Abstract
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.
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