This research explores the \((3+1)\) -dimensional nonlinear extended quantum Zakharov–Kuznetsov (EQZK) equation. This equation has significant applications in the study of nonlinear wave transmission in the quantum plasma or the magnetized plasma medium. In this connection, the exact traveling-wave solution of the equation play a key role in the explanation of energy transmission and dispersion phenomena in the high-dimensional plasma medium. In this regard, two efficient analytical approaches: the extended hyperbolic function method (EHFM) and the improved modified Sardar sub-equation method (IMSSEM) are used. A range of explicit solutions is derived, including singular, periodic, hyperbolic, rational, bright, dark, and kink soliton solutions. The physical characteristics of these solutions are presented using two-dimensional line graphs and three-dimensional surface representations. Besides solution derivation, a comprehensive bifurcation and multistability analysis is also conducted, which demonstrates that there are coexisting wave forms, as well as a great dependence of solutions upon system parameters and initial values. The obtained outcomes clearly indicate that the approach has a high level of efficiency and effectiveness for the investigation of the nonlinear partial differential equations of high dimension. The originality of the research is based upon the fact that the combined approach not only covers the solution region which is not attained using solutions, but also describes the stability, bifurcation, and multistableness of the extended quantum Zakharov–Kuznetsov model successfully. The symbolic calculations are carried out using the Maple software package.
