Exact Solutions For Non Linear Partial Differential Equations
Using Cosine Function Method
M.F. El-Sabbagh, R. Zait And R.M. Abdelazeem
Mathematics Department, Faculty of Science, Minia University, Egypt
Corresponding e-mail: rasha_math24@yahoo.com
__________________________________________________________________________
Abstract
In this paper, we establish new exact solutions for some nonlinear partial
differential equations of interest such as the general modified Benjamin–Bona–
Mahony (GMBBM) equation,the Boussinesq equation, the general biharmonic
equation, the ∅4 –equation and the Zakharov-Kuznetsov (ZK) equation. The cosine
method is also used to construct periodic and solitary wave solutions for the
considered equations.
Keywords: Nonlinear PDEs, Exact Solutions, Nonlinear Waves, Cosine-function
method and solitary wave solutions.
____________________________________________________________________
1. Introduction
The study of numerical methods for the solution of nonlinear partial differential
equations has enjoyed an intense period of activity over the last 40 years from both
theoretical and practical points of view. Improvements in numerical techniques, together
with the rapid advances in computer technology, have meant that many of the partial
differential equations arising from engineering and scientific applications, which
were previously intractable, can now, be mroutinely solved.
The aim of the present paper is to extend the cosine-function method (refs.[1], [2],
[3], [4], [5]) introduced for finding new solitary solutions to some important
equations such as the general modified Benjamin–Bona–Mahony (GMBBM)
equation, the Boussinesq equation, the general biharmonic equation, the ∅4 –equation
and the Zakharov-Kuznetsov (ZK) equation .
2. Cosine function method
Consider a nonlinear partial differential equation:
𝐹(𝑢, 𝑢𝑡 , 𝑢𝑥 , 𝑢𝑥𝑥 , 𝑢𝑥𝑥𝑡 , … … . . ) = 0
(1)
where u(x, t) is the solution of nonlinear partial differential equation Eq. (1). We use
the following transformations:
1
𝑢(x, t) = 𝑓(𝜉),
(2)
𝜉 = 𝑥 − 𝑐𝑡
This enable us to use the following change:
𝜕
𝜕𝑡
𝑑
𝜕
𝑑𝜉
𝜕𝑥
(·)=−c (·),
𝑑
𝜕2
𝑑2
𝑑𝜉
𝜕𝑥
𝑑𝜉 2
(·)= (·),
2 (·)=
(·),…………….
(3)
Use equation (3) to transform equation (1) from nonlinear partial deferential equation
to nonlinear ordinary differential equation:
𝑑𝑓 𝑑2 𝑓 𝑑3 𝑓
𝐺 (𝑓, , 2 , 3 , … … . . )
𝑑𝜉 𝑑𝜉 𝑑𝜉
=0
(4)
The solution of equation (4) can be expressed as
𝑢(x, t) = 𝑓(𝜉)= 𝜆𝑐𝑜𝑠 𝛽 (𝜇𝜉),
|𝜉| ≤
𝜋
(5)
2𝜇
where 𝜆, 𝛽 and 𝜇 are unknown parameters which will be determined.
Then we have
𝑑𝑓
𝑑𝜉
=−𝜆 𝛽 𝜇𝑐𝑜𝑠 𝛽−1 (𝜇𝜉) sin(𝜇𝜉)
𝑑2 𝑓
𝑑𝜉 2
2
= 𝜆 𝜇 𝛽(𝛽 − 1) 𝑐𝑜𝑠 𝛽−2 (𝜇𝜉)- 𝜆 𝜇 2 𝛽(𝛽 − 1)𝑐𝑜𝑠 𝛽 (𝜇𝜉)- 𝜆 𝜇 2 𝛽 𝑐𝑜𝑠 𝛽 (𝜇𝜉)
(6)
Substituting equation (6) into the nonlinear ordinary differential equation (4) gives a
trigonometric equation of 𝑐𝑜𝑠 𝛽 (μξ) terms. To determine the parameters first balancing the
exponents of each pair of cosine to determine 𝛽 . Then we collect all terms with the
same power in 𝑐𝑜𝑠 𝛽 (μξ) and put to zero their coefficients to get a system of algebraic
equations among the unknowns β, λ and μ.
Now, the problem is reduced to a system of algebraic equations that can be solved to
obtain the unknown parameters β, λ and μ. Hence, the solution considered in equation
(5) is obtained.
