Physics Letters A 449 (2022) 128355

Contents lists available at ScienceDirect

Physics Letters A
www.elsevier.com/locate/pla

A direct method for generating rogue wave solutions to the

(3+1)-dimensional Korteweg-de Vries Benjamin-Bona-Mahony
equation
Xiangyu Yang a , Zhao Zhang b , Abdul-Majid Wazwaz c , Zhen Wang a,∗
a
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
b
Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou, 510631, China
c
Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA

a r t i c l e i n f o a b s t r a c t

Article history: In this paper, we provide a generating mechanism to obtain rogue wave solutions from N soliton of
Received 17 June 2022 Hirota’s bilinear method. Based on the long wave limit method, the phase parameters are reconstructed
Received in revised form 25 July 2022 to generating rogue wave solutions of Korteweg–de Vries Benjamin-Bona-Mahony equation. The rogue
Accepted 31 July 2022
wave solutions are expressed explicitly in rational forms. Besides fundamental pattern of rogue waves,
Available online 5 August 2022
Communicated by B. Malomed
the triangle or pentagon pattern have been revealed. Both types of triangle and pentagon patterns are
resolved upon different choices of free parameters introduced.
Keywords: © 2022 Elsevier B.V. All rights reserved.
Rogue wave
Soliton
Hirota bilinear method
KdV-BBM equation

1. Introduction

Rogue waves are localized in both space and time while showing a large amplitude [1], which related to giant single waves appearing
in the ocean and “appear from nowhere”. Being unforeseen and massive, they are capable of leading to extreme damage to the affairs of
people [2]. Therefore, it is very important to study the emerging mechanism of rogue waves, and it also attracted significant attention
of many researchers [3–5]. Rogue waves have also been studied a lot in other areas, such as internal waves [6], nonlinear fiber optics
[7], surface plasma [8], microwaves [9], superfluids [10], Bose–Einstein condensates [11], inhomogeneous media [12]. These experimental
developments promoting theoretical studies. The mathematical study of rogue waves can be considered as the rational solution of the
Nonlinear Schrödinger equation (NLS), which is known as Peregrine soliton [13]. In 2009, Akhmediev et al. present explicit expressions
for rogue wave solutions from first to fourth order of NLS [14]. After that, many integrable equations were shown to exhibit the rogue
wave-type rational solutions, such as Ablowitz–Ladik equations [15], Davey-Stewartson I equation [16], coupled Maxwell–Bloch equation
[17], Boussinesq equation [18], complex modified Korteweg-de Vries equation [19], Sasa–Satsuma equation [20] and nonlocal Maccari
system [21]. Their methods are all about Darboux Transformation or Kadomtsev–Petviashvili hierarchy reduction method. Compared with
the Hirota’s direct method [22–24], they have more complex process and not easy to derive rogue wave solution. So how to derive rogue
wave solutions from N-soliton solution directly is an meaningful question.
As the rogue waves solution is rational solution. The classic method to obtain rational solution is the long wave limit method [25].
It only needs performing an appropriate limiting procedure on the soliton solutions of Hirota’s bilinear methods, without tedious trans-
formations [26]. Due to its universality, rational solutions of many (1+1) and (2+1) nonlinear evolution equations have been investigated
[25,27–29]. The new multisoliton solution (rogue waves like) of KP equation has been reported in [30] based on such limiting method.
Method proposed in this work is motivated by the above works.
In this work, we considering rogue wave solutions of the following (3 + 1)-dimensional Korteweg-de Vries Benjamin–Bona–Mahony
(KdV-BBM) equation [31]

* Corresponding author.
E-mail address: wangzmath@163.com (Z. Wang).

https://doi.org/10.1016/j.physleta.2022.128355
0375-9601/© 2022 Elsevier B.V. All rights reserved.
X. Yang, Z. Zhang, A.-M. Wazwaz et al. Physics Letters A 449 (2022) 128355

u xt + μ1 (uu x )x + μ2 u xxxx − μ3 u xxxt + μ4 u y y − μ5 u zz = 0, (1)

where the coefficients μ1 , μ2 , μ3 , μ4 , μ5 are real constants. It can be reduced to several famous integrable equations by different choices
of the coefficients. When μ3 = μ4 = μ5 = 0, Eq. (1) reduces to KdV equation

ut + μ1 uu x + μ2 u xxx = 0. (2)

