Ansatz method for analyzing an extended (4+1)-dimensional Fokas equation. Part (II): soliton and periodic solutions

<jats:p>This work develops a systematic ansatz framework to construct and classify exact nonlinear wave structures of a recently proposed extended (4+1)-dimensional Fokas equation [Wazwaz et al., Rom. Rep. Phys. 78, 109 (2026)], a higher-dimensional integrable model that remains robust under additional linear couplings in the transverse directions. Building on a fully integrable Fokas-type extension, the new extended model preserves its Painlevé property while incorporating additional mixed-derivative and dispersive terms relevant to multidimensional wave propagation across various interdisciplinary fields, including fluids, plasmas, and optical media. Within this setting, a logarithmic ansatz formula combined with mixed derivatives Φ=R(lnf)mn is introduced to the extended (4+1)-dimensional Fokas equation, which yields seven distinct closed-form families of soliton-like solutions. These solutions explicitly link the amplitude and geometric localization of the waves to the extended dispersion relation, thereby exposing how the added linear terms reshape the propagation and interaction of coherent structures in higher dimensions. Complementing this construction, various forms of the ansatz in hyperbolic and trigonometric forms are employed to derive additional hyperbolic and trigonometric solutions, producing both regular and singular localized pulses as well as spatially periodic wave trains within a single unified framework. The resulting catalogue of solutions, supported by numerical visualization of their dynamics, demonstrates that the extended Fokas model can sustain a rich hierarchy of multidimensional solitons and periodic patterns, thereby providing a versatile prototype for describing nonlinear wave phenomena in complex dispersive media.</jats:p>