# Rogue Wave and Multiple Lump Solutions of the (2+1)-Dimensional Benjamin-Ono Equation in Fluid Mechanics

Authors: Zhonglong Zhao; Lingchao He; Yubin Gao
Journal: Complexity
 (2019)

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## Abstract

In this paper, the bilinear method is employed to investigate the rogue wave solutions and the rogue type multiple lump wave solutions of the (2+1)-dimensional Benjamin-Ono equation. Two theorems for constructing rogue wave solutions are proposed with the aid of a variable transformation. Four kinds of rogue wave solutions are obtained by means of Theorem 1. In Theorem 2, three polynomial functions are used to derive multiple lump wave solutions. The 3-lump solutions, 6-lump solutions, and 8-lump solutions are presented, respectively. The 3-lump wave has a “triangular” structure. The centers of the 6-lump wave form a pentagram around a single lump wave. The 8-lump wave consists of a set of seven first order rogue waves and one second order rogue wave as the center. The multiple lump wave develops into low order rogue wave as parameters decline to zero. The method presented in this paper provides a uniform method for investigating high order rational solutions. All the results are useful in explaining high dimensional dynamical phenomena of the (2+1)-dimensional Benjamin-Ono equation.


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## Body

## 1. Introduction

Rogue wave is an isolated huge wave, which plays an important role in analyzing many science problems, such as ocean’s waves [1–3], optical fibers [4], Bose-Einstein condensates [5, 6], and financial markets [7, 8]. The Darboux transformation method [9, 10] is one of the most effective methods to construct the rogue wave solutions of the integrable systems. Rogue wave solutions are an interesting class of lump-type solutions [11]. It is interesting that lump functions can provide approximate prototypes to model rogue waves [12]. Based on the bilinear method, the bilinear forms can be used to construct the lump solutions by choosing appropriate multivariate positive substitution quadratic functions. In 2015, Ma investigated the lump solutions of the (2+1)-dimensional Kadomtsev-Petviashvili equation by means of the bilinear forms and positive quadratic functions [13]. Inspired by this work, many researchers studied the lump solutions of the integrable systems, such as the (2+1)-dimensional Sawada-Kotera equation [14], the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation [15], the (3+1)-dimensional B-type KP equation [16], and the generalized Calogero-Bogoyavlenskii-Schiff equation [17]. Furthermore, the generalized bilinear forms can also be used to derive the lump solutions [18].Searching for the interaction solutions between the lump wave and other types of wave is one of the hot issues. There are many methods to investigate the lump solutions and the interaction solutions among solitons, such as the inverse scattering transformation [19], the Painlevé analysis [20–22], the symmetry analysis [23–27], and the Darboux transformation [28, 29]. Fokas et al. studied the completely elastic collision between a lump wave and a line soliton of the DSII equation [30]. Tang et al. investigated the nonelastic collision between a lump wave and a stripe soliton of the (2+1)-dimensional Ito equation [31]. Then the interaction lump-soliton solutions [32, 33], lump-twin-stripe solutions [34], and lump-periodic wave solutions [35] were investigated.Compared with the construction of single lump wave solutions, multiple wave solutions are more difficult to obtain. Zhang et al. obtained the multiple lump solutions of a (3+1)-dimensional nonlinear evolution equation by taking a “long wave” limit for the corresponding N-solitons [36]. The dynamical behaviors of the interactions of multilumps within the Kadomtsev-Petviashvili-1 equation were analyzed by numerical simulation [37]. The multiple lump solutions of a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation in incompressible fluid were derived [38]. The high-order lumps of the (3+1)-dimensional KP-Boussinesq equation were presented by using Hirota’s bilinear method [39]. Zhaqilao et al. proposed a new method to construct the multiple lump solutions with a controllable center [40, 41]. We constructed the multiple lump solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation [42].In this paper, we consider the (2+1)-dimensional Benjamin-Ono equation(1)utt+au2xx+buxxxx+uyy=0,where u=ux,y,t. Eq.(1) is the reduction of a generalized (2+1)-dimensional Boussinesq equation [43, 44]. The (2+1)-dimensional Benjamin-Ono equation arises in the study of internal waves in deep stratified fluids [45].The organization of the paper is as follows. In Section2, multiple rogue wave solutions of (1) are constructed with the aid of bilinear method. A theorem is proposed to establish a unified approach to study multiple rogue wave solutions. In Section 3, three polynomial functions are introduced to construct the rogue type multiple lump wave solutions. The 3-lump, 6-lump, and 8-lump solutions are investigated in Sections 4, 5, and 6. Finally some conclusions are given in the last section.