When μ3 = μ5 = 0, Eq. (1) reduces to KP equation

u xt + μ1 (uu x )x + μ2 u xxxx + μ4 u y y = 0. (3)

When μ3 = 0, Eq. (1) reduces to 3D KP equation

u xt + μ1 (uu x )x + μ2 u xxxx + μ4 u y y − μ5 u zz = 0. (4)

The problem of describing physically relevant solutions of Eq. (1) is important. It can be regarded as a very useful model that can be used
to study the evolution of long waves with small amplitude propagating in plasmas whether to consider transverse dynamics. It also can
be utilized to analysis the dynamics of waves in fluids and weak dispersion environments. In [32], authors used polynomial functions to
construct rogue wave solution, which originates from [33]. So we derive rogue wave solution directly from N-soliton of Hirota’s bilinear
method is totally different from the above methods.
This work is organized as follows. In Section 2, we present the procedure of deriving first-order rogue wave solution by applying limits
on soliton solution directly. Moreover, profiles of rogue waves have been analyzed. In Section 3 and 4, through the ingenious selection of
phase factors, the explicit expressions of second- and third-order rogue waves are obtained respectively. Different types of rogue waves
are shown by changing free parameters in the solutions. Finally, we discuss and summarize the main results in Section 4.

2. First-order rogue wave

In this section, we start from the classic soliton solution from Hirota’s bilinear method and taking further limits to derive the rogue
wave solution. Similar to the cases of deriving lump solutions that widely used [34], we find a direct method to derive rogue wave
solution from soliton solution. The main idea is shown below. Supposing u (x, y , z, t ) = u (ξ, η), using the dimensionless variables ξ = x + t
and η = y + z in Eq. (1), it is rewritten as

1  
uξ ξ + μ2 u 2 + (μ2 − μ3 )u ξ ξ ξ ξ + (μ4 − μ5 )u ηη = 0. (5)
2 ξξ

By the independent variable transformation

μ2 − μ3 ∂ 2
u (ξ, η) = 12 ln f , (6)
μ1 ∂ξ 2
where f = f (ξ, η) is the auxiliary function, Eq. (5) is reduced to the following bilinear equation
     
f ξ ξ f 2 − f ξ2 + (μ2 − μ3 ) f ξ ξ ξ ξ f 2 − 4 f ξ ξ ξ f ξ + 3 f ξ2ξ + (μ4 − μ5 ) f ηη f − f η2 = 0. (7)

Based on the Hirota’s bilinear theory, Eq. (7) admits solutions as follows:

 
(N ) 
N 
fN = exp ρi θi + ρi ρ j A i j , (8)
ρ =0,1 j =1 1≤ j < i ≤ N

with

1
θi = ki ξ + w i η + ln(ζi ), w i = −σi (μ5 − μ4 )− 2 ki (μ2 − μ3 )k2i + 1,
 2  2  4
(μ5 − μ4 ) w i − w j − ki − k j + (μ3 − μ2 ) ki − k j
exp A i j =  2  2  4
,
(μ5 − μ4 ) w i + w j − ki + k j + (μ3 − μ2 ) ki + k j
σi = ±1, (i , j = 1, 2, · · · , N ).
and the sum taken all the possible combinations of ρi = 0, 1 (i = 1, 2, · · · , N), where ki , p i (i = 1, 2, · · · , N ) are free parameters. Then we
discuss how to generate rational solution from soliton solution by taking limits.
One soliton solution is expressed as

f 1 = 1 + exp(θ1 ). (9)

To obtain rational solution, we set k1 = K 1 , substituting this in the above equation and considering → 0, we have
 √  
f 1 = 1 + ζ1 + ζ1 K 1 ησ1 μ5 − μ4 + ξ(μ4 − μ5 ) (μ4 − μ5 )−1 + O 2
. (10)

Setting ζ1 = −1, the zero power term of has been eliminated

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X. Yang, Z. Zhang, A.-M. Wazwaz et al. Physics Letters A 449 (2022) 128355

 1
  
2
f1 = ησ1 (μ5 − μ4 ) 2 − ξ K 1 + O . (11)

Noticed that f 1 is the solution of Eq. (7) and we use the logarithmic transformation (6) to obtain u, then f 1 can be expressed as the
simple form

√
f 1  ησ1 μ5 − μ4 + ξ(μ4 − μ5 ), (12)

here f  g if f = eaξ +b g, a and b are independent of ξ . Setting μ4 = 2, μ5 = 1, then we obtain the rational solution

f 1  ξ + i σ1 η, (13)

which is a singular solution of polynomial forms.

