New Solutions of Benney-Luke Equation Using The (G’/G,1/G) Method
NGUYEN MINH TUAN1ID , SANOE KOONPRASERT1,3ID , SEKSON SIRISUBTAWEE1,3ID ,
PHAYUNG MEESAD2ID , NATTAWUT KHANSAI1ID
1Department of Mathematics, Faculty of Applied Science,
King Mongkut’s University of Technology North Bangkok,
1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800,
THAILAND
2Information Technology and Management Department,
King Mongkut’s University of Technology North Bangkok,
1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800,
THAILAND
3Centre of Excellence in Mathematics,
CHE, Si Ayutthaya Road, Bangkok 10400,
THAILAND
Abstract: The Benney-Luke equation has contributed to studying the propagation of the water wave surfaces.
This paper illustrates the (G’/G,1/G)-method to obtain the solutions of the Benney-Luke equation and an extension
of the Benney-Luke equation. The new types of solutions are also constructed to gather the performance and
visualization in three dimensions for observing the behaviors. The solutions are found in the expressions of
hyperbolic functions giving the general performance by selecting arbitrary constants.
Key-Words: (G’/G,1/G)-method; exact solution; Benney-Luke equation, extended Benney-Luke equation.
Received: November 7, 2023. Revised: March 6, 2024. Accepted: March 19, 2024. Published: April 24, 2024.
1 Introduction
Finding exact solutions has contributed to devel-
oping the studies in mechanics and dynamics, [1],
[2]. The important role of the (G’/G,1/G)-method
has been performed in finding solutions of nonlin-
ear partial differential equations (NPDEs). Using
the (G’/G,1/G)-method, various solutions of the fifth-
order nonlinear equation have been found and illus-
trated in three dimensions, [3]. Based on the travel-
ing wave solution, the NPDEs will be turned into or-
dinary differential equations and applied two variable
(G’/G,1/G)-expansion method using the mathemati-
cal application such as Maple, Mathematica to find
the suitable solutions corresponding to the case con-
struction, [4]. The solutions gained using the two-
term (G/G,1/G)-expansion method are closed-form
solutions demonstrating hyperbolic function, trigono-
metric, and rational function solutions, [5]. The
(G’/G,1/G)-expansion method is also useful in solv-
ing the nonlinear medium equal width (MEW) wave
equation that represents one-dimensional wave prop-
agation related to the dispersion process, [6].
The procedure of the (G’/G, 1/G)-method trans-
forms from NPDEs to ordinary differential equations
(ODE) based on the traveling wave and construct the
solutions with parameters, [7]. The (G’/G, 1/G)-
method is extended from the well-known (G’/G)-
method and has demonstrated more effectively and
more general than the (G’/G)-method, [8]. The
(G’/G, 1/G)-method is considered direct, concise, and
elementary for attaining the solutions of nonlinear
evolution equations (NLEEs), [9]. Compared to the
sine-Gordon expansion method (SGEM) obtaining
kink-type solutions, bell-shaped solitary wave solu-
tions, and anti-bell-shaped type soliton solutions, the
(G’/G, 1/G)-method supports the other types of the
solutions, [10]. Besides that, the (G’/G, 1/G)-method
is perfectly complemented for the special solutions
of the (3+1DJM) equation that performs a stationary
wave in physics, [11].
The Benney-Luke equation is one of the NPDEs
representing the two-way propagation of water ten-
sion. By applying the (G’/G)-method, hyperbolic,
and trigonometric solutions have been found replied
on the setting of shifting variables, [12]. The solu-
tions of the Benney-Luke equation have been found
in the expression of hyperbolic and trigonometric
solutions by choosing appropriate parameters, [13],
[14]. Another presentation banked on the variation
direct method (VDM), various solutions such as dark
solitary type solution, dark-like solitary type solu-
tion, kinky-dark solitary type solution, and periodic
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.29
Nguyen Minh Tuan, Sanoe Koonprasert,
Sekson Sirisubtawee, Phayung Meesad, Nattawut Khansai
E-ISSN: 2224-2880
267
Volume 23, 2024
wave type solutions have been built, [15]. More-
over, the (1/G’)-expansion method has supported the
new tools for getting solutions of nonlinear evolution
equations (NLEEs), and shown the necessary way to
gather other hyperbolic solutions, [16]. One more
successful performance, the modified simple equa-
tion method has straightforwardly contributed to find-
ing exact traveling wave solutions and the solitary
wave solutions of the Benney-Luke equation, [17].
