On Reliable Method for New Solitary

Wave Solutions of Some Non-linear

Partial Differential Equations

Anwar Ja'afar Mohamad Jawad

Computer Engineering Technique Department

Al -Rafidain University College, Baghdad, Iraq

anwar_jawad2001@yahoo.com ,

Abstract : In this paper, the tan function method and Sech method are proposed to establish a

traveling wave solution for nonlinear partial differential equations. The two methods are used to

obtain new solitary wave solutions for Bretherton equation, the generalized Rosenau-kdv equation, ,

and the Generalized-Pochhammer-Chree equation (GPC). which are important Soliton equations.

Methods have been successfully implemented to establish new solitary wave solutions for the nonlinear

PDEs.

Keywords: Nonlinear PDEs, Tan function method, Sech function method, Bretherton equation,

Rosenau-KdV equation. , and the Generalized-Pochhammer-Chree equation (GPC).

Classification: 35 Q , 35 D

1. INTRODUCTION

Nonlinear evolution equations (NLEEs) appear in various fields of applied and

nonlinear sciences [1–9]. These include Physics, Engineering and Biosciences. New

approaches for finding the exact solutions to nonlinear evolution equations have been

proposed for solving these NLEEs. These include the shock waves, solitary waves,

singular solitary waves, periodic waves as well as double periodic waves.

Recent Methods such as, Tanh-Sech method [2], extended tanh method [3], hyperbolic

function method [4], Jacobi elliptic function expansion method [5], F-expansion method

[6], and the sine-cos method [7], Variational Iteration Method VIM [8], The tan-cot

method [9] has been used to solve different types of nonlinear PDEs.

Bretherton equation [10-13] and the generalized Rosenau-KdV equation [14-16], and

the Generalized-Pochhammer-Chree equation (GPC) [17] are studied in this paper. The

modern methods of integrability Tan and Sech function methods will be applied to

integrate these equations. Methods will reveal solutions that will be useful in the

literature of NLEEs.

2. METHODOLOGY

Consider the nonlinear partial differential equation in the form:

( ) (1)

where u(x, y, t) is a traveling wave solution of nonlinear partial differential equation.

We use the transformations ( ) ( )

Where:

(2)

This enables us to use the following changes:

( )

( ) ,

( )

( ) ,

( )

( ) (3)

Using Eq. (2) to transfer the nonlinear partial differential equation Eq. (1) to nonlinear

ordinary differential equation

( ) (4)

The ordinary differential equation (4) is then integrated as long as all terms contain

derivatives, where we neglect the integration constants.

2. 1. Tan Method

The solutions of many nonlinear equations can be expressed in the form [9, 18]:

( ) ( ) | |

(5)

with the derivatives of Eq. (5) :

( ) 0 ( ) ( )1

( ) 0( ) ( ) ( ) ( ) ( )1

( ) 0( )( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )1

( )( ) 0( )( )( ) ( ) ( ),( )

- ( ) ( ) ( )

( ),( ) - ( ) ( )( )( ) ( )1

(6)

Where μ , and β are parameters to be determined, μ and are the wave number and

the wave speed, respectively. We substitute Eq. (5) into the reduced equation Eq. (4),

balance the terms of the tan functions and solve the resulting system of algebraic

equations by using computerized symbolic packages. We next collect all terms with the

same power in ( ) and set to zero their coefficients to get a system of algebraic

equations among the unknown's , μ and β, and solve the subsequent system.

2.2. Sech Method

The solutions of many nonlinear equations can be expressed in the form:

( ) ( ) (7)

Where μ and β are parameters to be determined, μ and are the wave number and

the wave speed, respectively. We use

( ) ( )

( ) ( ) ( ) (8)

( ) [( ) ( ) ( )]

( ) [( )( ) ( ) ( )] ( )

( )( ) [ ( ) ( ) ,( ) - ( )

( ) ( ) ( ) ( )]

and so on. We substitute (8) into the reduced equation (4), balance the terms of the sech

functions are used, and solve the resulting system of algebraic equations by using

computerized symbolic packages. We next collect all terms with the same power in

( ) and set to zero their coefficients to get a system of algebraic equations

among the unknown's , μ and β, and solve the subsequent system.

3. APPLICATIONS:

3.1 Generalized Bretherton equation

Bretherton equation (BE) is one of the recently studied nonlinear evolution equations

(NLEEs) [10-13]. The focus will be on obtaining the exact 1-soliton solutions for the

following generalized Bretherton equation:

(9)

where k , a , b and c are constants. The Sech method will calculate the closed form

solutions for any arbitrary values of the exponents n and m with m > n and n ≠1. It

needs to be noted that the traveling wave solutions of this NLEE with a positive

coefficient of the second order dispersion have recently been constructed by

Kudryashov et al [11], for the particular cases m=1 with any n , m ≠1 and n = 5 and

m = 2 , n = 3. Additionally, Esfahani[12] studied this equation in 2011. However, in that

paper the special case with m =1, n = 2 was considered. Triki et al [10] solved the (BE)

and considered any arbitrary values of m and n.