3. Applications
In order to illustrate the effectiveness of the proposed method examples of
mathematical interest are chosen as follows:
2
3.1. The general modified Benjamin-Bona-Mahony (GMBBM) equation
A great deal of research work has been invested for the analytical expressions of
traveling wave solutions of the nonlinear dispersive MBBM equations, written in the
form 𝑢𝑡 +𝑢𝑥 +a𝑢2 𝑢𝑥 +𝑢𝑥𝑥𝑥 = 0 (ref. [6]), which was first derived to describe an
approximations for surface long waves in nonlinear dispersive media. The equation
can also characterize the hydro magnetic waves in cold plasma, acoustic waves in
anharmonic crystals and acoustic-gravity waves in compressible fluids.
In this section we introduce solitary exact solution for a generalized MBBM
equation we call it GMBBM which is as follows:
𝑢𝑡 +𝑢𝑥 +a𝑢𝑝 𝑢𝑥 +𝑢𝑥𝑥𝑥 = 0
(7)
We use here the above mentioned cosine method
From equations (2) and (3), equation (7) becomes
𝑑𝑓(𝜉)
−c
𝑑𝜉
+
𝑑𝑓(𝜉)
𝑑𝜉
+
𝑑𝑓(𝜉)
𝑎(𝑓(𝜉))𝑃
𝑑𝜉
+
𝑑3 𝑓(𝜉)
𝑑𝜉 3
=0.
(8)
By integrating we have (with no constants for integration)
𝑎
(1 − 𝑐)𝑓(𝜉) +
𝑃+1
(𝑓(𝜉 ))𝑃+1 +
𝑑2 𝑓(𝜉)
=0.
𝑑𝜉 2
(9)
Substituting from equations (5) and (6) in equation (9), we have:
(1 − 𝑐)𝜆𝑐𝑜𝑠 𝛽 (𝜇𝜉) +
𝑎
𝑃+1
𝑃+1
𝜆𝑃+1 (𝑐𝑜𝑠 𝛽 (𝜇𝜉))
+ 𝜆 𝜇2 𝛽(𝛽 − 1) 𝑐𝑜𝑠 𝛽−2 (𝜇𝜉)-
𝜆 𝜇2 𝛽(𝛽 − 1)𝑐𝑜𝑠 𝛽 (𝜇𝜉)- 𝜆 𝜇2 𝛽 𝑐𝑜𝑠 𝛽 (𝜇𝜉) = 0
(10)
By balancing the exponents of each pair of cosine we have:
(1 − 𝑐)𝜆 − 𝜆 𝜇2 𝛽(𝛽 − 1) − 𝜆 𝜇2 𝛽=0,
𝑎
𝜆𝑃+1 + 𝜆 𝜇2 𝛽(𝛽 − 1) = 0
βP+2=0,
(11)
𝑃+1
Using MATHEMATICA package software for solving the system equations (11) we
obtain:
𝛽=
−2
1
𝑐−1
𝑃
2
1+𝑃
, 𝜇=− √
𝜆=
+ 𝑐𝑃 − 𝑃 − 𝑐 + 1 ,
3
1
1
−𝑃 2−2𝑐+𝑃−𝑐𝑃 𝑃
2 (
)
𝑎
(12)
Substitute in equation (5), we obtain new exact soliton solution of equation (7) in the
form:
𝑢(x, t) = 2
1
𝑃
−
1
−2
2−2𝑐+𝑃−𝑐𝑃 𝑃
𝑐𝑜𝑠 𝑃
(
)
𝑎
1
𝑐−1
(− 2 √1+𝑃 + 𝑐𝑃 − 𝑃 − 𝑐 + 1 ) (𝑥 − 𝑐𝑡), with
−𝜋
|𝑥 − 𝑐𝑡| ≤
(13)
𝑐−1
+𝑐𝑃−𝑃−𝑐+1
1+𝑃
√
As special case, if P= 2, we have the modified Benjamin-Bona-Mahony (MBBM)
equation in the form:
𝑢𝑡 +𝑢𝑥 +a𝑢2 𝑢𝑥 +𝑢𝑥𝑥𝑥 = 0
(14)
Putting P=2 in equation (13), we obtain new exact soliton solution of the special
MBBM (equation (14)) in the form:
𝑢(x, t) = √2√
1−𝑐
𝑎
1
𝑐−1
2
3
𝑐𝑜𝑠 −1 (− √
−𝜋
+ 𝑐 − 1 (x-ct)), 𝑤𝑖𝑡ℎ |𝑥 − 𝑐𝑡| ≤
√
𝑐−1
+𝑐−1
3
(15)
which is different from its solution obtained in ref. [6].