## 2. Multiple Rogue Wave Solutions

In this section, we shall investigate rogue wave solutions of (1). Eq.(1) can be reduced to the following equation:(2)m2uXX+au2XX+buXXXX+uyy=0,by setting X=x+mt. With the aid of variable transformation(3)u=6baln⁡fXX, (2) can be reduced to the bilinear equation(4)m2ffXX-fX2+bffXXXX-4fXfXXX+3fXX2+ffyy-fy2=0.This bilinear form can be used to construct multiple rogue wave solutions of (1).Theorem 1.
(2+1)-Dimensional Benjamin-Ono equation (1) has the multiple rogue wave solutions(5)unx,y,t=6baln⁡fXX,where(6)fX,y=FnX,y=∑k=0nn+1/2∑i=0kann+1-2k,2iy2iXnn+1-2k,X=x+mt, fX,y is the solution of bilinear equation (4) and the coefficients ann+1-2k,2i are the parameters to be determined.
Theorem1 provides a uniform method for constructing the multiple rogue wave solutions of (1). Using this method we obtain the following polynomials:(7)
F
1
X
,
y
=
X
2
+
m
2
y
2
-
3
b
m
2
,
F
2
X
,
y
=
X
6
-
25
b
X
4
m
2
+
3
m
2
y
2
X
4
+
-
125
b
2
m
4
-
90
b
y
2
+
3
m
4
y
4
X
2
-
1875
b
3
m
6
+
475
b
2
y
2
m
2
-
17
b
m
2
y
4
+
m
6
y
6
,
F
3
X
,
y
=
X
12
+
-
98
b
m
2
+
6
m
2
y
2
X
10
+
735
b
2
m
4
-
690
b
y
2
+
15
m
4
y
4
X
8
+
-
75460
3
b
3
m
6
+
18620
b
2
y
2
m
2
-
1540
b
m
2
y
4
+
20
m
6
y
6
X
6
+
-
5187875
3
b
4
m
8
-
220500
b
3
y
2
m
4
+
37450
b
2
y
4
-
1460
b
m
4
y
6
+
15
m
8
y
8
X
4
+
-
159786550
3
b
5
m
10
+
565950
b
4
y
2
m
6
+
14700
b
3
y
4
m
2
+
35420
m
2
b
2
y
6
-
570
b
m
6
y
8
+
6
m
10
y
10
X
2
+
878826025
9
b
6
m
12
-
300896750
3
b
5
y
2
m
8
+
16391725
3
b
4
y
4
m
4
-
798980
3
b
3
y
6
+
4335
b
2
m
4
y
8
-
58
b
m
8
y
10
+
m
12
y
12
,
F
4
X
,
y
=
X
20
-
270
b
X
18
m
2
+
10
m
2
y
2
X
18
+
16605
b
2
m
4
-
3030
b
y
2
+
45
m
4
y
4
X
16
+
-
351000
b
3
m
6
+
275400
b
2
y
2
m
2
-
13800
b
m
2
y
4
+
120
m
6
y
6
X
14
+
-
18877950
b
4
m
8
-
10621800
b
3
y
2
m
4
+
1367100
b
2
y
4
-
34440
b
m
4
y
6
+
210
m
8
y
8
X
12
+
-
2094264900
b
5
m
10
+
107534700
b
4
y
2
m
6
-
56586600
b
3
y
4
m
2
+
3308760
m
2
b
2
y
6
-
52500
b
m
6
y
8
+
252
m
10
y
10
X
10
+
-
178095030750
b
6
m
12
-
4871002500
b
5
y
2
m
8
+
1071960750
b
4
y
4
m
4
-
135286200
b
3
y
6
+
4513950
b
2
m
4
y
8
-
50820
b
m
8
y
10
+
210
m
12
y
12
X
8
+
-
6967194507000
b
7
m
14
-
521628471000
b
6
y
2
m
10
-
636363000
b
5
y
4
m
6
+
3824793000
b
4
y
6
m
2
-
151237800
m
2
b
3
y
8
+
3601080
b
2
m
6
y
10
-
31080
b
m
10
y
12
+
120
m
14
y
14
X
6
+
190578711448125
b
8
m
16
-
33286514625000
b
7
y
2
m
12
-
405853402500
b
6
y
4
m
8
-
45237339000
b
5
y
6
m
4
+
2667498750
b
4
y
8
-
94613400
m
4
b
3
y
10
+
1619100
b
2
m
8
y
12
-
11400
b
m
12
y
14
+
45
m
16
y
16
X
4
+
696163557521250
b
9
m
18
+
870343420196250
