Unraveling novel wave structures and modulation instability for the (3＋1)-dimensional fifth-order nonlinear Wazwaz equation

This study presents a comprehensive investigation of the (3＋1)-dimensional fifth-order nonlinear Wazwaz equation (3D-WAZWAZe), addressing its significant role in modeling complex wave phenomena in systems exhibiting strong nonlinearity, high-order dispersion, and multidimensional effects across fluid dynamics, plasma physics, optical fibers, and material science. We employ the improved modified extended tanh-function (IMETF) method – implemented here for the first time for this equation – to derive seven distinct classes of exact analytical solutions: dark solitons (intensity dips on finite backgrounds), singular solitons (unbounded amplitudes at discrete points), singular periodic solutions (combining periodicity with localized singularities), Jacobi elliptic solutions (sn, cn, dn functions), exponential solutions (decaying/growing wave profiles), polynomial solutions (algebraic wave structures), and rational solutions (polynomial ratios). Through detailed graphical representations (2D, 3D, and contour plots) and stability analysis, we demonstrate these solutions’ physical interpretability and identify precise conditions for modulation instability. The work’s novelty lies in developing a full analytical solution framework for the 3D-WAZWAZe, introducing new solution families driven by higher-order dispersion, and providing concrete stability criteria. Our results significantly advance the understanding of higher-dimensional nonlinear waves while offering practical tools for wave control in engineering and astrophysical applications, opening new research directions in nonlinear wave theory.