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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This work is supported by the NSF of China under Grant Nos. 11671219 and 11871446.
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Cao, Y., He, J., Cheng, Y. et al. Reduction in the \(\mathbf {(4+1)}\)-dimensional Fokas equation and their solutions. Nonlinear Dyn 99, 3013–3028 (2020). https://doi.org/10.1007/s11071-020-05485-x
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DOI: https://doi.org/10.1007/s11071-020-05485-x