b
8
y
2
m
14
-
90898176915000
b
7
y
4
m
10
+
1982064357000
b
6
y
6
m
6
-
31477666500
b
5
y
8
m
2
+
671510700
m
2
b
4
y
10
-
35645400
m
6
b
3
y
12
+
354600
b
2
m
10
y
14
-
2190
b
m
14
y
16
+
10
m
18
y
18
X
2
+
5917390238930625
b
10
m
20
-
3474517664913750
b
9
y
2
m
16
+
348683786758125
b
8
y
4
m
12
-
43199536653000
b
7
y
6
m
8
+
1200881855250
b
6
y
8
m
4
-
21813668100
b
5
y
10
+
360709650
b
4
m
4
y
12
-
3299400
m
8
b
3
y
14
+
23085
b
2
m
12
y
16
-
150
m
16
b
y
18
+
m
20
y
20
, which corresponded to the rogue wave solutions u1,u2,u3, and u4.Figures1, 2, 3, and 4 show the dynamical behaviors of the rogue wave solutions u1,u2,u3, and u4, respectively.Figure 1
Plots (colored online) of rogue waveu1 with m=2,a=b=-2. (a) Perspective view of the wave ux,y,0, (b) perspective view of the wave u0,y,t, and (c) perspective view of the wave ux,0,t.
(a) (b) (c)Figure 2
Plots (colored online) of rogue waveu2 with m=2,a=b=-2. (a) Perspective view of the wave ux,y,0, (b) perspective view of the wave u0,y,t, and (c) perspective view of the wave ux,0,t.
(a) (b) (c)Figure 3
Plots (colored online) of rogue waveu3 with m=2,a=b=-2. (a) Perspective view of the wave ux,y,0, (b) perspective view of the wave u0,y,t, and (c) perspective view of the wave ux,0,t.
(a) (b) (c)Figure 4
Plots (colored online) of rogue waveu4 with m=2,a=b=-2. (a) Perspective view of the wave ux,y,0, (b) perspective view of the wave u0,y,t, and (c) perspective view of the wave ux,0,t.
(a) (b) (c)Figures1(a) and 1(b) show that the rogue wave u1 has one peak which is higher than the water level. This kind of rogue wave is called lump wave that maintains the localization property in the planes x,y. The lump wave reaches the higher peak at the point -mt,0. The lump wave Figures 1(a) and 1(b) have two lower peaks. The coordinates of the lower peaks are -mt+3-b/m,0 and -mt-3-b/m,0. The center of the lump wave in the plane x,y is -mt,0. The one order rogue wave u1 develops into one-soliton solution in Figure 1(c).If we takem=2,α=β=-2,n=4 in rogue wave u2, we obtain the second order rogue wave described in Figures 2(a) and 2(b). The rogue wave u2 has the structure of two-line solitons in the plane x,t. Figures 3 and 4 plot the rogue waves u3 and u4. These show that the number of the higher peaks of the multiple rogue wave solutions is equivalent to the subscript of un. From Figures 1(a), 2(a), 3(a), and 4(a), we can conclude that the maxima of the solutions un all lie on the line y=0. Figures 3(c) and 4(c) exhibit the fact that the rogue wave un develops into the multiple soliton solutions in the plane x,t.