3.2. The Boussinesq equation
The Boussinesq equation describes propagation of waves in weakly nonlinear and
weakly dispersive media (ref. [6]). In fluid dynamics, the Boussinesq approximation
for water waves is an approximation valid for weakly non-linear and fairly long
waves. The approximation is named after Joseph Boussinesq, who first derived them
in response to the observation by John Scott Russell of the wave of translation (also
known as solitary wave or soliton). The 1872 paper of Boussinesq introduces the
equations now known as the Boussinesq equations (refs.[7], [8]).
The Boussinesq approximation for water waves takes into account the vertical
structure of the horizontal and vertical flow velocity. This results in non-linear partial
differential equations, called Boussinesq-type equations, which incorporate frequency
dispersion (as opposite to the shallow water equations, which are not frequencydispersive). In coastal engineering, Boussinesq-type equations are frequently used in
computer models for the simulation of water waves in shallow seas and harbours.
In this section we introduces new exact solution of the Boussinesq equation via
the cosine method as follows:
Consider the nonlinear partials differential equation:
4
𝑢𝑡𝑡 –𝑢𝑥𝑥 − 𝑢𝑥𝑥𝑥𝑥 − 3(𝑢𝑃 )𝑥𝑥 = 0
(16)
From equations (2) and (3), equation (16) become
𝑐2
𝑑 2 𝑓(𝜉)
𝑑𝜉 2
−
𝑑 2 𝑓(𝜉)
𝑑𝜉 2
−
𝑑 4 𝑓(𝜉)
𝑑𝜉 4
− 3P((P − 1)(𝑓(𝜉))𝑃−2 (
𝑑𝑓(𝜉) 2
𝑑𝜉
) + (𝑓(𝜉))𝑃−1
𝑑2 𝑓(𝜉)
𝑑𝜉2
(17)
=0
Substituting equations (5), and (6) in equation (17), we have:
(𝑐 2 − 1)𝜆2 𝜇2 𝛽(𝛽 − 1)𝑐𝑜𝑠𝛽−2 (𝜇𝜉) − (𝑐 2 − 1)𝜆𝜇 2 𝛽(𝛽 − 1)𝑐𝑜𝑠𝛽 (𝜇𝜉) − (𝑐 2 − 1)𝜆𝜇 2 𝛽𝑐𝑜𝑠𝛽 (𝜇𝜉) −
3𝑃𝜆𝑃 𝜇2 𝛽(𝛽 − 1)𝑐𝑜𝑠𝑃𝛽−2 (𝜇𝜉) + 3𝑃𝜆𝑃 𝜇2 𝛽(𝛽 − 1)𝑐𝑜𝑠𝑃𝛽 (𝜇𝜉) + 3𝑃𝜆𝑃 𝜇 2 𝛽𝑐𝑜𝑠𝑃𝛽 (𝜇𝜉) −
3𝑃(𝑃 − 1)𝜆𝑃 𝜇2 𝛽 2 𝑐𝑜𝑠𝑃𝛽−2 (𝜇𝜉) + 3𝑃(𝑃 − 1)𝜆𝑃 𝜇 2 𝛽 2 𝑐𝑜𝑠𝑃𝛽 (𝜇𝜉) − 𝜆𝜇 4 𝛽(𝛽 − 1)(𝛽 − 2)(𝛽 −
3)𝑐𝑜𝑠𝛽−4 (𝜇𝜉) + 𝜆𝜇 4 𝛽(𝛽 − 1)(𝛽 − 2)(𝛽 − 3)𝑐𝑜𝑠𝛽−2 (𝜇𝜉) + 𝜆𝜇 4 𝛽(𝛽 − 1)(𝛽 − 2)𝑐𝑜𝑠𝛽−2 (𝜇𝜉) +
𝜆𝜇 4 𝛽 2 (𝛽 − 1)2 𝑐𝑜𝑠𝛽−2 (𝜇𝜉) − 𝜆𝜇 4 𝛽 2 (𝛽 − 1)2 𝑐𝑜𝑠𝛽 (𝜇𝜉) − 𝜆𝜇 4 𝛽 2 (𝛽 − 1)𝑐𝑜𝑠𝛽 (𝜇𝜉) +
𝜆𝜇 4 𝛽 2 (𝛽 − 1)𝑐𝑜𝑠𝛽−2 (𝜇𝜉) − 𝜆𝜇 4 𝛽 2 (𝛽 − 1)𝑐𝑜𝑠𝛽 (𝜇𝜉) − 𝜆𝜇 4 𝛽 2 𝑐𝑜𝑠𝛽 (𝜇𝜉) = 0.