For N = 2, one can derive

f 2 = 1 + exp(θ1 ) + exp(θ2 ) + exp(θ1 + θ2 + A 12 ) (14)

where θ1 , θ2 and A 12 are determined by Eq. (8). Then we consider the substitution k1 = K 1 , k2 = K 2 , σ1 = 1, σ2 = −1, when → 0, we
have
 1

f 2 = (ζ1 + 1) (ζ2 + 1) + (ζ1 ζ2 K 1 + ζ1 ζ2 K 2 + K 1 ζ1 + K 2 ζ2 )ξ − (ζ1 ζ2 K 1 − ζ1 ζ2 K 2 + K 1 ζ1 − K 2 ζ2 )(μ5 − μ4 )− 2 η + O ( 2 ). (15)

Solving coefficients simultaneous equations of 0 and 1 , unknown variables ζ1 and ζ2 in the phase factor are obtained when the coeffi-
cients equal zero. Through calculating, their values are given as ζ1 = −1 and ζ2 = −1. Substituting them into Eq. (15) and calculating the
coefficient of 2

  (μ2 − 1)( K 14 + K 24 )
f2 = ξ 2 + (μ4 − μ5 )−1 η2 + 3(μ3 − 1) K 1 K 2 + 2
+ O ( 3) (16)
2K 1 K 2

Due to the transformation Eq. (6), when μ2 = 1, f 2 can be rewritten as

f 2  ξ 2 + (μ4 − μ5 )−1 η2 + 3(μ3 − 1). (17)

The general first order rogue wave solution can be generated by Eq. (17) via transformation (6)

24(μ2 − μ3 )(μ4 − μ5 ) (μ5 − μ4 )(ξ 2 − 3μ3 + 3) + η2
u=  . (18)
μ1 (μ4 − μ5 )(ξ 2 − 3μ3 + 3) + η2
Plugging in Eq. (18) the following set of parameters:

μ1 = −6, μ2 = 1, μ3 = 2, μ4 = 1, μ5 = 1, (19)

the solution of the bilinear equation (7) is given with

f 2 = ξ 2 + η 2 + 3, (20)

and the first order rogue wave solution u can be obtained through the transformation (6), where

4(η2 − ξ 2 + 3)
u= . (21)
(η2 + ξ 2 + 3)2
This solution demonstrates a fundamental rogue wave solution. Compared with the rogue wave solutions of Boussinesq equation in
[18,33,35], we have verified the expression of solution share the same form except for the coefficients of each term. To guarantee the
rogue wave solution is nonsingular, μ4 − μ5 > 0 and μ3 − μ2 > 0 must be satisfied. As shown in Fig. 1(a) and (c), it is symmetric
and has a single maximum. The amplitude of the first order rogue wave solution is 43 and the minimum is − 16 and located at (±3, 0).
Fig. 1(b) and (d) depicted the dark rogue wave solution. From Eq. (18) one can see the specific rogue wave solution is determined by five
arbitrary parameters μi (i = 1 · · · 5). So the dynamics property of rogue wave can be analyzed through different parameter classification in
Eq. (18). Meanwhile, the locality of rogue wave still remains unchanged. μ1 and μ2 can determine the properties of wave profile about
local excitation is bright or not, see Fig. 2(a) and (b). Furthermore, they also affect the amplitude or depth of the rogue wave. Specifically,
when |μ1 | < 0, the wave amplitude will decrease gradually as μ1 increase; when μ1 > 0, the wave depth will decrease gradually as
|μ1 | increase. And μ2 has the opposite effect on the amplitude of rogue wave compared to μ1 , with |μ2 | increase gradually, the wave
amplitude or depth are increased as well. While Fig. 2(c) shows that with the increase of μ3 , the width of rogue wave will widen and
amplitude invariance. Here μ4 and μ5 have the same effect on the profile. With the increase of μ4 , the width of rogue wave will widen
and amplitude invariance in (u , η)-plane.