In the present study, the Benney-Luke equation, [13],
given with a=1,b=1,n=1 as follows:
utt −uxx +uxxxx −uxxtt +utuxx +2uxuxt =0,(1.1)
and an extended Benney-Luke equation given as the
following will be considered to seek the exact ana-
lytic solutions:
utt −uxx +uxxxx −uxxtt +utuxx +2uxuxt +utuxx =0.(1.2)
The Benney-Luke equations have played a significant
function in studying the two-way propagation of wa-
ter wave surfaces, and the (G’/G,1/G)-method will be
applied to find the solutions of equation (1.1), and
equation (1.2).
2 Methodology
Given an NPDE formed
F(u,ut,ux,uxx,utt ) = 0.(2.1)
The (G’/G,1/G)-method will be constructed as the
following steps:
Step 1. Setting the traveling wave variable
u(x,t) = U(
ξ
),
ξ
=x−Vt.(2.2)
Substituting (2.2) into (2.1) to get an ODE formed
Pu,u′,u′′,...=0,(2.3)
where Vis a constant, u′=du
d
ξ
.
Step 2. Given solutions of (2.3) can be expressed in
form:
u(
ξ
) =
N
∑
i=0
aifi+
N
∑
i=1
bifi−1g,(2.4)
where aiand bibe the constants will be achieved
later.
Step 3. Find the value of Nin (2.4) by balancing the
greatest order of derivative term and nonlinear term
in (2.3).
Step 4. Substitute (2.4) into (2.3) along with case
1 (shown below) to find the value ai,bi,V,
µ
,A1,A2,
and
λ
to gather the exact analytic solutions expressed
in the types of hyperbolic solutions, trigonometric
solutions, and rational solutions.
• Now, considering the second-order linear ordi-
nary differential equation (LODE):
G′′(
ξ
) +
λ
G(
ξ
) =
µ
.(2.5)
Using the transform f=G′
G,g=1
Gto get the sys-
tem of equations
f′=−f2+
µ
g−
λ
,g′=−f g.
Case 1. With
λ
<0, the specific solution of LODE
(2.5) is formed of hyperbolic functions:
G(
ξ
) = A1sinh−
λξ
+A2cosh−
λξ
+
µ
λ
.
and we gather
g2=−
λ
λ
2
σ
+
µ
2f2−2
µ
g+
λ
.
where A1,A2are the given constants and
σ
=A2
1−A2
2.
Case 2. With
λ
>0, the specific solution of LODE
(2.5) is formed of trigonometry functions as
G(
ξ
) = A1sin√
λξ
+A2cos√
λξ
+
µ
λ
.
and we have
g2=
λ
λ
2
σ
−
µ
2f2−2
µ
g+
λ
.
where A1,A2are the given constants and
σ
=A2
1+A2
2.
Case 3.
λ
=0 then the general solution of LODE
(2.5) is formed of rational functions:
G(
ξ
) =
µ
2
ξ
2+A1
ξ
+A2.
and we have
g2=
λ
A2
1−2
µ
A2f2−2
µ
g.
where A1,A2are the given constants.
Step 5. Using the same way by replacing (2.4) by
(2.3) coming along with cases 1, 2, 3 (shown above)
to gain the exact solutions of hyperbolic type solu-
tion, trigonometry type solutions, and rational type
solutions.
3 Appication
3.1 (G/G,1/G)-expansion method applied
for Benney-Luke equation
Apply two variable (G′/G,1/G)-expansion method
to find the traveling wave solutions of nonlinear
utt −uxx +uxxxx −uxxtt +utuxx +2uxuxt =0.(3.1)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.29
Nguyen Minh Tuan, Sanoe Koonprasert,
Sekson Sirisubtawee, Phayung Meesad, Nattawut Khansai
E-ISSN: 2224-2880
268
Volume 23, 2024
Step 1. Setting the traveling wave variable
u(x,t) = U(
ξ
),where
ξ
=x−Vt.