(18)
By balancing the exponents of each pair of cosine we have:
−(𝑐 2 − 1)𝜆𝜇2 𝛽(𝛽 − 1) − (𝑐 2 − 1)𝜆𝜇2 𝛽 − 𝜆𝜇4 𝛽 2 (𝛽 − 1)2 − 𝜆𝜇4 𝛽 2 (𝛽 − 1)
− 𝜆𝜇4 𝛽 2 (𝛽 − 1) − 𝜆𝜇4 𝛽 2 = 0,
(𝑐 2 − 1)𝜆2 𝜇2 𝛽(𝛽 − 1) + 𝜆𝜇4 𝛽(𝛽 − 1)(𝛽 − 2)(𝛽 − 3) + 𝜆𝜇4 𝛽(𝛽 − 1)(𝛽 − 2) +
𝜆𝜇4 𝛽 2 (𝛽 − 1)2 +𝜆𝜇4 𝛽 2 (𝛽 − 1) = 0,
3𝑃𝜆𝑃 𝜇2 𝛽(𝛽 − 1) + 3𝑃𝜆𝑃 𝜇2 𝛽 + 3𝑃(𝑃 − 1)𝜆𝑃 𝜇2 𝛽2 = 0
(19)
𝛽 ( 𝑃 − 1) + 2 = 0 ,
−3𝑃𝜆𝑃 𝜇2 𝛽(𝛽 − 1) − 3𝑃 (𝑃 − 1)𝜆𝑃 𝜇2 𝛽2 − 𝜆𝜇4 𝛽 (𝛽 − 1)(𝛽 − 2)(𝛽 − 3) = 0
Using MATHEMATICA package software for solving the system equations (19) we
obtain:
𝛽=
−2
𝑃−1
, 𝜇 = √1 − 𝑐 2 − 2𝑃 + 2𝑐 2 𝑃 + 𝑃 2 − 𝑐 2 𝑃2 ,
𝜆 = −𝑃2 − 1.
(20)
Substitute in equation (5), we obtain exact soliton solution of equation (16) in the
form:
−2
𝑢(x, t) = (−𝑃2 − 1)𝑐𝑜𝑠 𝑃−1 (√1 − 𝑐 2 − 2𝑃 + 2𝑐 2 𝑃 + 𝑃2 − 𝑐 2 𝑃2 )(𝑥 − 𝑐𝑡), with
|𝑥 − 𝑐𝑡| ≤
𝜋
2√1−𝑐 2 −2𝑃+2𝑐 2 𝑃+𝑃2 −𝑐 2 𝑃2
As special case, if P= 2, then we have the Bounssinesq equation in the form:
5
( 21)
𝑢𝑡𝑡 −𝑢𝑥𝑥 -𝑢𝑥𝑥𝑥𝑥 − 3(𝑢2 )𝑥𝑥 = 0
(22)
Putting P=2 in equation (21), we thus obtain new exact soliton solution for the
Bounssinesq equation in the form:
𝑢(x, t) = −5𝑐𝑜𝑠 −1 √1 − 𝑐 2 (𝑥 − 𝑐𝑡), 𝑤𝑖𝑡ℎ |𝑥 − 𝑐𝑡| ≤
𝜋
(23)
2√1−𝑐 2
which is different from the solution obtained in ref. [8].
3.3.The general biharmonic equation
Consider the nonlinear partial differential equation (known as the general
biharmonic equation ( ref. [9] )):
𝑢𝑡 + 𝑢𝑥 + 𝑢𝑝 𝑢𝑥 − 𝑢𝑥𝑥𝑡 = 0
(24)
From equations (2) and (3), equation (24) become:
𝑑𝑓(𝜉)
−c
𝑑𝜉
+
𝑑𝑓(𝜉)
𝑑𝜉
+
𝑑𝑓(𝜉)
(𝑓(𝜉))𝑃
𝑑𝜉
𝑑3 𝑓(𝜉)
+c
𝑑𝜉 3
=0.
(25)
By integrating we have
(1 − 𝑐)𝑓(𝜉) +
1
𝑃+1
(𝑓(𝜉))𝑃+1 + c
𝑑 2 𝑓(𝜉)
𝑑𝜉 2
=0.