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X. Yang, Z. Zhang, A.-M. Wazwaz et al. Physics Letters A 449 (2022) 128355

Fig. 1. (Color online) First order rogue wave solution with parameter selections (19). (a) 3d plot; (c) density plot. Dark rogue wave solution with (19) except μ1 = 6. (b) 3d
plot; (d) density plot.

Fig. 2. (Color online) Profile of first order rogue wave solution. (a) With parameter selection (19) except μ1 = −6 (blue), μ1 = −3 (orange),μ1 = 2 (yellow). (b) With
parameter selection (19) except μ1 = 1 (blue), μ1 = 3 (orange),μ1 = 6 (yellow). (c) With parameter selection (19) except μ3 = 2 (blue), μ3 = 4 (orange),μ3 = 6( yellow).
(d) With parameter selection (19) except μ4 = 2 (blue), μ4 = 4 (orange),μ4 = 6 (yellow).

3. Second-order rogue wave

We have extended the traditional method to derive rational solution by generalizing some factors in soliton solution in this section.
And compared with Sec. 2, the factors have a more complicated form. For simplicity, we setting μ1 = −6, μ2 = 1, μ3 = 2, μ4 = 2, μ5 = 1
firstly and it will not affect the solution of bilinear equation. As N = 4,

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X. Yang, Z. Zhang, A.-M. Wazwaz et al. Physics Letters A 449 (2022) 128355

Fig. 3. (Color online) Second order fundamental rogue wave solution with μ1 = −6, μ2 = 1, μ3 = 2, μ4 = 2.μ5 = 1. (a) 3d plot; (c) density plot. Dark rogue wave solution
with μ1 = 6, μ2 = 1, μ3 = 2, μ4 = 2.μ5 = 1. (b) 3d plot; (d) density plot.
⎛ ⎞

4 
4 
4
 4 
4
f4 = 1 + exp (θi ) + exp(θi + θ j + A i j ) + exp θi + θ j + θk + A i j + A ik + A jk + exp ⎝ θi + Ai j ⎠ . (22)
i =1 i< j i < j <k i =1 i< j

We rewrite θi as
2 3
θi = ki ξ + w i η + ln(ζi ) + αi + βi , (23)
and other variable remain unchanged in Eq. (8).
Firstly, we consider the case of βi = 0. Letting k1 = K 1 , k2 = K 2 , k3 = K 3 , k4 = K 4 , σ1 = 1, σ2 = 1, σ3 = −1, σ4 = −1, when → 0,
f 4 can be expressed as the following series


6
(i ) i
f4 = f4 + O ( 7 ). (24)
i =0

(0) (1) (2) (3) (4) (5)

By solving the six simultaneous equations f4 = 0, f 4 = 0, f 4 = 0, f 4 = 0, f 4 = 0, f 4 = 0 , ζi and αi have the following form

K1 + K2 K3 + K4 K1 K2 K3 K4
ζ1 = −ζ2 = , ζ3 = −ζ4 = , α1 = α2 = , α3 = α4 = . (25)
K1 − K2 K3 − K4 6 6
Substituting Eq. (25) into Eq. (24) yields
K 1 K 2 K 3 K 4 ( K 1 + K 2 )( K 3 + K 4 ) 
f4 = η6 + (3ξ 2 + 17)η4 + (3ξ 4 + 90ξ 2 + 475)η2 + ξ 6 + 25ξ 4 − 125ξ 2 + 1875 6
+ O ( 7 ). (26)
36
Due to the transformation Eq. (6), f 4 can be rewritten as

f 4  η6 + (3ξ 2 + 17)η4 + (3ξ 4 + 90ξ 2 + 475)η2 + ξ 6 + 25ξ 4 − 125ξ 2 + 1875. (27)