We construct the following terms
utt =V2U
ξ ξ
;uxx =U
ξ ξ
;
uxxxx =U
ξ ξ ξ ξ
;uxxtt =V2U
ξ ξ ξ ξ
;
utuxx =−VU
ξ
U
ξ ξ
;2uxuxt =−2VU
ξ
U
ξ ξ
.(3.2)
Substituting (3.2) into (3.1) to get the ODE formed
V2U
ξ ξ
−U
ξ ξ
+U
ξ ξ ξ ξ
−V2U
ξ ξ ξ ξ
−VU
ξ
U
ξ ξ
−2VU
ξ
U
ξ ξ
=0.(3.3)
Simplify on both sides Eq. (3.3), we have
V2−1U
ξ ξ
+1−V2U
ξ ξ ξ ξ
−3VU
ξ
U
ξ ξ
=0.
Integrate both sides of the equation and putting con-
stant equals zero, we have
V2−1U
ξ
+1−V2U
ξ ξ ξ
−3
2VU
ξ
2=0,(3.4)
where Vis a constant, U
ξ
=dU
d
ξ
.
Step 2. Given solutions of (3.4) will be illustrated as
follows:
U(
ξ
) =
N
∑
i=0
aifi+
N
∑
i=1
bifi−1g.
where aiand bibe the constants will be achieved later.
Step 3. Balancing both terms U
ξ ξ ξ
, and U
ξ
2:
N+3=2(N+1),
then N=1.
Step 4. So the solution is formed of
U(
ξ
) = a0+a1f(
ξ
) + b1g(
ξ
),(3.5)
where a0,a1,and b1are constants. Using the trans-
form f=G′
G,g=1
Gto get the system of equations as
follows
f′=−f2+
µ
g−
λ
,g′=−f g.(3.6)
Case 1. With
λ
<0, the specific solution of LODE
(3.4) is attained
G(
ξ
) = A1sinh−
λξ
+A2cosh−
λξ
+
µ
λ
.(3.7)
and we have
g2=−
λ
λ
2
σ
+
µ
2f2−2
µ
g+
λ
.(3.8)
where A1,A2are the given constants and
σ
=A2
1−A2
2.
For
λ
<0, substituting Eq. (3.5) into Eq. (3.4) and by
using Eq. (3.6) and Eq. (3.7)-(3.8) yields a set of al-
gebraic equations for a0,a1,b1,
µ
,
λ
, and V. Solving
the constructed simultaneous equation, the solutions
of algebraic values will be collected. We substitute
the values to Eq. (1.1) and select the exact solutions.
These systems are expressed in terms of
1,f(
ξ
)g(
ξ
),f(
ξ
)2,f(
ξ
),g(
ξ
),g(
ξ
)f(
ξ
)3,
g(
ξ
)f(
ξ
)2,g(
ξ
)f(
ξ
)4,f(
ξ
)3.(3.9)
corresponding to the following:
1 : −2
λ
λ
2A2
1−A2
2+
µ
23a1
−2(A1−A2)V2−3
4Va1−1
×(A1+A2)
λ
3+V2−1A2
1
+−V2+1A2
2
λ
2+
µ
2V2−1
λ
+
µ
2V2−1.
g(
ξ
)f(
ξ
)3:12b1V2−1
2Va1−1
λ
2A2
1−A2
2+
µ
24.
f(
ξ
)g(
ξ
):−14−5(A1−A2)V2−3
5Va1−1
×(A1+A2)
λ
3
7
+(A1−A2)(A1+A2)(V−1)(V+1)
λ
2
7
+
µ
2V2−3
7Va1−1
λ
+V2
µ
2
7−
µ
2
7
×b1
λ
2A2
1A2
2+
µ
23.
f(
ξ
)2:108(A1−A2)a1V2−3
8Va1−1
(A1+A2)
λ
3
5
+−(A1−A2)(A1+A2)(V−1)(V+1)a1
5
+3V b2
1
10
λ
2+a1V2−3
10Va1−1
µ
2
λ
−
µ
2a1(V−1)(V+1)
5
×
λ
2A2
1−A2
2+
µ
23.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.29
Nguyen Minh Tuan, Sanoe Koonprasert,
Sekson Sirisubtawee, Phayung Meesad, Nattawut Khansai
E-ISSN: 2224-2880
269
Volume 23, 2024
f(
ξ
):12b1
λ
2V2−1
2Va1−1
λ
2A2
1−A2
2+
µ
23
µ
.