(26)
Substituting from equations (5) and (6) in (26), we have:
(1 − 𝑐)𝜆𝑐𝑜𝑠 𝛽 (𝜇𝜉) +
1
𝑃+1
𝑃+1
𝜆𝑃+1 (𝑐𝑜𝑠 𝛽 (𝜇𝜉))
+ 𝑐( 𝜆 𝜇2 𝛽(𝛽 − 1) 𝑐𝑜𝑠 𝛽−2 (𝜇𝜉)-
𝜆 𝜇2 𝛽(𝛽 − 1)𝑐𝑜𝑠 𝛽 (𝜇𝜉)- 𝜆 𝜇2 𝛽 𝑐𝑜𝑠 𝛽 (𝜇𝜉)) = 0.
(27)
By balancing the exponents of each pair of cosine we have:
(1 − 𝑐)𝜆 − 𝑐𝜆 𝜇 2 𝛽(𝛽 − 1) − 𝑐𝜆 𝜇 2 𝛽=0,
βP+2=0,
1
𝑃+1
𝜆𝑃+1 + 𝑐𝜆 𝜇 2 𝛽(𝛽 − 1) = 0
(28)
Using MATHEMATICA package software for solving the system equations (28) we
obtain:
𝛽=
−2
√𝑃2 −𝑐𝑃2
𝑃
2√𝑐
, 𝜇=−
, 𝜆=2
−1⁄
𝑃 (2𝑐
− 3𝑃 + 3𝑃𝑐 − 𝑃2 + 𝑐𝑃2 − 2)
1⁄
𝑃
Substitute in equation (5), we obtain new exact soliton solution of equation (24)
takes the form:
6
(29)
𝑢(x, t) = 2
−1⁄
𝑃 (2𝑐
with |𝑥 − 𝑐𝑡| ≤ −
− 3𝑃 + 3𝑃𝑐 − 𝑃2 + 𝑐𝑃2 − 2)
1⁄
𝑃
−2
𝑐𝑜𝑠 𝑃 (−
√𝑃2 −𝑐𝑃2
2√𝑐
(𝑥 − 𝑐𝑡)),
𝜋 √𝑐
(30)
√𝑃2 −𝑐𝑃2
As special case, if P=1 , we have the biharmonic equation in the form:
𝑢𝑡 + 𝑢𝑥 +u𝑢𝑥 - 𝑢𝑥𝑥𝑡 = 0
(31),
Putting P=1 in equation (30), we obtain new exact soliton solution of the biharmonic
equation in the form:
𝑢(x, t) = 3(𝑐 − 1)𝑐𝑜𝑠 −2 (−
√1−𝑐
(x-ct)),
2 √𝑐
𝑤𝑖𝑡ℎ |𝑥 − 𝑐𝑡| ≤ −
𝜋 √𝑐
√1−𝑐
(32)
3.4. The ∅𝟒 –equation
Consider the following nonlinear partial differential equation (known as ∅4 –equation
(refs. [10], [11] ):
∅𝑡𝑡 − ∅𝑥𝑥 − ∅ + ∅3 = 0
(33)
From equations (2) and (3), equation (33) becomes in the following form:
𝑐2
𝑑 2 𝑓(𝜉)
𝑑𝜉 2
−
𝑑 2 𝑓(𝜉)
𝑑𝜉 2
− 𝑓(𝜉) + (𝑓(𝜉))3 =0
(34)
Substituting from equations (5) and (6) in equation (34), we have:
(𝑐 2 − 1)[−𝜆𝛽𝜇2 𝑐𝑜𝑠 𝛽 (𝜇𝜉) + 𝜆𝜇2 𝛽(𝛽 − 1)𝑐𝑜𝑠𝛽−2 (𝜇𝜉) − 𝜆𝜇2 𝛽(𝛽 − 1)𝑐𝑜𝑠 𝛽 (𝜇𝜉)] −
𝜆𝑐𝑜𝑠 𝛽 (𝜇𝜉) + 𝜆3 𝑐𝑜𝑠 3𝛽 (𝜇𝜉) = 0
(35)
By balancing the exponents of each pair of cosine we have:
(𝑐 2 − 1)(−𝜆𝛽𝜇2 − 𝜆𝛽𝜇2 (𝛽 − 1)) − 𝜆 = 0,
𝛽 + 1 = 0,
(36)
(𝑐 2 − 1)𝜆𝜇2 𝛽(𝛽 − 1) + 𝜆3 = 𝟎
Using MATHEMATICA package software for solving the system equations (36) we
obtain:
𝜆 = −√2,
𝛽 = −1,
𝜇=
−𝟏
√𝟏−𝒄𝟐
(37)
Substitute in equation (5), we obtain new exact soliton solution of the ∅4 –equation
in the form:
7
∅(𝑥, 𝑡) = −√2 cos −1 (
−(x−ct)
π√1−c2
√1−c
2
), 𝑤𝑖𝑡ℎ |x − ct| ≤ −
2
(38)
3.5 The Zakharov -Kuznetsov (ZK) equation
This ZK appears in many areas of physics, applied Mathematics, and Engineering.