This is the solution of bilinear equation (7) which is polynomial of degree 6 in ξ and η . The second order rogue wave solution can be
given by Eq. (6). This is the fundamental pattern of second order rogue wave, see Fig. 3.
(6)
If βi = 0, substituting Eq. (25) into Eq. (24), and letting Im f 4 = 0, we can obtain the following equation
   
K 13 K 2 A + K 3 K 43 β2 − K 33 K 4 β2 − B K 23 K 1 + K 2 K 3 K 4 β1 (3C )−1 = 0 (28)

where

A = β4 K 3 − K 4 β3 , B = ( K 3 − K 4 )( K 3 + K 4 ), C = ( K 1 − K 2 )( K 3 − K 4 ).

By solving the above equation, a set of values to βi are given as

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X. Yang, Z. Zhang, A.-M. Wazwaz et al. Physics Letters A 449 (2022) 128355

Fig. 4. (Color online) Second order triangle rogue wave with μ1 = −6, μ2 = 1, μ3 = 2, μ4 = 2, μ5 = 1. (a) 3d plot; (c) density plot. Dark rogue wave solution with μ1 =
6, μ2 = 1, μ3 = 2, μ4 = 2.μ5 = 1. (b) 3d plot; (d) density plot.

 
(K1 + K2)K1 K2
β1 = β2 = β4 , β3 = β4 (29)
(K3 + K4)K3 K4
(6)
and β4 is a free parameter, substituting Eq. (29) into f 4 ,

K1 K2(K1 + K2)    
f 4trig = f 4 (2K 33 K 43 + K 34 K 42 + K 32 K 44 ) − 24ξ 3η2 − ξ 2 + 1 K 3 K 4 ( K 3 + K 4 )β4 , (30)
36K 3 K 4 ( K 3 + K 4 )
where f 4 in Eq. (27) can generate the solution of second-order fundamental rogue wave solution. f 4trig can form triangle structure of
rogue wave when β4 is large enough, see Fig. 4. This also reveals a certain connection between the two types rogue waves. Recently, the
rogue wave patterns of NLS equation have been systematically studied in [36]. Speaking from the more specific aspect, setting different
free parameters in the rogue wave solutions large enough, they have shown that the patterns of rogue waves can be described by the
root structure of the Yablonskii–Vorob’ev polynomial hierarchy by studying the asymptotic behaviors of solutions. Also, the second-order
rogue wave consisting three spaced apart fundamental rogue waves which are away from the center have shown in their work. It is
worth emphasizing that this dispersed phenomenon also occurs in the second-order rogue wave solutions of the KdV-BBM equation, see
Fig. 4. As shown in Fig. 5, the second-order rogue waves contain three fundamental rogue waves which separate from each other when
β4 gradually increase.

4. Third-order rogue wave

For the third-order rogue wave solution, we started from N = 6


6 
6 
6

f 6 =1 + exp (θi ) + exp(θi + θ j + A i j ) + exp θi + θ j + θk + A i j + A ik + A jk
i =1 i< j i < j <k


6

+ exp θi + θ j + θk + θl + A i j + A ik + A il + A jk + A jl + A kl
i < j <k<l


6

+ exp θi + θ j + θk + θl + θm + A i j + A ik + A il + A im + A jk + A jl + A jm + A kl + A km + A lm
i < j <k<l<m
⎛ ⎞
6 
6
+ exp ⎝ θi + Ai j ⎠ (31)
i =1 i< j

with

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X. Yang, Z. Zhang, A.-M. Wazwaz et al. Physics Letters A 449 (2022) 128355

Fig. 5. (Color online) Evolution of second order rogue wave solution with different β4 .