## 3. Rogue Type Multiple Lump Wave Solutions

Theorem 2.
(2+1)-Dimensional Benjamin-Ono equation (1) has the rogue type multiple lump wave solutions(8)ux,y,t=6baln⁡fXX,where(9)fX,y=Fn+1X,y+2αyPnX,y+2βXQnX,y+α2+β2Fn-1X,y,n≥2,(10)FnX,y=∑k=0nn+1/2∑i=0kann+1-2k,2iy2iXnn+1-2k,PnX,y=∑k=0nn+1/2∑i=0kbnn+1-2k,2iX2iynn+1-2k,QnX,y=∑k=0nn+1/2∑i=0kcnn+1-2k,2iy2iXnn+1-2k,X=x+mt, fX,y is the solution of bilinear equation (4), the coefficients ann+1-2k,2i, bnn+1-2k,2i and cnn+1-2k,2i are the parameters to be determined, and α and β are arbitrary constants.Remark 3.
Theorem2 gives a uniform method for constructing the rogue type multiple lump wave solutions of (1). One can obtain the multiple wave solutions more than six lump waves by selecting n≥3. The parameters α and β determine the shape of the multiple lump wave. The multiple lump wave tends to a low order rogue wave as α2+β2 declines to 0. All peaks of the lump wave tend to the same height when α2+β2 is sufficiently large. In what follows, we shall construct the multiple lump solutions of (1).


## 4. 3-Lump Solutions

Based on Theorem2, we construct the 3-lump solutions with the aid of following polynomial functions:(11)fX,y=F2X,y+2αyP1X,y+2βXQ1X,y+α2+β2,=X6+a4,0X4+a4,2X4y2+a2,0+a2,2y2+a2,4y4X2+a0,0+a0,2y2+a0,4y4+a0,6y6+2αyb0,0+b2,0y2+b0,2X2+2βXc0,0+c0,2y2+c2,0X2+α2+β2,Substituting (11) into (4) and setting all the coefficients of the different polynomials of X and y to zero yield a set of determining equations. Solving these equations, one has(12)a0,0=-16875b3+9m6α2+9m6β2-α2m4b0,22-9m6β2c2,029m6,a0,2=475b2m2,a0,4=-17bm2,a0,6=m6,a2,0=-125b2m4,a2,2=-90b,a2,4=3m4,a4,0=-25bm2,a4,2=3m2,b0,0=-5b0,2b3m2,b2,0=-m2b0,23,c0,0=c2,0bm2,c0,2=-3c2,0m2.Then we obtain the 3-lump solutions(13)ux,y,t=6baln⁡fXX,where X=x+mt, the coefficients of f are determined by (12), and m, a, b, α, and β are arbitrary real constants. Figure 5 plots the 3-lump wave (13) when m=2,a=b=-2, b0,2=c2,0=1, α=3000,β=4000. The 3-lump wave is the arrangement of three first order rogue waves in the planes x,y and y,t, which forms a structure of triangular. From Figures 5(c) and 5(f), we observe that 3-lump wave solution (13) develops into two-soliton solution in the plane x,t.Figure 5
Plots (colored online) of 3-lump waveu (13) with m=2,a=b=-2, b0,2=c2,0=1, α=3000,β=4000. (a) Perspective view of the wave ux,y,0, (b) perspective view of the wave u0,y,t, (c) perspective view of the wave ux,0,t, (d) overhead view of the wave (a), (e) overhead view of the wave (b), and (f) overhead view of the wave (c).
(a) (b) (c) (d) (e) (f)