g(
ξ
):2−5(A1−A2)V2−3
5Va1−1
×(A1+A2)
λ
3+V2−1A2
1
+−V2+1A2
2
λ
2
+
µ
2V2−1
λ
+
µ
2V2−1
λ
2A2
1−A2
2+
µ
23a1
µ
.
g(
ξ
)f(
ξ
)2:−24(A1−A2)a1V2−1
4Va1−1
×(A1+A2)
λ
2
+V b2
1
λ
4+a1V2−1
4Va1−1
µ
2
λ
2A2
1−A2
2+
µ
23
µ
.
f(
ξ
)4:12(A1−A2)a1V2−1
4Va1−1
(A1+A2)
λ
2
+V b2
1
λ
4+a1V2−1
4Va1−1
µ
2
λ
2A2
1−A2
2+
µ
23.
f(
ξ
)3:12b1
λ
V2−1
2Va1−1
λ
2A2
1−A2
2+
µ
23
µ
.
Case 2.
λ
>0 then the general solution of LODE
(3.4) is formed
G(
ξ
) = A1sin√
λξ
+A2cosh√
λξ
+
µ
λ
.
and we attain
g2=
λ
λ
2
σ
−
µ
2f2−2
µ
g+
λ
,
where A1,A2are the given constants and
σ
=A2
1+A2
2.
There is no solution existing in this case.
Case 3.
λ
=0 then the specific solution of LODE
(3.4) is formed
G(
ξ
) =
µ
2
ξ
2+A1
ξ
+A2.
and we have
g2=
λ
A2
1−2
µ
A2f2−2
µ
g,
where A1,A2are the given constants.
There is no solution existing in this case.
Step 5: will be moved to the next section.
3.2 (G/G,1/G)-expansion method applied
for extended Benney-Luke equation
Apply two variable (G′/G,1/G)-expansion method
to find the traveling wave solutions of the extended
Benney-Luke equation as follows:
utt −uxx +uxxxx −uxxtt +utuxx +2uxuxt
+utuxx =0.(3.10)
Step 1. Setting the shifting wave variable
u(x,t) = U(
ξ
),where
ξ
=x−Vt.
where Vis a given nonzero value.
We establish the following terms
utt =V2U
ξ ξ
;uxx =U
ξ ξ
;
uxxxx =U
ξ ξ ξ ξ
;uxxtt =V2U
ξ ξ ξ ξ
;
utuxx =−VU
ξ
U
ξ ξ
;2uxuxt =−2VU
ξ
U
ξ ξ
.(3.11)
Substituting (3.11) into (3.10) to get the ODE formed
V2U
ξ ξ
−U
ξ ξ
+U
ξ ξ ξ ξ
−V2U
ξ ξ ξ ξ
−2VU
ξ
U
ξ ξ
−2VU
ξ
U
ξ ξ
=0.(3.12)
Simplify on both sides equation (3.12), we have
V2−1U
ξ ξ
+1−V2U
ξ ξ ξ ξ
−4VU
ξ
U
ξ ξ
=0.
Integrate both sides of the equation and putting a con-
stant equal to zero, we have
V2−1U
ξ
+1−V2U
ξ ξ ξ
−2VU
ξ
2=0,
(3.13)
where Vis a constant, U
ξ
=dU
d
ξ
.
Step 2. Given solutions of (3.13) can be expressed in
form:
U(
ξ
) =
N
∑
i=0
aifi+
N
∑
i=1
bifi−1g.
where aiand bibe the constants will be achieved later.
Step 3. Balancing both sides equation (3.13), two
terms U
ξ ξ ξ
, and U
ξ
2:
N+3=2(N+1),
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.29
Nguyen Minh Tuan, Sanoe Koonprasert,
Sekson Sirisubtawee, Phayung Meesad, Nattawut Khansai
E-ISSN: 2224-2880
270
Volume 23, 2024
then N=1.
