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Nonlinear Dyn
https://doi.org/10.1007/s11071-022-07509-0
ORIGINAL PAPER
Analysis of lumps, single-stripe, breather-wave, and
two-wave solutions to the generalized perturbed-KdV
equation by means of Hirota’s bilinear method
Marwan Alquran ·Rahaf Alhami
Received: 15 March 2022 / Accepted: 4 May 2022
© The Author(s), under exclusive licence to Springer Nature B.V. 2022
Abstract In this paper, we implement the Hirota’s
bilinear method to extract diverse wave profiles to
the generalized perturbed-KdV equation when the
test function approaches are taken into consideration.
Several novel solutions such as lump-soliton, lump-
periodic, single-stripe soliton, breather waves, and two-
wave solutions are obtained to the proposed model. We
conduct some graphical analysis including 2D and 3D
plots to show the physical structures of the recovery
solutions. On the other hand, this work contains a cor-
rection of previous published results for a special case
of the perturbed KdV. Moreover, we investigate the sig-
nificance of the nonlinearity, perturbation, and disper-
sion parameters being acting on the propagation of the
perturbed KdV. Finally, our obtained solutions are ver-
ified by inserting them into the governing equation.
Keywords Perturbed-KdV equation ·Hirota bilinear ·
Lump soliton ·Breather waves
Mathematics Subject Classification 26A33 ·35F25 ·
35C10
M. Alquran (B)·R. Alhami
Department of Mathematics and Statistics, Jordan
University of Science & Technology, Irbid 22110, Jordan
e-mail: marwan04@just.edu.j
R. Alhami
e-mail: rfalhami20@sci.just.edu.jo
1 Introduction
Finding exact solutions of nonlinear equations plays
an imperative role in understanding the processes and
phenomena of many nonlinear models arising in flu-
ids, dynamics, physical science, and nonlinear optical
fibers. In the theory of solitons, different types of solu-
tions such as bell-shaped, kink, cusp, periodic and oth-
ers are identified by using suggested forms of solutions
either in terms of exponential, trigonometric, or hyper-
bolic functions as in the cases of simplified bilinear
method, tanh expansion, (G/G)expansion, Riccati
expansion, sine–cosine function, sech–csch function,
Kudryashov expansion, unified expansion, Lie symme-
try, and many other methods ( [1–10]).
Recently, new types of solitons are produced by
combining the Hirota bilinear method and the Cole–
Hopf transformation u=a(ln f)xor u=b(ln f)xx,
see ( [11–20]). If fis chosen to be a polynomial, then
the resulting solution uis identified as the lump soliton.
If fis the combination of polynomial and sine/cosine,
then uis of periodic-lump type. The breather-soliton
waves are obtained if fis a combination of sine/cosine
and the exponential functions. Finally, the two-wave
soliton type can be obtained by combining sin–sinh or
cos–cosh with the exponential functions.
In this paper, we investigate new features of soli-
tary wave solutions to the generalized perturbed-KdV
equation which reads as
ut+αux+βuux+γuxxx =0,(1.1)
123
M. Alquran, R. Alhami
where u=u(x,t)represents the free surface advance-
ment, αis the perturbation parameter known as the
Coriolis effect, and β, γ are the nonlinearity and
dispersion factors, respectively. The perturbed-KdV
model describes the physical mechanism of sound
propagation in fluid and appears in the applications
of aerodynamics, acoustics, and medical engineering.
Special case of (1.1) has been discussed in [21], for
β=3
2and γ=1
6. The authors extracted differ-
ent lumps and breather solutions but upon assign-
ing a wrong choice of the Cole–Hopf transformation
u=R(ln f(x,t))xx by taking R=2.
The motivation of the current work is threefold:
First, we derive the correct value of Rthat covers,
in particular, the case of [21] and, in general, the
case of (1.1). Second, we assign different choices of
f(x,t), to construct new lump-soliton, lump-periodic,
single-stripe soliton, breather waves, and two-wave
solutions to (1.1). Finally, we investigate the impact
of the involved model’s parameters on the propagation
form of the retrieved solutions to the proposed model.
The paper is organized as follows: Section (2) deals
with the construction of Hirota’s bilinear form to the
perturbed-KdV equation. Then, we derive both lump
and periodic-lump solutions in Sect. (3). The single-
stripe soliton and breather-wave solutions are extracted
in Sects. (4) and (5), respectively. The two-wave solu-
tions are investigated in Sect. (6), and some dynam-
ical aspects are discussed in Sect. (7). Finally, some
concluding remarks based on the obtained results are
presented in Sect. (8).