## 5. 6-Lump Solutions

In order to obtain the 6-lump solutions of (1), we choose(14)fX,y=F3X,y+2αyP2X,y+2βXQ2X,y+α2+β2F1X,y,where(15)F3X,y=X12+a10,0+a10,2y2X10+a8,0+a8,2y2+a8,4y4X8+a6,0+a6,2y2+a6,4y4+a6,6y6X6+a4,0+a4,2y2+a4,4y4+a4,6y6+a4,8y8X4+a2,0+a2,2z2+a2,4y4+a2,6y6+a2,8y8+a2,10y10X2+a0,0+a0,2y2+a0,4y4+a0,6y6+a0,8y8+a0,10y10+a0,12y12,P2X,y=b0,0+b2,0+b2,2X2+b2,4X4y2+b4,0+b4,2X2y4+y6+b0,2X2+b0,4X4+b0,6X6,Q2X,y=c0,0+c0,2y2+c0,4y4+c0,6y6+c2,0+c2,2y2+c2,4y4X2+c4,0+c4,2y2X4+X6,F1X,y=X2+a0,2y2+a0,0.Substituting (14) into (4) and setting all the coefficients of the different polynomials of X and y to zero yield a set of determining equations. Solving these equations, one has(16)a0,0=b878826025m4b2-27α2-27m14β29m16α2+β2+1,a0,2=-300896750m4b5-3α2-3m14β23m12α2+β2+1,a0,4=16391725b43m4,a0,6=-798980b33,a0,8=4335b2m4,a0,10=-58bm8,a0,12=m12,a2,0=--3α2+3m14α2+159786550m4b53m14,a2,2=565960b4m6,a2,4=14700b3m2,a2,6=35420b2m2,a2,8=-570bm6,a2,10=6m10,a4,0=-5187875b43m8,a4,2=-220500b3m4,a4,4=37450b2,a4,6=-1460bm4,a4,8=15m8,a6,0=-75460b33m6,a6,2=18620b2m2,a6,4=-1540bm2,a6,6=20m6,a8,0=735b2m4,a8,2=-690b,a8,4=15m4,a10,0=--98bm2,a10,2=6m2,b0,0=-18865b33m12,b0,2=-665b2m10,b0,4=-105bm8,b0,6=5m6,b2,0=-245b2m8,b2,2=190bm6,b2,4=-5m4,b4,0=7bm4,b4,2=-9m2,c0,0=-12005b33m6,c0,2=535b2m2,c0,4=-45bm2,c0,6=5m6,c2,0=-245b2m4,c2,2=230b,c2,4=-5m4,c4,0=-13bm2,c4,2=-9m2.Then we obtain the 6-lump solutions(17)ux,y,t=6baln⁡fXX,where X=x+mt and the coefficients of f are determined by (16), m, a, b, α, and β are arbitrary real constants.Figures6(a) and 6(e) plot 6-lump wave (17) when m=2,a=b=-2, α=8,β=10. This case illustrates that as α2+β2 tends to 0 the six lump wave becomes a third order rogue wave u3 in Section 2. Figures 6(b), 6(c), 6(f), and 6(g) exhibit the fact that three more peaks appear as α and β increase. As α and β become more large, six lump waves appear and all the centers of the single lump waves form a structure of pentagram. Figure 6(d) shows that all the peaks of the 6-lump wave tend to the same hight when α2+β2 is sufficiently large. Figure 7 plots the 6-lump wave (17) when m=2,a=b=-2, α=8×105,β=10×105. The corresponding 6-lump wave is composed of six single lump waves, which array a pentagon. Figure 7(c) shows that the 6-lump wave solutions have the structure of three solitons in the plane x,t.Figure 6
Plots (colored online) of 6-lump waveu (17) with m=2,a=b=-2. (a) Perspective view of the wave ux,y,0 with α=8,β=10, (b) perspective view of the wave ux,y,0 with α=800,β=1000, (c) perspective view of the wave ux,y,0 with α=3000,β=4000, (d) perspective view of the wave ux,y,0 with α=3×104,β=4×104, (e) overhead view of the wave (a), (f) overhead view of the wave (b), (g) overhead view of the wave (c), and (h) overhead view of the wave (d).
(a) (b) (c) (d) (e) (f) (g) (h)Figure 7
Plots (colored online) of 6-lump waveu (17) with m=2,a=b=-2, α=8×105,β=10×105. (a) Perspective view of the wave ux,y,0, (b) perspective view of the wave u0,y,t, (c) perspective view of the wave ux,0,t, (d) overhead view of the wave (a), (e) overhead view of the wave (b), and (f) overhead view of the wave (c).
(a) (b) (c) (d) (e) (f)


