 # Multiple rogue wave solutions of a generalized (3+1)-dimensional variable-coefficient Kadomtsev--Petviashvili equation

From the journal Open Physics

## Abstract

Kadomtsev–Petviashvili equation is used for describing the long water wave and small amplitude surface wave with weak nonlinearity, weak dispersion, and weak perturbation in fluid mechanics. Based on the modified symbolic computation approach, the multiple rogue wave solutions of a generalized (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation are investigated. When the variable coefficient selects different functions, the dynamic properties of the derived solutions are displayed and analyzed by different three-dimensional graphics and contour graphics.

## 1 Introduction

With the development of the world economy, maritime safety has become an important research topic in the field of marine engineering. Strong nonlinear waves can cause nonlinear problems such as wave climbing and slamming of offshore structures, which bring great wave loads and are a huge threat to the safety of offshore structures . A rogue wave is such a strong nonlinear extreme wave with remarkable characteristics such as extremely high wave height, prominent wave crest, concentrated energy, and large destructive power. The concept of rogue wave was first proposed by Draper  in 1965, and since then, the phenomenon of rogue waves and its influence on offshore structures have received extensive attention. People called the rogue wave “deep sea monster,” describing it like a wall of water, appearing suddenly, and then quickly disappearing without a trace. Currently, it has been found that rogue wave widely exist in various sea areas of the world, which can appear in the deep sea and offshore sea, in stormy weather and general weather, but most of them are stormy weather. Therefore, the study of rogue wave is of great significance [3,4,5].

In this work, under investigation is a generalized (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation (vcKPe) :

(1) θ 1 u x 2 + θ 1 u u x x + θ 2 u x x x x + θ 3 u x x + θ 4 u x z + θ 5 u y y + θ 6 u z z + θ 7 u x y + u x t = 0 ,

where u = u ( x , y , z , t ) . Eq. (1) represents the long water wave and small amplitude surface wave with weak nonlinearity, weak dispersion, and weak perturbation in fluid mechanics, as well as the electrostatic wave potential in plasma physics. θ i = θ i ( t ) ( i = 1 , 2 , , 7 ) is arbitrary function, describing the nonlinearity, dispersion, perturbed effect, and disturbed wave velocity along the y -and z -directions, respectively. Jaradat et al.  obtained the multiple soliton solutions and singular multiple soliton solutions for Eq. (1). Xie et al.  investigated the soliton collision and Bäcklund transformation of Eq. (1). Chai et al.  and Yin et al.  studied the rouge wave solutions for Eq. (1). Chai et al.  discussed the fusion and fission phenomena of Eq. (1). Chen and Tian  given the Gramian solutions and soliton interactions of Eq. (1). Liu and Zhu  presented the lump-type, breather wave, and kink-solitary wave solutions for Eq. (1). However, the multiple rogue wave solutions have not been derived, which will become our main work.

This article is organized as follows. Section 2 gives the one-order rogue wave solution; Section 3 presents the three-order rogue wave solution; Section 4 derives the six-order rogue wave solution; Section 5 presents a conclusion.