Step 4. So the solution is built in the form
U(
ξ
) = a0+a1f(
ξ
) + b1g(
ξ
).(3.14)
where a0,a1,and b1are constants.
Using the transform f=G′
G,g=1
Gto get the system
of equations
f′=−f2+
µ
g−
λ
,g′=−f g.(3.15)
Case 1.
λ
<0 then the general solution of LODE
(2.5) is formed of
G(
ξ
) = A1sinh−
λξ
+A2cosh−
λξ
+
µ
λ
.
(3.16)
and we have
g2=−
λ
λ
2
σ
+
µ
2f2−2
µ
g+
λ
.(3.17)
where A1,A2are the given constants and
σ
=A2
1−A2
2.
For
λ
<0, substituting Eq. (3.14) into Eq. (3.13) and
by using Eq. (3.15) and Eq. (3.16)-(3.17) leads to
a set of algebraic equations for a0,a1,b1,
µ
,
λ
, and
V. Solving the constructed simultaneous equation,
we select the exact solutions satisfied Eq. (1.2) These
systems of
1,f(
ξ
)g(
ξ
),f(
ξ
)2,f(
ξ
),g(
ξ
),f(
ξ
)4,f(
ξ
)3,
g(
ξ
)f(
ξ
)2,g(
ξ
)f(
ξ
)3.
corresponding to the following systems
1 : −2
λ
a1
λ
2A2
1−A2
2+
µ
23−2(A1−A2)
(A1+A2)V2−Va1−1
λ
3+V2−1A2
1
+−V2+1A2
2
λ
2+
µ
2V2−1
λ
+
µ
2V2−1.
f(
ξ
)g(
ξ
):12
λ
2A2
1−A2
2+
µ
24V2−2
3Va1−1b1.
f(
ξ
)2:−24(A1+A2)(A1−A2)V2−1
3Va1−1
a1
λ
2+V
λ
b2
1
3+
µ
2V2−1
3Va1−1a1
λ
2A2
1−A2
2+
µ
23
µ
.
f(
ξ
):−14
λ
2A2
1−A2
2+
µ
23
−5V2−4
5Va1−1(A1+A2) (A1−A2)
λ
3
7
+(A1−A2)(A1+A2)(V−1)(V+1)
λ
2
7
+
µ
2V2−4
7Va1−1
λ
+V2
µ
2
7−
µ
2
7b1.
g(
ξ
):10
λ
2A2
1−A2
2+
µ
23×
8(A1+A2)V2−1
2Va1−1×
(A1−A2)a1
λ
3
5+−(A1−A2)(A1+A2)
(V−1)(V+1)a1
5+2V b2
1
5
λ
2
+V2−2
5Va1−1
µ
2a1
λ
−
µ
2a1(V−1)(V+1)
5.
f(
ξ
)4:12
λ
λ
2A2
1−A2
2+
µ
23
V2−2
3Va1−1
µ
b1.
f(
ξ
)3:12(A1+A2)(A1−A2)V2−1
3Va1−1
a1
λ
2+V
λ
b2
1
3+
µ
2V2−1
3Va1−1
a1
λ
2A2
1−A2
2+
µ
23.
g(
ξ
)f(
ξ
)2:12(A1−A2)V2−1
3Va1−1
a1(A1+A2)
λ
2+V
λ
b2
1
3
+
µ
2V2−1
3Va1−1a1
λ
2A2
1−A2
2+
µ
23.
g(
ξ
)f(
ξ
)3:12V2−2
3Va1−1
µ
λ
2A2
1−A2
2+
µ
23
λ
b1.
Case 2.
λ
>0, there is no solution existing in this
case.
Case 3.
λ
=0, there is no solution existing in this
case.