## 6. 8-Lump Solutions

To find the 8-lump solutions of (1), we investigate polynomial function solutions to (4) with the assumption(18)fX,y=F4X,y+2αyP3X,y+2βXQ3X,y+α2+β2F2X,y,where(19)
F
4
X
,
y
=
X
20
+
a
18,0
+
a
18,2
y
2
X
18
+
a
16,0
+
a
16,2
y
2
+
a
16,4
y
4
X
16
+
a
14,0
+
a
14,2
y
2
+
a
14,4
y
4
+
a
14,6
y
6
X
14
+
a
12,0
+
a
12,2
y
2
+
a
12,4
y
4
+
a
12,6
y
6
+
a
12,8
y
8
X
12
+
a
10,0
+
a
10,2
y
2
+
a
10,4
y
4
+
a
10,6
y
6
+
a
10,8
y
8
+
a
10,10
y
10
X
10
+
a
8,0
+
a
8,2
y
2
+
a
8,4
y
4
+
a
8,6
y
6
+
a
8,8
y
8
+
a
8,10
y
10
+
a
8,12
y
12
X
8
+
a
6,0
+
a
6,2
y
2
+
a
6,4
y
4
+
a
6,6
y
6
+
a
6,8
y
8
+
a
6,10
y
10
+
a
6,12
y
12
+
a
6,14
y
14
X
6
+
a
4,0
+
a
4,2
y
2
+
a
4,4
y
4
+
a
4,6
y
6
+
a
4,8
y
8
+
a
4,10
y
10
+
a
4,12
y
12
+
a
4,14
y
14
+
a
4,16
y
16
X
4
+
a
2,0
+
a
2,2
y
2
+
a
2,4
y
4
+
a
2,6
y
6
+
a
2,8
y
8
+
a
2,10
y
10
+
a
2,12
y
12
+
a
2,14
y
14
+
a
2,16
y
16
+
a
2,18
y
18
X
2
+
a
0,0
+
a
0,2
y
2
+
a
0,4
y
4
+
a
0,6
y
6
+
a
0,8
y
8
+
a
0,10
y
10
+
a
0,12
y
12
+
a
0,14
y
14
+
a
0,16
y
16
+
a
0,18
y
18
+
a
0,20
y
20
,
P
3
X
,
y
=
b
12,0
y
12
+
b
10,0
+
b
10,2
X
2
y
10
+
b
8,0
+
b
8,2
X
2
+
b
8,4
X
4
y
8
+
b
6,0
+
b
6,2
X
2
+
b
6,4
X
4
+
b
6,6
X
6
y
6
+
b
4,0
+
b
4,2
X
2
+
b
4,4
X
4
+
b
4,6
X
6
+
b
4,8
X
8
y
4
+
b
2,0
+
b
2,2
X
2
+
b
2,4
X
4
+
b
2,6
X
6
+
b
2,8
X
8
+
b
2,10
X
10
y
2
+
b
0,0
+
b
0,2
X
2
+
b
0,4
X
4
+
b
0,6
X
6
+
b
0,8
X
8
+
b
0,10
X
10
+
b
0,12
X
12
,
Q
3
X
,
y
=
c
12,0
X
12
+
c
10,0
+
c
10,2
y
2
X
10
+
c
8,0
+
c
8,2
y
2
+
c
8,4
y
4
X
8
+
c
6,0
+
c
6,2
y
2
+
c
6,4
y
4
+
c
6,6
y
6
X
6
+
c
4,0
+
c
4,2
y
2
+
c
4,4
y
4
+
c
4,6
y
6
+
c
4,8
y
8
X
4
+
c
2,0
+
c
2,2
y
2
+
c
2,4
y
4
+
c
2,6
y
6
+
c
2,8
y
8
+
c
2,10
y
10
X
2
+
c
0,0
+
c
0,2
y
2
+
c
0,4
y
4
+
c
0,6
y
6
+
c
0,8
y
8
+
c
0,10
y
10
+
c
0,12
y
12
,
F
2
X
,
y
=
X
6
+
a
4,0
+
a
4,2
y
2
X
4
+
a
2,0
+
a
2,2
y
2
+
a
2,4
y
4
X
2
+
a
0,0
+
a
0,2
y
2
+
a
0,4
y
4
+
a
0,6
y
6
, Submitting (18) into (4), we obtain a set of constraining equations for the parameters (20)a0,0=a18,035727449317681a18,07m2+3874204890000000α2b0,122+189836039610000000m2β2c12,021992822809409936000000m21+α2+β2,a0,2=a18,02490732619400000α2b0,122+1681494968699a18,07m2+24045898350600000m2β2c12,0236904126100184000001+α2+β2,a0,4=m2a18,0337491467677a18,07m2+35126124336000α2b0,122+1721180092464000m2β2c12,02273363897038400001+α2+β2,a0,6=m421092893290000m2β2c12,02+8711017679a18,07m2+430467210000α2b0,122210928932900001+α2+β2,a0,8=17790842357395628000a18,06m8,a0,10=242374091594323000m10a18,05,a0,12=80157711809800m12a18,04,a0,14=6113645m14a18,03,a0,16=1960m16a18,02,a0,18=59m18a18,0,a0,20=m20,a2,0=a18,02336908783393a18,07m2+129140163000000α2b0,122+6327867987000000m2β2c12,023690412610018400000m21+α2+β2,a2,2=a18,0421203235453a18,07m2+92980917360000α2b0,122+4556064950640000m2β2c12,02136681948519200001+α2+β2,a2,4=37967207922000m2β2c12,02+10997557207a18,07m2+774840978000α2b0,122m2126557359740001+α2+β2,a2,6=7340979114348907000m6a18,06,a2,8=2331679106288200m8a18,05,a2,10=7461235904900m10a18,04,a2,12=66