2 Hirota bilinear form of the perturbed-KdV
equation
To find the Hirota’s form to (1.1), we apply the simpli-
fied bilinear method. First, we start with the following
function
u(x,t)=esx−rt.(2.2)
Then, we substitute (2.2) in the linear terms of (1.1)to
obtain the dispersion relation as
r=αs+γs3.(2.3)
The second step is to bring the following function
k(x,t)=1+Ce
sx−αs+γs3t,(2.4)
and to apply one of the Cole–Hopf transformations. In
particular, we consider
u(x,t)=R(ln(k(x,t)))xx .(2.5)
To find R, we insert (2.5)in(1.1) to get that
R=12γ
β.(2.6)
As the third step, we update the assumption of the func-
tion uto take the following action:
u(x,t)=ψxx(x,t). (2.7)
Substituting (2.7)in(1.1) and simplifying by integra-
tion with respect to x, we reach at the following relation
regarding the new function ψ
ψxt +αψxx +β
2(ψxx)2+γψ
xxxx =0.(2.8)
Then, we choose ψas
ψ(x,t)=12γ
βln(f(x,t)). (2.9)
Finally, we insert (2.9)in(2.8) to deduce the following
relation:
ff
xt −fxft+αff
xx −αf2
x+γff
xxxx
−4γfxfxxx +3γf2
xx =0.(2.10)
By using Hirota’s bilinear operators, (2.10) is written
as
(DxDt+αD2
x+γD4
x)f.f=0,(2.11)
where Drepresents the Hirota bilinear operator defined
as
Dl
xDk
tf.g=∂
∂x−∂
∂xl∂
∂t
−∂
∂tk
f(x,t)g(x,t)|x=x,t=t,(2.12)
and f,g∈C∞(R2).
3 Lump-type solutions
In this section, we derive two types of lump solutions to
(1.1) by choosing fto be either quadratic function, or a
combination of quadratic function and cosine function.
3.1 Lump soliton
To obtain lump soliton, we consider the following
assumption
f=XTAX +u0,(3.13)
123
Analysis of lumps, single-stripe
where X=(1,x,t)T,A=(ai,j)3×3is a symmetric
matrix, and aij,u0are real constants to be determined.
By expanding (3.13), we get
f(x,t)=a1,1+a2,1t+a3,1x+x(a1,2+a2,2t+a3,2x)
+t(a1,3+a2,3t+a3,3x)+u0.(3.14)
Next, we insert (3.14)in(2.10) and solve for the
unknowns aij,u0. By doing so, we obtain two cases:
Case I:
a2,3=−αa2,2−αa3,3,
u0=αa2
1,2+a3,1(a1,3+a2,1+αa3,1)+a1,2(a1,3+a2,1+2αa3,1)−a1,1(a2,2+a3,3)
a2,2+a3,3
,
a3,2=0,
where a1,1,a1,2,a1,3,a2,1,a2,2,a3,1and a3,3are
free parameters. Accordingly,
f=(a1,2+a3,1+(a2,2+a3,3)t)(αa1,2+a1,3+a2,1+αa3,1+(a2,2+a3,3)(x−αt))
a2,2+a3,3
.(3.15)
Recalling (2.7), the first lump soliton to (1.1)is
u1(x,t)
=− 12γ(a2,2+a3,3)2
β(αa1,2+a1,3+a2,1+αa3,1+(a2,2+a3,3)(x−αt))2.
(3.16)
Case II:
a2,1=−αa1,2−a1,3−αa3,1,
a2,3=a3,2=0,
a3,3=−a2,2.
Thus,
f=u0+a1,1+(a1,2+a3,1)(x−αt), (3.17)
with a1,1,a1,2,a1,3,a2,2,a3,1and u0being free
parameters. By this case, the second lump soliton is
u2(x,t)=− 12γ(a1,2+a3,1)2
β(u0+a1,1+(a1,2+a3,1)(x−αt))2.
(3.18)
In Fig. 1, we show the physical structure of the first
lump soliton (3.16), which is similar in shape to (3.18).
3.2 Lump-periodic solution
To construct lump-periodic solution to (1.1), fis to be
chosen as a linear combination of quadratic and cosine
functions, i.e.
f=XTAX +ωcos (p1x+p2t+p3)+σ. (3.19)
We substitute (3.19)in(2.10) and look up for the coeffi-
cients of different polynomials of x,tand trigonomet-
ric functions. Then, we set each coefficient to zero and
solve the resulting system to get the following output:
ω=∓
a1,2+a3,1
p1
,
p2=−αp1+γp3
1,
a3,2=a2,3=0,
a3,3=−a2,2,
a1,3=−αa1,2+3γp2
1a1,2−a2,1−αa3,1+3γp2
1a3,1.
(3.20)
Let = p1(x−αt)+γp3
1t+p3,=σ+a1,1+
a1,2(x−αt)+a3,1x−αa3,1tand T=a1,2+a3,1.