In particular, it shows up in the areas of Plasma Physics. The ZK govern the behavior of
weakly nonlinear ion-acoustics waves in a plasma comprising of cold ion and hot
isothermal electron in the presence of a uniform magnetic field (refs. [12], [13], [14],
[15], [16], [17]).
The ZK equation is given by
𝑢𝑡 + 𝑎𝑢𝑛 𝑢𝑥 + 𝑏(𝑢𝑥𝑥 + 𝑢𝑦𝑦 )𝑥 = 0.
(39)
setting
𝑢(𝑥, 𝑦, 𝑡) = 𝑓(𝜉) = 𝜆𝑐𝑜𝑠 𝛽 (𝜇𝜉), 𝑤ℎ𝑒𝑟𝑒 𝜉 = 𝑥 + 𝑦 − 𝑐𝑡, |𝜉| ≤
𝜋
2𝜇
(40)
Using the transformation (40), equation (39) transform to ordinary differential
equation in the form:
−𝑐
𝑑𝑓(𝜉)
𝑑𝜉
+ 𝑎(𝑓(𝜉 ))𝑛
𝑑𝑓(𝜉)
𝑑𝜉
+ 𝑏
𝑑
𝑑𝜉
(2
𝑑 2 𝑓(𝜉)
𝑑𝜉 2
)=0
(41)
By integration we have:
−𝑐 𝑓(𝜉 ) +
𝑎
𝑛+1
(𝑓 (𝜉 ))
𝑛+1
+ 2𝑏
𝑑 2 𝑓(𝜉)
𝑑𝜉 2
=0
(42)
Thus from the transformation (40), we have
−𝑐𝜆𝑐𝑜𝑠 𝛽 (𝜇𝜉) +
𝑎
𝜆𝑛+1 𝑐𝑜𝑠 (𝑛+1)𝛽 (𝜇𝜉) + 2𝑏𝜆𝜇2 𝛽(𝛽
𝑛+1
𝛽 (𝜇𝜉)
2
𝛽 (𝜇𝜉)
2𝑏𝜆𝜇2 𝛽(𝛽 − 1)𝑐𝑜𝑠
− 2𝑏𝜆𝜇 𝛽𝑐𝑜𝑠
− 1)𝑐𝑜𝑠 𝛽−2 (𝜇𝜉) −
=0
(43)
By balancing the exponents of each pair of cosine we have:
2
−𝑐𝜆 − 2𝑏𝜆𝜇2 𝛽(𝛽 − 1) − 2𝑏𝜆𝜇2 𝛽 = 0,
𝑎
𝑛+1
𝛽 + = 0,
𝑛
𝜆𝑛+1 + 2𝑏𝜆𝜇2 𝛽(𝛽 − 1) = 0
(44)
Using MATHEMATICA package software for solving the equation (44) we obtain:
8
𝟐
𝛽=− ,
𝜇=
𝒏
−𝑖𝑛√𝑐
,
2√2𝑏
1
−1 𝑐(2+3𝑛+𝑛2
𝑛
(
)
𝑛
𝑎
𝜆=2
(45)
Substituting in equation (40), we obtain new exact soliton solution of ZK equation is
in the form:
1
2
−1 𝑐(2+3𝑛+𝑛2
𝜋√2𝑏
− −𝑖𝑛√𝑐
(
)𝑛 𝑐𝑜𝑠 𝑛 (
(𝑥+𝑦−𝑐𝑡)) , 𝑤𝑖𝑡ℎ |𝑥+𝑦−𝑐𝑡| ≤
𝑛
𝑎
−𝑖𝑛√𝑐
2√2𝑏
𝑢(𝑥, 𝑦, 𝑡) = 2
(46)
which is different from the solution obtained in refs. [13], [14].
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