## 2 One-order rogue wave solution

Under the assumption

(2) θ 1 = 6 θ 2 , κ = x + η z Θ ( t ) , u = 2 [ ln ψ ] κ κ , ψ = ψ ( κ , y ) ,

where η is real constant, Θ ( t ) is undetermined function. Eq. (1) becomes

(3) ϕ θ 1 [ ( 2 ψ κ 3 3 ψ ψ κ κ ψ κ + ψ 2 ψ κ κ κ ) 2 + ( ψ κ 2 ψ ψ κ κ ) [ 6 ψ κ 4 12 ψ ψ κ κ ψ κ 2 + 4 ψ 2 ψ κ κ κ ψ κ + ψ 2 ( 3 ψ κ κ 2 ψ ψ κ κ κ κ ) ] ] θ 2 [ 120 ψ κ 6 360 ψ ψ κ κ ψ κ 4 + 120 ψ 2 ψ κ κ κ ψ κ 3 30 ψ 2 ( ψ ψ κ κ κ κ 9 ψ κ κ 2 ) ψ κ 2 + 6 ψ 3 ( ψ ψ κ κ κ κ κ 20 ψ κ κ ψ κ κ κ ) ψ κ + ψ 3 [ 30 ψ κ κ 3 + 5 ψ ( 2 ψ κ κ κ 2 + 3 ψ κ κ ψ κ κ κ κ ) ψ 2 ψ κ κ κ κ κ κ ] ] + [ 6 ψ κ 4 + 12 ψ ψ κ κ ψ κ 2 4 ψ 2 ψ κ κ κ ψ κ + ψ 2 ( ψ ψ κ κ κ κ 3 ψ κ κ 2 ) ] ψ 2 ( θ 3 + η ( θ 4 + η θ 6 ) Θ ( t ) ) + θ 5 [ ( 2 ψ ψ κ κ 6 ψ κ 2 ) ψ y 2 2 ψ ( ψ ψ κ κ y 4 ψ κ ψ κ y ) ψ y + ψ [ ψ y y ( 2 ψ κ 2 ψ ψ κ κ ) + ψ [ ψ ψ κ κ y y 2 ( ψ κ y 2 + ψ κ ψ κ y y ) ] ] ] ψ 2 + θ 7 [ ψ [ 6 ψ κ y ψ κ 2 3 ψ ψ κ κ y ψ κ + ψ ( ψ ψ κ κ κ y 3 ψ κ y ψ κ κ ) ] ψ y ( 6 ψ κ 3 6 ψ ψ κ κ ψ κ + ψ 2 ψ κ κ κ ) ] ψ 2 = 0 .

Based on the modified symbolic computation approach [13,14,15], we suppose that Eq. (3) has the following solution:

(4) ψ = ( κ ρ ) 2 + ε 1 ( y ϱ ) 2 + ε 0 ,

where ρ , ϱ , ε 0 , and ε 1 are unknown constants. Substituting Eqs. (4) into (3), we obtain

(5) Θ ( t ) = [ θ 3 + η θ 4 ε 1 θ 5 + η 2 θ 6 ] d t , θ 7 = 0 , θ 5 = 3 θ 2 ε 0 ε 1 .

Substituting Eqs. (4) and (5) into Eq. (2), the one-order rogue wave solution of Eq. (1) is written as follows:

(6) u = 4 3 θ 2 ε 0 + θ 3 + η ( θ 4 + η θ 6 ) d t + ρ x η z 2 + ε 1 ( y ϱ ) 2 + ε 0 ] ] / 3 θ 2 ε 0 + θ 3 + η ( θ 4 + η θ 6 ) d t + ρ x η z 2 + ε 1 ( y ϱ ) 2 + ε 0 2 .

When θ 3 = θ 4 = θ 6 = θ 2 = 1 , Eq. (6) represents a rogue wave, which is shown in Figure 1. When θ 3 = θ 6 = θ 2 = 1 , θ 4 = t , Eq. (6) has two rogue waves, which is shown in Figure 2. When θ 2 = θ 3 = 1 , θ 4 = sin ( 2 t ) , θ 6 = 3 cos t , Eq. (6) shows a periodic-type rogue wave, which is shown in Figure 3. Figure 1

η = 2 , ε 0 = 3 , ε 1 = 1 , ρ = ϱ = x = z = 0 , (a) 3D plot and (b) contour plot. Figure 2

η = 2 , ε 0 = 3 , ε 1 = 1 , ρ = ϱ = x = z = 0 , (a) 3D plot and (b) contour plot. Figure 3

η = 2 , ε 0 = 3 , ε 1 = 1 , ρ = ϱ = x = z = 0 , (a) 3D plot and (b) contour plot.

## 3 Three-order rogue wave solution

To obtain the three-order rogue wave solution of Eq. (1), we choose

(7) ψ ( κ , y ) = ρ 2 + ϱ 2 + κ 6 + y 6 ε 17 + y 4 ε 16 + 2 ρ κ ( y 2 ε 23 + κ 2 ε 24 + ε 22 ) + 2 ϱ y ( y 2 ε 20 + κ 2 ε 21 + ε 19 ) + κ 4 y 2 ε 11 + y 2 ε 15 + κ 2 ( y 4 ε 14 + y 2 ε 13 + ε 12 ) + κ 4 ε 10 + ε 18 ,

where ε i ( i = 10 , , 24 ) is unknown constant. Substituting Eq. (7) into Eq. (3), we derive

