Multiple Lump Solutions of the (4 + 1)-Dimensional Fokas Equation

Nonlinear evolution equations can be used to simulate many nonlinear phenomena in the real world, which appear in many areas, especially in physical [1], engineering sciences [2], applied mathematics [3], chemistry, and biology [4]. Recently, it is well known that rogue waves play an essential role in helping us apprehending the qualitative properties of many phenomena; it is interesting that lump functions can provide approximate fitting prototypes to model rogue waves. In addition to appearing in the ocean [5], lump waves also actually appear in many other fields, such as atmosphere [6], superfluids [7], and capillary waves [8]. Further study of lump waves will help us interpret some unknown fields more deeply. Certain ways have been arranged to solve the lump wave solutions of some equations; they are inclusive of the Hirota bilinear method [9, 10], the inverse scattering transformation [11], the Darboux transformation [12], the Bäcklund transformation [13], the functional variable method [14], the reduced differential transform method [15], and so on. Many integrable equations which have lump wave solutions are enumerated here, for example, the (3 + 1)-dimensional KPI equation [16], the Davey-Stewartson I equation [17], the (3 + 1)-dimensional nonlinear evolution equation [18, 19], and the nonlinear Schrödinger equation [20]. Generally speaking, it is easier to solve the lower order rational solutions than to solve the multiple lump waves of the nonlinear evolution equation. In this paper, we mainly work on a (4 + 1)-dimensional Fokas equation.


Introduction
Nonlinear evolution equations can be used to simulate many nonlinear phenomena in the real world, which appear in many areas, especially in physical [1], engineering sciences [2], applied mathematics [3], chemistry, and biology [4]. Recently, it is well known that rogue waves play an essential role in helping us apprehending the qualitative properties of many phenomena; it is interesting that lump functions can provide approximate fitting prototypes to model rogue waves. In addition to appearing in the ocean [5], lump waves also actually appear in many other fields, such as atmosphere [6], superfluids [7], and capillary waves [8]. Further study of lump waves will help us interpret some unknown fields more deeply. Certain ways have been arranged to solve the lump wave solutions of some equations; they are inclusive of the Hirota bilinear method [9,10], the inverse scattering transformation [11], the Darboux transformation [12], the Bäcklund transformation [13], the functional variable method [14], the reduced differential transform method [15], and so on. Many integrable equations which have lump wave solutions are enumerated here, for example, the (3 + 1)-dimensional KPI equation [16], the Davey-Stewartson I equation [17], the (3 + 1)-dimensional nonlinear evolution equation [18,19], and the nonlinear Schrödinger equation [20]. Generally speaking, it is easier to solve the lower order rational solutions than to solve the multiple lump waves of the nonlinear evolution equation. In this paper, we mainly work on a (4 + 1)-dimensional Fokas equation.
which was first derived by Fokas by the generalization of two critical nonlinear evolution equations, which are the integrable KP equation and DS equation [21]. The (4 + 1)-dimensional Fokas equation could be applied to portray nonelastic and elastic interactions [21,22]. In nonlinear wave theory, KP and DS equations can be used to characterize the surface waves and internal waves in straits or channels of varying depth and width, respectively [23][24][25][26]. The significance of the (4 + 1)-dimensional Fokas equation follows naturally from the physical applications of the KP and DS equations. Therefore, the (4 + 1)-dimensional Fokas equation could be adopted to represent a number of phenomena in fluid mechanics, optical fiber communications, ocean engineering, and many others. More recently, ( (1)) are researched when the subscript of f n is equal to 1 in Equation (10). In Section 4, the 6lump waves of Equation (1) are studied when the subscript of f n is equal to 2 in Equation (10). Section 5 is devoted to a short conclusion and discussion.

1-Lump Solutions
The fundamental desire of this section is to investigate the 1lump solutions of the new (4 + 1)-dimensional Fokas equation. Firstly, setting X = x + my + nt and Z = z + cw in Equation (1) yields where m, n, c are all real parameters. With the help of variable transformation we can convert the Equation (2) into a bilinear form that reads where f is a real function with regard to variable X, Z. D 2 X , D 4 X , and D 2 Z are called Hirota bilinear D operators. By applying the symbolic computation approach, assuming where α, β, a 1 , and a 0 are constants to be determined. Substituting (5) into (4) and equating the coefficients of all powers of X i 1 Z i 2 to 0, one has Solving these equations, one has Therefore, we can get a solution of Equation (4) as By using variable transformation (3), the 1-lump wave solutions of Equation (1) read where X = x + mt, Z = z + cw, m, n, c, α, β are arbitrary real constants. The 1-lump wave (9) has the structure for three wave peaks. One peak is higher than the water level, and the other two are opposite. Figure 1 presents the three dimensional plot, the density plot, and the corresponding contour plot of the 1-lump wave solution of Equation (1). From 2 Advances in Mathematical Physics Figure 1, we can see that the 1-lump wave has one center ðβ, αÞ. Furthermore, at the point ððð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −3mnðn 2 − 1Þ p + 2mβÞ/2mÞ, αÞ in the plane ðX, ZÞ, the maximum amplitude of the 1lump wave is ðu 0 − ðð2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −3mnðn 2 − 1Þ p Þ/ð3nðn 2 − 1ÞÞÞÞ.

3-Lump Solutions
In the section, in order to construct the multiple lump solutions, we propose the notation just like this: where a i,j , b i,j , and c i,j are arbitrary constants. Then, we take n = 1, where F 2 X, Z ð Þ= X 6 + a 4,0 X 4 + a 4,2 Z 2 X 4 + a 2,0 + a 2,2 Z 2 + a 2,4 Z 4 À Á X 2 + a 0,0 + a 0,2 Z 2 + a 0,4 Z 4 + a 0,6 Z 6 , Substituting Equation (11) into Equation (4) and collecting all the coefficients of X i 1 Z i 2 , we can get a group of constraining relationships for the parameters. Dealing with these equations, one gets where f is given in Equation (11), in which X = x + my + nt, Z = z + cw. By increasing the value of α and β, 3-lump wave merge and their centers form a triangle (see Figure 2). The 3-lump wave is the arrangement of three 1-lump waves in the plane ðX, ZÞ:

Conclusions
In this work, we have analytically established and analyzed novel multiple lump solutions of a (4 + 1)-dimensional Fokas equation based on the bilinear equation and a new ansatz. A series of rational solutions including the 1-lump wave solutions, the 3-lump wave solutions, and the 6-lump wave solutions are obtained. The 1-lump wave has one positive peak and two negative peaks. In order to search the 3-lump and the 6-lump solutions, three polynomial functions F n , P n , and Q n are utilized. It is notable that these lump waves all have the properties lim x→±∞ u = u 0 , lim y→±∞ u = u 0 , and lim z→±∞ u = u 0 . The 3-lump and 6-lump waves consist of three and six independent single 1-lump waves, respectively. All the peaks of the multiple lump waves tend to the same height when α and β are large enough. The results of this paper enrich the types of solutions of the (4 + 1)-dimensional Fokas equation. Comparing with the existing results in the literature, our results are new. We expect these results to provide some values for researching the dynamics of multiple waves in the deep ocean and nonlinear optical fibers.
And it is very helpful for us to obtain the soliton molecules in the future.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.

Acknowledgments
The work is supported by the National Natural Science