013645m12a18,03,a2,14=39481m14a18,02,a2,16=739m16a18,0,a2,18=10m18,a4,0=a18,0184461370093a18,07m2+51656065200000α2b0,122+2531147194800000m2β2c12,0227336389703840000m21+α2+β2,a4,2=161090293a18,07m2+30993639120α2b0,122+1518688316880m2β2c12,025062294389601+α2+β2,a4,4=-60126435739562800a18,06m4,a4,6=167545753144100m6a18,05,a4,8=79037157464m8a18,04,a4,10=175213645a18,03m10,a4,12=179981a18,02m12,a4,14=3809m14a18,0,a4,16=45m16,a6,0=-63278679870000m2α2-1291401630000α2b0,122-63278679870000m2β2c12,02+63278679870000m2β2-4214722603a18,07m2163278679870000m2,a6,2=-1931957314348907000a18,06m2,a6,4=2356953144100m4a18,05,a6,6=141659196830m6a18,04,a6,8=280073645m8a18,03,a6,10=20006405m10a18,02,a6,12=10369m12a18,0,a6,14=120m14,a8,0=-2638444957395628000a18,06,a8,2=7216321257640a18,05m2,a8,4=158809787320m4a18,04,a8,6=83511215a18,03m6,a8,8=10031162m8a18,02,a8,10=16949m10a18,0,a8,12=210m12,a10,0=23269611594323000a18,05,a10,2=1194835904900a18,04m2,a10,4=34931215m4a18,03,a10,6=18382405m6a18,02,a10,8=17509m8a18,0,a10,10=252m10,a12,0=-4195111809800a18,04,a12,2=19673645a18,03m2,a12,4=151981m4a18,02,a12,6=11489m6a18,0,a12,8=210m8,a14,0=13729a18,03,a14,2=349m2a18,02,a14,4=4609m4a18,0,a14,6=120m6,a16,0=41180a18,02,a16,2=1019m2a18,0,a16,4=45m4,a18,2=10m2,b0,0=37349344373768000b0,12a18,06,b0,2=933013188646000b0,12a18,05,b0,4=-609723619600b0,12a18,04,b0,6=2321870b0,12a18,03,b0,8=5972b0,12a18,02,b0,10=13b0,12a18,0,b2,0=-118370858800b0,12a18,05m2,b2,2=524311809800b0,12a18,04m2,b2,4=-13121870b0,12a18,03m2,b2,6=-23243b0,12a18,02m2,b2,8=-m2b0,12a18,0,b2,10=-2m2b0,12,b4,0=-703323619600m4b0,12a18,04,b4,2=20921870m4b0,12a18,03,b4,4=-113486m4b0,12a18,02,b4,6=-14645b0,12m4a18,0,b4,8=-9m4b0,12,b6,0=-19153090m6b0,12a18,03,b6,2=13078505m6b0,12a18,02,b6,4=2663m6b0,12a18,0,b6,6=-367m6b0,12,b8,0=12268m8b0,12a18,02,b8,2=131189b0,12a18,0m8,b8,4=257m8b0,12,b10,0=-1189m10b0,12a18,0.b10,2=187m10b0,12,b12,0=-17m12b0,12,c0,0=-1999271721868840000c12,0a18,06,c0,2=-8780033188646000c12,0a18,05m2,c0,4=-8432923619600m4a18,04c12,0,c0,6=-772430m6a18,03c12,0,c0,8=-203972m8a18,02c12,0,c0,10=-4945c12,0a18,0m10,c0,12=-7m12c12,0,c2,0=579671062882000a18,05c12,0,c2,2=75112361960c12,0a18,04m2,c2,4=1612430m4a18,03c12,0,c2,6=497243m6a18,02c12,0,c2,8=7c12,0a18,0m8,c2,10=14m10c12,0,c4,0=-19814723920c12,0a18,04,c4,2=-37121870c12,0a18,03m2,c4,4=-49486m4c12,0a18,02,c4,6=1829c12,0a18,0m6,c4,8=63m8c12,0,c6,0=4721870c12,0a18,03,c6,2=-4431215a18,02c12,0m2,c6,4=-589c12,0m4a18,0,c6,6=36m6c12,0,c8,0=-11972a18,02c12,0,c8,2=-18727m2a18,0c12,0,c8,4=-25m4c12,0,c10,0=37135c12,0a18,0,c10,2=-18m2c12,0,b=-1270m2a18,0.This set leads to a 8-lump solution to (1)(21)ux,y,t=6baln⁡fXX,where X=x+mt, the coefficients of f are determined by the above constraining equations, and m, a, b, α, and β are arbitrary real constants. When setting m=2,a=-2, α=3×109,β=4×109, b0,12=c12,0=1,a18,0=600, we find that solution (21) (Figures 8(a), 8(b), 8(d), and 8(e)) possesses eight lump waves, namely, seven single lump waves and one second