(8) ε 20 = 1 9 ε 11 ε 21 , Θ ( t ) = ( 25 θ 2 + ε 10 θ 3 + η ε 10 θ 4 + η 2 ε 10 θ 6 ) d t ε 10 , θ 5 = 75 θ 2 ε 10 ε 11 , ε 24 = 25 ε 22 ε 10 , ε 14 = ε 11 2 3 , ε 16 = 17 225 ε 10 ε 11 2 , θ 7 = 0 , ε 17 = ε 11 3 27 , ε 15 = 19 75 ε 10 2 ε 11 , ε 13 = 6 ε 10 ε 11 5 , ε 12 = ε 10 2 5 , ε 19 = ε 10 ε 21 15 , ε 23 = 25 ε 11 ε 22 ε 10 , ε 18 = ρ 2 ϱ 2 + 625 ρ 2 ε 22 2 ε 10 2 + ϱ 2 ε 21 2 3 ε 11 + 3 ε 10 3 25 .

Substituting Eqs. (7) and (8) into Eq. (2), the three-order rogue wave solution of Eq. (1) is obtained as follows:

(9) κ = x + η z Θ ( t ) , u = 2 [ ln ψ ] κ κ , ψ = ψ ( κ , y ) ,

When θ 3 = θ 4 = θ 6 = θ 2 = 1 in Eq. (9), we can see the interaction between three rogue waves in Figure 4. When θ 3 = θ 6 = θ 2 = 1 , θ 4 = t in Eq. (9), we can see the interaction between five rogue waves in Figure 5. Figure 4

η = 1 , ε 10 = 6 , ε 11 = ε 21 = ε 22 = 1 , ρ = ϱ = 10 , x = z = 0 , (a) 3D plot and (b) contour plot. Figure 5

η = 1 , ε 10 = 6 , ε 11 = ε 21 = ε 22 = 1 , ρ = ϱ = 10 , x = z = 0 , (a) 3D plot and (b) contour plot.

## 4 Six-order rogue wave solution

To present the six-order rogue wave solution of Eq. (1), we have

(10) ψ ( κ , y ) = κ 12 + y 8 ε 48 + y 6 ε 47 + y 4 ε 46 + ( ρ 2 + ϱ 2 ) ( κ 2 + y 2 ε 1 + ε 0 ) + ε 51 + κ 10 ( y 2 ε 26 + ε 25 ) + y 2 ε 45 + κ 8 ( y 4 ε 29 + y 2 ε 28 + ε 27 ) + y 12 ε 50 + y 10 ε 49 + 2 ρ κ [ κ 6 + y 6 ε 64 + y 4 ε 63 + κ 4 ( y 2 ε 69 + ε 68 ) + y 2 ε 62 + κ 2 ( y 4 ε 67 + y 2 ε 66 + ε 65 ) + ε 61 ] + 2 ϱ y [ y 6 + y 4 ( κ 2 ε 57 + ε 56 ) + y 2 ( κ 4 ε 55 + κ 2 ε 54 + ε 53 ) + κ 6 ε 60 + κ 4 ε 59 + κ 2 ε 58 + ε 52 ] + κ 6 [ y 6 ε 33 + y 4 ε 32 + y 2 ε 31 + ε 30 ] + κ 4 [ y 8 ε 38 + y 6 ε 37 + y 4 ε 36 + y 2 ε 35 + ε 34 ] + κ 2 ( y 10 ε 44 + y 8 ε 43 + y 6 ε 42 + y 4 ε 41 + y 2 ε 40 + ε 39 ) ,

where ε i ( i = 25 , , 69 ) is undetermined constant. Substituting Eq. (10) into Eq. (3), we obtain

Θ ( t ) = 1 6 [ 6 θ 3 + 6 η θ 4 ε 26 θ 5 + 6 η 2 θ 6 ] d t , θ 2 = ε 26 4 ε 59 θ 5 136080 , ε 32 = 11 ε 26 5 ε 59 5832 , ε 31 = 19 ε 26 7 ε 59 2 3149280 , ε 37 = 73 ε 26 6 ε 59 244944 , ε 36 = 107 ε 26 8 ε 59 2 52907904 , ε 45 = ε 1 ( ρ 2 ϱ 2 ) + ρ 2 ε 26 6 + 46656 ϱ 2 ε 26 6 + 3509 ε 59 5 ε 26 16 1259660837552947200 , ε 42 = 253 ε 26 9 ε 59 2 793618560 , ε 41 = ε 26 11 ε 59 3 28570268160 , ε 52 = 11 ε 26 6 ε 59 3 94478400 , ε 33 = 5 ε 26 3 54 , ε 55 = 180 ε 26 2 , ε 38 = 5 ε 26 4 432 , ε 35 = ε 26 10 ε 59 3 317447424 , ε 43 = 19 ε 26 7 ε 59 979776 , ε 29 = 5 ε 26 2 12 ,