Then, fhas the following form
f=σ+a1,1+a2,1t+Tx +a2,2xt ±Tcos ()
p1
+t((3γp2
1−α)T−a2,1−a2,2x). (3.21)
Therefore, the lump-periodic solution to (1.1)is
123
M. Alquran, R. Alhami
Fig. 1 2D and 3D plots of
u1(x,t)where
α=β=−1, γ=1,
a2,2=2, a3,3=a1,2=
a1,3=a2,1=a3,1=1
u3(x,t)=
12γT−T(sin ()∓1)2∓p1cos ()±cos ()T
p1+3γp2
1Tt
β(±Tcos()
p1+3γp2
1Tt)2.(3.22)
In Fig. 2, we present the physical structure of the lump-
periodic solution (3.22).
4 Single-stripe soliton solutions
The approach for finding single-stripe solitons is known
as a simplified bilinear method. They are similar to
those steps illustrated earlier and given by (2.2)-(2.6).
However, it can be derived directly using (2.9) and
assume fas
f=1+ced1x+d2t+d3,(4.23)
where di,i=1,2,3 and c= 0 are unknown real
constants to be determined. Substituting of (4.23)in
(2.10) gives that d2=−αd1−γd3
1, where d1and d3
are arbitrary constants. Thus, the single-stripe soliton
solution of (1.1)is
u4(x,t)=12cγe(x−αt)d1+γtd3
1+d3d2
1
β(eγtd3
1+ce
(x−αt)d1+d3)2.(4.24)
5 Breather-wave solution
To find some families of breather-wave solutions, we
consider the following test function
f=1cos (p2(x+b2t))+2ep1(x+b1t)+e−p1(x+b1t).
(5.25)
where i,pi,bi:i=1,2 are real constants to
be determined later. Substituting (5.25) in the bilinear
form (2.10) and equating the coefficients of exponen-
tials or trigonometric functions to zero, we get the fol-
lowing nonlinear algebraic system:
0=αp2
112+b1p2
112+γp4
112−αp2
212
−b2p2
212−6γp2
1p2
212+γp4
212,
0=αp2
11+b1p2
11+γp4
11−αp2
21−b2p2
21
−6γp2
1p2
21+γp4
21,
0=2αp1p212+b1p1p212+b2p1p212
+4γp3
1p212−4γp1p3
212,
0=−2αp1p21−b1p1p21
−b2p1p21−4γp3
1p21+4γp1p3
21,
0=4αp2
12+4b1p2
12+16γp4
12−αp2
22
1
−b2p2
22
1+4γp4
22
1.
Solving the above system leads to
b1=−α−γp2
1+3γp2
2,
b2=−α−3γp2
1+γp2
2,
2=−p2
22
1
4p2
1
,
with 1,pi:i=1,2 being free parameters. Let λ1=
p1(−x+αt+γtp2
1−3γtp2
2)and λ2=p2(x−αt−
3γtp2
1+γtp2
2), and we get
f=1cos (λ2)+eλ1−p2
22
1e−λ1
4p2
1
.(5.26)
123
Analysis of lumps, single-stripe
Fig. 2 2D and 3D plots of
u3(x,t)where
α=β=γ=1,
a1,1=a1,2=a3,1=σ=
p1=p3=1
Accordingly, the breather-wave solution to (1.1)is
u5(x,t)=(12γ(−(p1eλ1+p21sin (λ2)+p2
22
1e−λ1
4p1
)2
+(eλ1+1cos (λ2)−p2
22
1e−λ1
4p2
1
)( p2
1eλ1+p2
21
4
(−4cos(λ2)−1e−λ1))))/(β (eλ1+1cos (λ2)
−p2
22
1e−λ1
4p2
1
)2).
(5.27)
In Fig. 3, we present the physical structure of the
breather-wave solution (5.27).
6 Two-wave solution
To find the two-wave solution to the perturbed-KdV
equation, we consider the following test function:
f=ω1eμt+x+ω2e−(μt+x)+ω3sin (c1t+x)
+ω4sinh (c2t+x). (6.28)
To get information about the values of ωj:(j=
1,2,3,4), c1,c2and μ, we substitute (6.28)inEq.
(2.10). Then, we collect the coefficients of same terms
and set each to zero to obtain the following system:
0=−2αω1ω3−μω1ω3−c1ω1ω3,
0=−4γω
1ω3+μω1ω3−c1ω1ω3,
0=2αω1ω4+8γω
1ω4+μω1ω4+c2ω1ω4,
0=2αω2ω3+μω2ω3+c1ω2ω3,
0=−4γω
2ω3+μω2ω3−c1ω2ω3,
0=2αω2ω4+8γω
2ω4+μω2ω4+c2ω2ω4,
0=−2αω3ω4−c1ω3ω4−c2ω3ω4,
0=−4γω
3ω4−c1ω3ω4+c2ω3ω4,
0=4αω1ω2+16γω
1ω2+4μω1ω2−αω2
4−4γω
2
4
−c2ω2
4−αω2
3+4γω
2
3−c1ω2
3.
