Wave structures of the (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov equation: analytical insights utilizing two high impact methods

In this manuscript, we employ the unified method and the Sardar subequation method to systematically analyze various wave structures within the (3+1)-dimensional extended quantum nonlinear Zakharov–Kuznetsov equation, incorporating test function approaches. The equation, integral to understanding the intricate dynamics of quantum plasma in diverse scenarios like astrophysical environments, fusion devices, space plasma, and quantum fluids, serves as a foundation model for studying nonlinear waves, structures, and the collective quantum behavior of particles. Our study yields a spectrum of solutions, including bright, dark, singular, periodic, kink, and anti-kink solutions, and we conduct a modulation instability analysis through linear stability techniques. Furthermore, we elucidate the physical implications of our findings using 3-dimensional, 2-dimensional, and contour profiles with relevant parameters, exclusively implementing the symbolic software Mathematica for computations. The obtained results exhibit exceptional merit when compared to existing literature, showcasing the effectiveness of our approach in providing a robust and intensive mathematical framework for addressing complex nonlinear wave phenomena.