ε 50 = ε 26 6 46656 , ε 54 = 19 63 ε 26 ε 59 , ε 57 = 54 ε 26 , ε 49 = 29 ε 26 8 ε 59 88179840 , ε 63 = ε 26 5 ε 59 18144 , ε 48 = 289 ε 26 10 ε 59 2 44442639360 , ε 40 = 11 ε 26 13 ε 59 4 30855889612800 , ε 61 = 7 ε 26 9 ε 59 3 20407334400 , ε 64 = 5 ε 26 3 216 , ε 47 = 5707 ε 26 12 ε 59 3 53997806822400 , ε 27 = ε 26 6 ε 59 2 699840 , ε 66 = 23 ε 26 4 ε 59 13608 , ε 53 = ε 26 4 ε 59 2 58320 , ε 60 = 1080 ε 26 3 , ε 69 = 3 ε 26 2 , ε 65 = ε 26 6 ε 59 2 2099520 , ε 28 = 23 ε 26 4 ε 59 4536 , ε 25 = 7 ε 26 3 ε 59 1620 , ε 56 = 1 540 ε 26 2 ε 59 , ε 46 = 13381 ε 26 14 ε 59 4 23327052547276800 , ε 67 = 5 ε 26 2 36 ,

(11) ε 34 = 121 ε 26 12 ε 59 4 18513533767680 , ε 68 = 13 ε 26 3 ε 59 22680 , ε 58 = 19 ε 26 3 ε 59 2 68040 , ε 44 = ε 26 5 1296 , ε 30 = 11 ε 26 9 ε 59 3 5101833600 , ε 39 = ϱ 2 + 279936 ϱ 2 ε 26 7 + 1331 ε 59 5 ε 26 15 149959623518208000 , ε 62 = 107 ε 26 7 ε 59 2 617258880 , ε 51 = ε 0 ( ρ 2 ϱ 2 ) + ε 59 ( 279936 ϱ 2 + ρ 2 ε 26 7 ) 7560 ε 26 4 + 14641 ε 59 6 ε 26 18 20406505568357744640000 , ϱ ( t ) = 136080 β ( t ) ε 26 4 ε 59 , θ 7 = 0 .

Substituting Eqs. (10) and (11) into Eq. (2), the six-order rogue wave solution of Eq. (1) is obtained as follows:

(12) κ = x + η z Θ ( t ) , u = 2 [ ln ψ ] κ κ , ψ = ψ ( κ , y ) .

When θ 3 = θ 4 = θ 6 = θ 2 = 1 in Eq. (12), we can see the interaction between six rogue waves in Figure 6. When θ 3 = θ 6 = θ 2 = 1 , θ 4 = t in Eq. (12), we can see the interaction between eight rogue waves in Figure 7. Figure 6

η = 1 , ε 0 = 2 , ε 1 = ε 59 = ε 26 = 1 , ρ = ϱ = 10 , x = z = 0 , (a) 3D plot and (b) contour plot. Figure 7

η = 1 , ε 0 = 2 , ε 1 = ε 59 = ε 26 = 1 , ρ = ϱ = 10 , x = z = 0 , (a) 3D plot and (b) contour plot.

## 5 Conclusion

In this work, a generalized (3+1)-dimensional vcKPe are studied. Based on the modified symbolic computation approach and symbolic computation [16,17,18, 19,20,21, 22,23,24, 25,26,27, 28,29,30, 31,32,33, 34,35,36, 37,38,39, 40,41,42, 43,44], we obtain the multiple rogue wave solutions of the generalized (3+1)-dimensional vcKPe. By selecting different functions of the variable coefficient, the dynamic properties of the derived solutions are shown in Figures 17. In Figure 1, we can find a rogue wave spreading. When θ 4 = t , two rogue waves can be seen in Figure 2. When θ 4 and θ 6 are selected as trigonometric functions, a periodic-type rogue wave is shown in Figure 3. In Figure 4, we can observe the interaction between three rogue waves. The interaction between five rogue waves is shown in Figure 5. Further, the interaction between six rogue waves is shown in Figure 6, and the interaction between eight rogue waves is shown in Figure 7.

1. Funding information: The authors state no funding involved.

2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

3. Conflict of interest: The authors state no conflict of interest.

4. Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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