2 3 4 5
θi = ki ξ + w i η + ln(ζi ) + αi + βi + γi + δi . (32)

Letting ki = K i (i = 1, 2, · · · , 6) and σ1 = σ2 = σ3 = 1, σ4 = σ5 = σ6 = −1. To make the calculation simpler, we set K 1 = 1, K 2 = 2, K 3 =

3, K 4 = 4, K 5 = 5, K 6 = 6 in the following process, we find that the final result will not be affected through calculation. When → 0, f 6
can be expressed as


12
(i ) i 13
f6 = f6 + O( ). (33)
i =0

By setting the unknown variables in phase factor as

(K1 + K3) (K1 + K2) (K2 + K3) (K1 + K2) (K2 + K3) (K1 + K3)
ζ1 = − , ζ2 = , ζ3 = − ,
(K1 − K3) (K1 − K2) (K2 − K3) (K1 − K2) (K2 − K3) (K1 − K3)
(K4 + K6) (K4 + K5) (K5 + K6) (K4 + K5) (K5 + K6) (K4 + K6)
ζ4 = − , ζ5 = , ζ6 = − ,
(K4 − K6) (K4 − K5) (K5 − K6) (K4 − K5) (K5 − K6) (K4 − K6)
K1 K2 K3 (K1 + K2 − K3) K4 K5 K6 (K4 + K5 − K6)
α1 = α2 = , α3 = , α4 = α5 = , α6 = ,
6 6 6 6
7 (K1 K2 + K1 K3 + K2 K3) K1 K2 K3
γ1 = γ2 = γ3 = − ,
36( K 1 + K 2 + K 3 )
7 (K4 K5 + K4 K6 + K5 K6) K4 K5 K6
γ4 = γ5 = γ6 = − . (34)
36( K 4 + K 5 + K 6 )
And βi and δi will be determined later, they are parameters that affecting wave patterns of third-order rogue wave solution. Substituting
Eq. (34) into Eq. (33), the fundamental third-order rogue wave solution is given by Eq. (6) with βi = 0, δi = 0 and
   
f 6 =9(ξ 12 + η12 ) + 882ξ 10 + 54ξ 2 + 522 η10 + 6615ξ 8 + 135ξ 4 + 5130ξ 2 + 39015 η8 + 226380ξ 6
  
+ 180ξ 6 + 13140ξ 4 + 318780ξ 2 + 2396940 η6 − 15563625ξ 4 + 135ξ 8 + 13860ξ 6 + 337050ξ 4
 
−132300ξ 2 + 49175175 η4 + 479359650ξ 2 + 54ξ 10 + 6210ξ 8 + 167580ξ 6 + 1984500ξ 4

+5093550ξ 2 + 902690250 η2 + 878826025. (35)

The degree of the above polynomial corresponding to the third-order rogue wave solutions is 12. The fundamental third-order rogue wave
solution can be seen from Fig. 6.

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X. Yang, Z. Zhang, A.-M. Wazwaz et al. Physics Letters A 449 (2022) 128355

Fig. 6. (Color online) Third order fundamental rogue wave solution with μ1 = −6, μ2 = 1, μ3 = 2, μ4 = 2.μ5 = 1. (a) 3d plot; (c) density plot. Dark rogue wave solution with
μ1 = 6, μ2 = 1, μ3 = 2, μ4 = 2.μ5 = 1. (b) 3d plot; (d) density plot.

If setting δi = 0, a set of solution of βi can be solved through the above process

 
15 5 3
β3 = −5β1 + 4β2 , β5 = −15β1 + β2 + β4 , β6 = −40β1 + 20β2 + β4 (36)
2 4 2
with free parameters β1 , β2 , β4 , by setting β1 = 0, β4 = 0, then substituting these relations to Eq. (33), we can obtain the third-order
triangle pattern rogue wave solution, see Fig. 7. The degree of corresponding polynomial to the third-order rogue wave solutions is also
12, comparing with Eq. (35), it only introduced a variable β2 to form the structure. The third-order rogue waves contain six individual
first-order rogue wave solution, and the six individual first-order rogue wave form a stable structure which formed a symmetric triangle.
The analytic proof of these patterns is given in [36], and the root structure of the Yablonskii–Vorob’ev polynomial hierarchy is utilized to
investigating rogue wave solutions of the NLS equation by taking certain parameter large enough in their work. Our results are consistent
with their results respectively. When β2 gradually increase, the bound state of rogue wave will gradually disperse to six individual rogue
waves. Eventually, they formed a triangular structure, see Fig. 8.
If setting βi = 0, a set of solution of δi can be solved through the similar process
 
