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

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 highorder 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.

Analytical and Numerical Solutions for a Kind of High-Dimensional Fractional Order Equation

“…According to the specific expression of pr α X in Equation ( 25), substituting Equation (24) into Equation (22), and then after the calculation, the following formula can be obtained:…”

Bilinear Bäcklund transformations, lump solutions and interaction solutions for (4+1)-dimensional variable-coefficient Fokas equation

“…Because of the significance and wide applications of higher-dimensional equations in the field of mathematical physics, lots of researchers have paid attention to equation (2). Solitons ( [5], [6]), quasi-periodic solutions ( [7]), lumps ( [8], [9]), and lump-soliton solutions ( [10], [11]), bilinear Bäcklund transformation( [12]), high-order rational and semi-rational solutions( [13]), traveling wave solutions( [14]), Lie symmetry analysis and exact invariant solutions( [15]) for the 4D Fokas equation have been investigated. Although there are many studies for 4D Fokas equation ( 2), but few results on variable-coefficient equation (1).…”

Phase Portraits and Traveling Wave Solutions of Fokas System in Monomode Optical Fibers

“…The parameters a 1 , a 2 , a 3 , and a 4 are the nonzero constants. In recent years, many experts and scholars have studied the exact solution of Fokas system, many important methods have been proposed, such as the Jacobi elliptic function expansion method [22], the Hirota's bilinear method [23][24][25][26], the Painlevé analysis method [27], the bilinear Bäcklund transformation method [28], the exp ð−ψðkÞÞ-expansion method [29], the improved F-expansion method [30], the extended rational sine-cosine method [31], and the Exp-function method [32]. Although some important methods of constructing Fokas system have been established, the bifurcation of the dynamic system of Fokas system and the more general traveling wave solution have not been studied.…”