By solving the above system, we retrieve three solu-
tion’s sets:
Set(I):ω2=−ω2
4
4ω1,ω
3=0,c2=−2α−8γ−μ.
Then, fexplicitly is
f=ω1ex+μt
−ω2
4e−x−μt
4ω1
+ω4sinh (x−t(2α+8γ+μ)).
(6.29)
Thus, the sixth recovery solution to (1.1)is
u6(x,t)=12γ
β−12γ(ω
1ex+μt+ω4cosh (x−t(2α+8γ+μ)) +ω2
4e−x−μt
4ω1)2
β(ω1ex+μt+ω4sinh (x−t(2α+8γ+μ)) −ω2
4e−x−μt
4ω1)2
,
=− 96γω
1ω4e2(t(α+4γ)+x)
βω4e2t(α+4γ) −2e2xω12.(6.30)
Set(II):ω2=−ω2
3
4ω1,ω
4=0,μ=−α+2γ, c1=
−α−2γ. Then, fexplicitly is
f=ω1eζ−ω2
3e−ζ
4ω1
+ω3sin (η), (6.31)
where ζ=x−t(α −2γ) and η=x−t(α +2γ).
Hence, the seventh recovery solution to (1.1)is:
123
M. Alquran, R. Alhami
Fig. 3 2D and 3D plots of
u5(x,t)where
β=γ=−1,
α=p1=p2=1=1
Fig. 4 Propagations of u6(x,t)for different values of: aThe perturbation parameter αwhere t=β=γ=ω1=ω4=1. bThe
nonlinearity parameter βwhere t=α=γ=ω1=ω4=1. cThe dispersion parameter γwhere t=α=β=ω1=ω4=1
u7(x,t)=12γ(ω
1eζ−ω3sin (η) −ω2
3e−ζ
4ω1)
β(ω1eζ+ω3sin (η) −ω2
3e−ζ
4ω1)
−12γ(ω
1eζ+ω3cos (η) +ω2
3e−ζ
4ω1)2
β(ω1eζ+ω3sin (η) −ω2
3e−ζ
4ω1)2
,
=−
96γω
1ω3eαt+2γt+x4ω3ω1eαt+2γt+x+4ω2
1e4γt+2xcos(η) +ω2
3e2αtcos(η)
β−ω2
3e2αt+4ω3ω1eαt+2γt+xsin(η) +4ω2
1e4γt+2x2.(6.32)
Set(III):ω1=0,ω
2=0,ω
3=−ω4,c1=−α−
2γ, c2=−α+2γ. Then, fexplicitly is:
f=−ω4sin (η) +ω4sinh (ζ ). (6.33)
Accordingly, the eighth recovery solution to (1.1)is
u8(x,t)=−
12γ(−ω4cos (η) +ω4cosh (ζ ))2
β(−ω4sin (η) +ω4sinh (ζ ))2
123
Analysis of lumps, single-stripe
+12γ(ω
4sin (η) +ω4sinh (ζ ))
β(−ω4sin (η) +ω4sinh (ζ )) (6.34)
7 Dynamics of the perturbed KdV
In this section, we study the impact of the perturbation,
nonlinearity, and dispersion parameters, α,β,γ, being
acting on the propagation of the perturbed KdV. To
achieve this goal, we consider the obtained solution
depicted earlier as the function u6(x,t). We investigate
the physical structures to this function by plotting some
curves for different values of the assigned parameters.
Figure 4shows the dynamics of propagating u6, and
three observations can be drawn:
•The propagation is symmetric when αchanges its
sign.
•The propagation has a reflexive relation when β
changes its sign.
•The propagation is reflexive due to the sign of γ.
8 Conclusion
In this study, we derived the Hirota bilinear form for the
generalized perturbed-KdV equation. Then, the Cole–
Hopf transformations are used, and different selections
of the involved test function are elaborated to retrieve
novel types of solitons such as lumps, breather-wave,
and multi-wave solutions. Also, the dynamics of the
model’s parameters, perturbation, nonlinearity, and dis-
persion are investigated.
For future work, we aim to extend the use of Hirota’s
bilinear methods to study other important nonlinear
applications arising in physical and engineering fields
and higher-dimensional models.
Funding The authors have not disclosed any funding.
Data availibility Data sharing was not applicable to this article
as no datasets were generated or analysed during the current
study.
Declarations
Conflict of interest The authors declare that they have no con-
flict of interest
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