solution existence are rigorously analyzed. This work not only extends the known solution space of the BK equation but also underscores the versatility of the MEDAM technique in tackling variable-coefficient NLPDEs, and leverages bifurcation insights to deepen understanding of nonlinear stability and phase transitions. The findings provide a comprehensive analytical framework for modeling nonlinear phenomena in dispersive media, plasma physics, ocean dynamics, and optical communications, offering both quantitative solutions and qualitative stability criteria that are essential for practical applications in these fields.