11δ4 8 δ5
δ1 = δ1 , δ2 = δ2 , δ3 = δ3 , δ4 = δ4 , δ5 = δ5 , δ6 = 55δ1 − 44δ2 + 11δ3 − + (37)
6 3
with free parameters δ1 , δ2 , δ3 , δ4 , δ5 , by setting δ2 = 0, δ3 = 0, δ4 = 0, δ5 = 0 and substituting these parameters constraints to Eq. (33)
then the third-order pentagon pattern rogue wave solution is obtained. One can observe the pentagon structure when βi = 0 and δi large
enough, the third-order pentagon pattern rogue wave solution consists of the outer circular shell of five first-order rational solutions,
while the center is a first-order rogue wave solution as illustrated in Fig. 9. When δ1 gradually increase, the bound state of rogue wave
will gradually transform to a pentagon structure, see Fig. 10.

5. Conclusions

In this paper, the rogue wave solutions of KdV-BBM equation are constructed from the N-soliton solutions of Hirota’s bilinear method.
We introduced some free parameters in the phase factor and employing long wave limits. Rogue wave solutions are presented in explicit
forms by selecting appropriate phase parameters. Comparing to the widely used methods (DT, KP hierarchy reduction method), we have
confirmed our method is a simple and effective approach, because rogue waves can be generated directly from N-soliton and the method
reduces the difficulty of constructing rogue wave solutions. The results obtained are coinciding with the results obtained from other
methods. Compared with [32], rogue wave solution generated from our method has more parameters, they have the same expression
under special parameter restrictions. In addition, method in this work don’t need to provide the expressions of f in advance. The first
order rogue wave produced by a 2 degree rational polynomial functions which is the solution of corresponding bilinear equation, we
have shown that how the parameter μi can affect the profile of first-order rogue wave, such as bright-dark, width and amplitude. The
second-order rogue wave solutions are produced by a 6 degree rational polynomial functions. There are two structures of second-order
rogue waves consisting fundamental and triangle structure. The third-order rogue wave solutions are produced by a 12 degree rational
polynomial functions. Except for the fundamental structure, one pattern is a triangle whereas the other is a regular pentagon. Moreover,

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X. Yang, Z. Zhang, A.-M. Wazwaz et al. Physics Letters A 449 (2022) 128355

Fig. 7. (Color online) Third order triangle rogue wave solution with μ1 = −6, μ2 = 1, μ3 = 2, μ4 = 2.μ5 = 1. (a) 3d plot; (c) density plot. Dark rogue wave solution with
μ1 = 6, μ2 = 1, μ3 = 2, μ4 = 2.μ5 = 1. (b) 3d plot; (d) density plot.

Fig. 8. (Color online) Evolution of third order triangle rogue wave with different β2 .

the structures of rogue waves can be determined when one of free parameters in the solutions is large enough. We expect that the results
obtained in this work could help us better understand rogue waves.

CRediT authorship contribution statement

Xiangyu Yang: Writing – original draft, Software, Methodology, Formal analysis. Zhao Zhang: Writing – review & editing, Software,
Methodology. Abdul-Majid Wazwaz: Supervision. Zhen Wang: Writing – review & editing, Investigation, Formal analysis, Conceptualiza-
tion.

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X. Yang, Z. Zhang, A.-M. Wazwaz et al. Physics Letters A 449 (2022) 128355

Fig. 9. (Color online) Third order pentagon rogue wave solution with μ1 = −6, μ2 = 1, μ3 = 2, μ4 = 2.μ5 = 1. (a) 3d plot; (c) density plot. Dark rogue wave solution with
μ1 = 6, μ2 = 1, μ3 = 2, μ4 = 2.μ5 = 1. (b) 3d plot; (d) density plot.

Fig. 10. (Color online) Evolution of third order triangle rogue wave with different δ1 .

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.

Data availability

No data was used for the research described in the article.

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X. Yang, Z. Zhang, A.-M. Wazwaz et al. Physics Letters A 449 (2022) 128355

Acknowledgements

Project supported by National Natural Science Foundation of China (Grant Nos. 52171251, U2106225), Liaoning Revitalization Talents
Program (XLYC1907014) and the Fundamental Research Funds for the Central Universities (DUT21ZD205).

References

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