Reduction in the $(4+1)$-dimensional Fokas equation and their solutions

An integrable extension of the Kadomtsev–Petviashvili (KP) and Davey–Stewartson (DS) equations is investigated in this paper. We will refer to this integrable extension as the $(4+1)$-dimensional Fokas equation. The determinant expressions of soliton, breather, rational, and semi-rational solutions of the $(4+1)$-dimensional Fokas equation are constructed based on the Hirota's bilinear method and the KP hierarchy reduction method. The complex dynamics of these new exact solutions are shown in both three-dimensional plots and two-dimensional contour plots. Interestingly, the patterns of obtained high-order lumps are similar to those of rogue waves in the $(1+1)$-dimensions by choosing different values of the free parameters of the model. Furthermore, three kinds of new semi-rational solutions are presented and the classification of lump fission and fusion processes is also discussed. Additionally, we give a new way to obtain rational and semi-rational solutions of $(3+1)$-dimensional KP equation by reducing the solutions of the $(4+1)$-dimensional Fokas equation. All these results show that the $(4+1)$-dimensional Fokas equation is a meaningful multidimensional extension of the KP and DS equations. The obtained results might be useful in diverse fields such as hydrodynamics, nonlinear optics, and photonics, ion-acoustic waves in plasmas, matter waves in Bose–Einstein condensates, and sound waves in ferromagnetic media.