Step 5: will be transferred to the next section.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.29
Nguyen Minh Tuan, Sanoe Koonprasert,
Sekson Sirisubtawee, Phayung Meesad, Nattawut Khansai
E-ISSN: 2224-2880
271
Volume 23, 2024
4 Solution Benney-Luke equation
Using (G’/G,1/G)
4.1 Solution of the Benney-Luke Eq. (1.1)
By solving the system of simultaneous equations, the
solutions are obtained as follows:
Case 1:
Using the condition of the first case we have the con-
stant
λ
=−1
4,
µ
=0,a1=4(V2−1)
V,b1=0,
we calculate the term of the variant equation estab-
lished as the following
U(
ξ
) = a0+a1f(
ξ
),
where a0is arbitrary. Using
G(
ξ
) = A1sinh(
ξ
2) + A2cosh(
ξ
2).
and f(
ξ
) = G′
G.We have the hyperbolic solution
u11 =a0−2(V2−1)(A1coshVt
2−x
2−A2sinhV t
2−x
2)
V(A1sinhV t
2−x
2−A2coshV t
2−x
2),
where V,A1,A2are arbitrary constants. The solution
u11, kink type solution, is depicted in Figure 2 (Ap-
pendix).
Case 2:
Using the condition of the first case we have the con-
stant
λ
=−1,{
µ
=0,
µ
=
µ
},a1=2(V2−1)
V,
{b1=±2A2
1−A2
2V2−1
V,
b1=±4A2
1−4A2
2+4
µ
2(V2−1)
V}.
We calculate the term of the variant equation estab-
lished as the following
U(
ξ
) = a0+a1f(
ξ
) + b1g(
ξ
),
where a0is arbitrary. Using
G(
ξ
) = A1sinh
ξ
+A2cosh
ξ
+
µ
.
and
f(
ξ
) = G′
G,g(
ξ
) = 1
G.
We have the hyperbolic solution
u12 =a0+a1
A1cosh(Vt −x)−A2sinh(Vt −x)
−A1sinh(Vt −x) + A2cosh(Vt −x) +
µ
−b11
−A1sinh(Vt −x) + A2cosh(Vt −x) +
µ
,
where
a1=2V2−1
V,b1=±2A2
1−A2
2V2−1
V,
{
µ
=0,
µ
=
µ
},b1=±4A2
1−4A2
2+4
µ
2(V2−1)
V.
and V,A1,A2are arbitrary constants. The solution
u12, traveling wave solution, is performed in Figure
3, and Figure 1 (Appendix).
4.2 Solution of the extended Benney-Luke
equation (1.2)
By solving the system of simultaneous equations, the
solutions are obtained as the following:
Case 1:
Using the condition of the first case we have the con-
stant
λ
=−1
4,
µ
=0,a1=3(V2−1)
V,b1=0.
We calculate the term of the variant equation estab-
lished as the following
U(
ξ
) = a0+a1f(
ξ
),
a0is arbitrary. Using
G(
ξ
) = A1sinh(
ξ
2) + A2cosh(
ξ
2),
and f(
ξ
) = G′
G, we have the exact solution
u21 =a0−3(V2−1)(A1coshVt
2−x
2−A2sinhV t
2−x
2)
2V(A1sinhVt
2−x
2−A2coshV t
2−x
2),
where V,A1,A2are arbitrary constants. The solution
u21, kink type solution, is delineated in Figure 4 (Ap-
pendix).
Case 2:
Using the condition of the first case we have the con-
stant
λ
=−1,
µ
=0,a1=3(V2−1)
2V,
b1=±9A2
1−9A2
2+9
µ
2V2−1
2V.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.29
Nguyen Minh Tuan, Sanoe Koonprasert,
Sekson Sirisubtawee, Phayung Meesad, Nattawut Khansai
E-ISSN: 2224-2880
272
Volume 23, 2024
We calculate the term of the variant equation estab-
lished as the following
U(
ξ
) = a0+a1f(
ξ
) + b1g(
ξ
),
a0is arbitrary. Using
G(
ξ
) = A1sinh
ξ
+A2cosh
ξ
−
µ
,
and
f(
ξ
) = G′
G,g(
ξ
) = 1
G.
We have the exact solution
u22 =a0+a1
A1cosh(Vt −x) + A2sinh(V t −x)
A1sinh(Vt −x) + A2cosh(Vt −x)
−c11
−A1sinh(Vt −x) + A2cosh(Vt −x),
where
a1=3V2−1
2V,c1=±3A2
1−A2
2+
µ
2V2−1
2V.
where V,A1,A2are arbitrary constants. The solution
u22, Traveling wave solution, is illustrated in Figure
5 (Appendix).