order rogue wave u2 (see Figures 2(a) and 2(b)). The seven single lump waves array a heptagon around the second order rogue wave. If we set y=0, the 8-lump solution develops into four-soliton solution, which is plotted in Figures 8(c) and 8(f). Figures 9(a) and 9(e) show that the 8-lump wave develops into fourth order rogue wave u4 of Section 2 when α2+β2 declines to 0. From Figures 9(b)–9(d) and 9(f)–9(h), we can observe that the fouth order rogue wave splits into a set of seven first order rogue waves and one second order rogue as the center with parameters α and β gradually become large enough.Figure 8
Plots (colored online) of 8-lump waveu (21) with m=2,a=-2, α=3×109,β=4×109, b0,12=c12,0=1,a18,0=600. (a) Perspective view of the wave ux,y,0, (b) perspective view of the wave u0,y,t, (c) perspective view of the wave ux,0,t, (d) overhead view of the wave (a), (e) overhead view of the wave (b), and (f) overhead view of the wave (c).
(a) (b) (c) (d) (e) (f)Figure 9
Plots (colored online) of 8-lump waveu (21) with m=2,a=-2, b0,12=c12,0=1,a18,0=600. (a) Perspective view of the wave ux,y,0 with α=30,β=40, (b) perspective view of the wave ux,y,0 with α=1×107,β=2×107, (c) perspective view of the wave ux,y,0 with α=5×107,β=7×107, (d) perspective view of the wave ux,y,0 with α=8×108,β=1×109, (e) overhead view of the wave (a), (f) overhead view of the wave (b), (g) overhead view of the wave (c), and (h) overhead view of the wave (d).
(a) (b) (c) (d) (e) (f) (g) (h)


## 7. Conclusions

In this paper, we focus on investigating the rogue wave solutions and the rogue type multiple lump solutions of the (2+1)-dimensional Benjamin-Ono equation by means of the bilinear method. Based on the reduced bilinear equations, four kinds of rogue wave solutions are obtained. The number of the higher peaks of the multiple rogue wave solutions is equivalent to the subscript ofun. The maxima of the solutions un all lie on the line y=0. The multiple rogue waves develop into the multiple stripe solitons in the plane t,x. A theorem for constructing multiple lump solutions has been presented by introducing three polynomial functions Fn, Pn, and Qn (10) of degree nn+1. The multiple lump waves tend to the low order rogue waves as the parameter α2+β2 is near 0. Numerical experiments show that the multiple lump solutions will produce new peaks with the increase of parameters. The number of the lumps will be fixed and all peaks will tend to the same height when α2+β2 is sufficiently large. The method presented in this paper shows that Hirota’s bilinear method is a powerful approch to construct the high order rational solutions to nonlinear integrable equations. Based on the bilinear forms, one can construct different types of rational solutions by choosing different polynomial functions. The results might be helpful in explaining nonlinear wave phenomena in fluid mechanics.



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*Source: 8249635-2019-08-07.xml*