All the analytic solutions attained corresponding to
Eq. (1.1), and Eq. (1.2) have been exactly substituted
for checking satisfaction.
5 Conclusion
By utilizing the prevalent (G’/G,1/G)-method, the hy-
perbolic function solutions of the Benney-Luke equa-
tion, and the extended Benney-Luke equation have
been found and illustrated in both two and three di-
mensions. The (G’/G,1/G)-method has supported the
variety of solutions of the Benney-Luke equation that
are meaningful in studying the propagation of the wa-
ter wave surface. Compared to previous studies, the
present method has attained a new form of kink-type
solutions, and traveling wave solutions that play a sig-
nificant role in studying the water surface tensions.
Acknowledgment:
The authors acknowledge the anonymous reviewer’s
comments, which improve the manuscript’s quality,
and kindly provide advice and valuable discussions.
References:
[1] Wazwaz, A.-M. (2009). Partial differential
equations and solitary waves theory. Higher
Education Press; Springer.
[2] Peng, L.-J. (2022). Dynamics investigation on a
KadomtsevPetviashvili equation with variable
coefficients. Open Physics, 20(1), 10411047.
https://doi.org/10.1515/phys-2022-0207
[3] Inan, I. E., Ugurlu, Y., & Inc, M. (2015). New
Applications of the (G/G,1/G)-Expansion
Method. Acta Physica Polonica A, 128(2),
245252.
https://doi.org/10.12693/APhysPolA.128.245
[4] Miah, M. M., Ali, H. M. S., Akbar, M. A., &
Seadawy, A. R. (2019). New applications of the
two variable (G’/G,1/G)-expansion method for
closed-form traveling wave solutions of
integro-differential equations. Journal of Ocean
Engineering and Science, 4(2), 132143.
https://doi.org/10.1016/j.joes.2019.03.001
[5] Mamun Miah, M., Shahadat Ali, H. M., Ali
Akbar, M., & Majid Wazwaz, A. (2017). Some
applications of the (G’/G,1/G)-expansion
method to find new exact solutions of NLEEs.
The European Physical Journal Plus, 132(6),
252. https://doi.org/10.1140/epjp/i2017-11571-0
[6] Huda, M. (2019). The two variables (G /G,1/G)
expansion method for investigating exact
solutions to nonlinear medium equal width
equation. JOURNAL OF MECHANICS OF
CONTINUA AND MATHEMATICAL
SCIENCES, 14(2).
https://doi.org/10.26782/jmcms.2019.04.00003
[7] Zayed, E. M. E., & Alurrfi, K. A. E. (2014). The
( G / G, 1 / G ) -Expansion Method and Its
Applications for Solving Two Higher Order
Nonlinear Evolution Equations. Mathematical
Problems in Engineering, 2014, 120.
https://doi.org/10.1155/2014/746538
[8] Zayed, E. M. E., & Hoda Ibrahim, S. A. (2013).
The two Variable (G/G,1/G) -expansion Method
for Finding Exact Traveling Wave Solutions of
the (3+1) -Dimensional Nonlinear Potential
Yu-Toda-Sasa-Fukuyama equation. Proceedings
of the 2013 International Conference on
Advanced Computer Science and Electronics
Information. 2013 International Conference on
Advanced Computer Science and Electronics
Information, Beijing, China.
https://doi.org/10.2991/icacsei.2013.98
[9] Li, L., Li, E., & Wang, M. (2010). The
(G’/G,1/G)-expansion method and its
application to traveling wave solutions of the
Zakharov equations. Applied Mathematics-A
Journal of Chinese Universities, 25(4), 454462.
https://doi.org/10.1007/s11766-010-2128-x
[10] Iatkliang, T., Kaewta, S., Tuan, N. M., &
Sirisubtawee, S. (2023). Novel Exact Traveling
Wave Solutions for Nonlinear Wave Equations
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.29
Nguyen Minh Tuan, Sanoe Koonprasert,
Sekson Sirisubtawee, Phayung Meesad, Nattawut Khansai
E-ISSN: 2224-2880
273
Volume 23, 2024
with Beta-Derivatives via the sine-Gordon
Expansion Method. WSEAS TRANSACTIONS
ON MATHEMATICS, 22, 432450.
https://doi.org/10.37394/23206.2023.22.50
[11] Yoku, A., & Durur, H. (2021).
(G/G,1/G)-expansion method for analytical
solutions of Jimbo-Miwa equation. Cumhuriyet
Science Journal, 42(1), 8898.
https://doi.org/10.17776/csj.689759
[12] Islam, S. M. R., Khan, K., & Woadud, K. M. A.
A. (2018). Analytical studies on the BenneyLuke
equation in mathematical physics. Waves in
Random and Complex Media, 28(2), 300309.
https://doi.org/10.1080/17455030.2017.1342880
[13] Gundogdu, H., & Gozukizil, O. F. (2021). On
the new type of solutions to Benney-Luke
equation. Boletim Da Sociedade Paranaense de
Matemática, 39(5), 103111.
https://doi.org/10.5269/bspm.41244
[14] Hossain, A. K. M. K. S., & Akbar, M. A.
(2021). Traveling wave solutions of Benny Luke
equation via the enhanced ( G / G )-expansion
method. Ain Shams Engineering Journal, 12(4),
41814187.
https://doi.org/10.1016/j.asej.2017.03.018
[15] Wang, K., & Wang, G. (2021). Study on the
explicit solutions of the BenneyLuke equation
via the variational direct method. Mathematical
Methods in the Applied Sciences, 44(18),
1417314183. https://doi.org/10.1002/mma.7683
[16] Durur, H., & Yoku, A. (2021). Exact solutions
of the BenneyLuke equation via (1/G)-expansion
method. Bilecik eyh Edebali Üniversitesi Fen
Bilimleri Dergisi, 8(1), 5664.
https://doi.org/10.35193/bseufbd.833244
[17] Akter, J., & Ali Akbar, M. (2015). Exact
solutions to the BenneyLuke equation and the
Phi-4 equations by using modified simple
equation method. Results in Physics, 5, 125130.
https://doi.org/10.1016/j.rinp.2015.01.008
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Nguyen Minh Tuan: Conceptualization, data cu-
ration, investigation, methodology, software, visu-
alization, writing-original draft and writing-review
and editing, validation, visualization, writing-original
draft and writing-review and editing.
Sanoe Koonprasert: Conceptualization, formal analy-
sis, methodology, resources, supervision, validation,
visualization, and writing review and editing.
Sekson Sirisubtawee: Conceptualization, formal
analysis, methodology, resources, supervision, vali-
dation, visualization, and writing review and editing.
Phayung Meesad: Conceptualization, formal analy-
sis, methodology, resources, supervision, validation,
visualization, and writing review and editing.
Nattawut Khansai: Employed for extended Benney-
Luke equation consisting of conceptualization, in-
vestigation, methodology, project administration, and
writing review and editing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself No
funding was received for conducting this study.
Conflicts of Interest The authors
have no conflicts of interest to
declare that are relevant to the content of this
article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
APPENDIX
Figure 1: Traveling wave solution, u12, when V=
0.5;a1=0.5;;A1=0.5;A2=0.5;a0=0.5;b1=
0.5,
µ
=0.2.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.29
Nguyen Minh Tuan, Sanoe Koonprasert,
Sekson Sirisubtawee, Phayung Meesad, Nattawut Khansai
E-ISSN: 2224-2880
274
Volume 23, 2024
Figure 2: Kink type solution, u12, performance when
V=0.5;a1=0.5;a0=0.5;A1=0.5;A2=0.5;b1=0.5.
Figure 3: Traveling wave solution performance, u12,
when V=−0.5;a1=0.5;a0=0.5;A1=0.5;A2=
0.5;b1=0.5.
Figure 4: Kink type solution, u21, when V=1
3√10 −
1/3;a1=3(V2−1)
2V;a0=0.5;A1=0.7;A2=0.5.
.
Figure 5: Traveling wave solution, u22, when V=
0.5;a1=0.2;a0=0.5;A1=0.7;A2=0.5;b1=0.5;b1=
0.5;
µ
=0.5;.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.29
Nguyen Minh Tuan, Sanoe Koonprasert,
Sekson Sirisubtawee, Phayung Meesad, Nattawut Khansai
E-ISSN: 2224-2880
275
Volume 23, 2024