Introducing the transformations

(10)

where λ is real constant. Hence,

Substitute (7) in Eq.(6) , we get the following ODEs

, - (11)

3.1.1. Tan method

Seeking the tan method to Eq.(11)

, - 0( ) ( ) ( ) ( ) ( )1

[

( )( )( ) ( ) ( ),( ) - ( )

( ) ( ) ( ),( ) - ( )

( )( )( ) ( )

]

( ) ( )

(12)

From Eq.(12), Equating the power of tan of two terms

β , then β

( ) (13)

also for

β , then β

( ) (14)

therefore

( )

( ) (15)

Then

(16)

Equating the coefficients of the same power in Eq. (12) then a system of equation is:

,( ) -

( )

( )( )( )

( ),( ) - (17)

Solving system (17) to get:

√ ( )

( )√

( )

( ) , 0

( )( )

( ) 1

( )

√

( )√

( )

( ) (18)

then:

( ) 0

( )( )

( ) 1

2 ( )

√

( )√

( )

( ) . √

( )

( )√

( )

( ) /3

(19)

Where:

( )

( )√

( )

( ) (20)

for , , .

√

√

, 0

1

, √

√

(21)

Then:

( ) 0

1

2√

√

. √

√

/3 (22)

provided that :

√

(23)

Clearly this solution exists provided that a < 0 , b >0 and c > 0. Also the solution

(22) exists under the condition (16) with n > m >1.

3.1.2. Sech method

Assume the following solution in (8) for Eq.(11)

, -[( ) ( ) ( )] [ ( )

( ) ,( ) - ( ) ( ) ( ) ( ) ( )]

( ) ( ) (24)

From eq.(9), Equating the power of sech term of two terms

β , then β

( ) (25)

also for

β , then β

( ) (26)

therefore

( )

( ) (27)

Then

(28)

Equating the coefficients of the same power in Eq. (24) then a system of equation is:

( ), - ( ) ( ( ) )

( ) ( ) ( )

, - (29)

Solving system (29) we get:

√

( )√

( )

, *

( )

( ) √

( )

+

(30)

√

( ) √

( )

(31)

for β

( ) √√

2√

( )√

. √

√

/3 (32)

provided that

√

Clearly this solution exists provided that a >0 , b >0 and c >0. Also the solution (32)

exists under the condition (28) with n > m >1.

3.2. The generalized Rosenau-KdV equation

Consider the generalized Rosenau-KdV equation[14-16],

( ) (33)

where a , b , c , and d are real valued constants while for the exponent we assume that n

≠ 0,1. This equation was studied before and the non-topological 1-soliton solution was

already obtained [14].

The starting hypothesis is given by introducing the transformations

(34)

where λ is real constant. Hence,

( ) ( ) (35)

Integrating Eq. (35) once with zero constant of integration

( ) ( ) (36)

3.2.1. Tan method

( ) ( ) 0( ) ( ) ( ) ( ) ( )1

[

( )( )( ) ( ) ( ),( ) - ( )

( ) ( ) ( ),( ) - ( )

( )( )( ) ( )

]

( )

(37)

From eq.(37), Equating the power of tan terms

β , then β

( ) (38)

system of equation exists for the same powers of tan terms:

( ) ( )

,( ) -

,( ) -

( )( )( ) (39)

Solving system (39) for the following two cases:

Case 1

n = 5

, √

, √ √

(40)

then:

( ) √ √

,√

.

/- (41)

where:

[ √ ] (42)

This solution exists provided that a >0 , b >0 , c < 0, and d>0 Also the solution (41)

exists under the condition n >1.

Case 2

n = 3

, √

, √

(43)

where:

√ (44)

Then

( ) √

,√

.

/- (45)

provided that:

(46)

This solution exists provided that a >0 , b >0 , c > 0, and d>0 Also the solution (45)

exists under the condition n >1.

3.2.2. Sech method

Seeking the solution in (8) Hence Eq.(36) becomes

( ) ( ) [( ) ( ) ( )]

* ( ) ( ) ,( ) - ( )

( ) ( ) ( ) ( )+

( )

(47)

Equating the exponents β , then

β

( ) (48)

which exist provided that n >1. Hence setting their respective coefficients to zero yields

the following system of algebraic equations:

,( ) -

( ) ( ) ( )

( ) (49)

To solve system (49), consider the following two cases:

Case 1

Let β , This yields β and therefore n = 5 , substitution of this value into

system (49) gives

√ [ √ ]

, ⁄

, 0

1

(50)

where:

( ) (51)

Then

( ) 0

( √ )1

{

√ [ √ ]

( ⁄

[ √ ] )} (52)

Case 2

Let β , This yields β and therefore n = 3 , substitution of this value into

system (49) gives

0

√ ( ) 1

,

0 √ ( ) 1 ,

0

. √ ( ) /1

(53)

such that

( ) (54)

Then

( )

[

. √ ( ) / ]

{

0

√

1

.

0 √ ( ) 1 /

}

(55)

3.3. Generalized-Pochhammer-Chree e quation (GPC):

Consider the following Generalized-Pochhammer-Chree equation [17],

(

) (56)

Where and are non-zero constants,

Some exact solitary solutions of the generalized Pochhammer-Chree equation was

obtained by Kourosh Parand et al [17] when using the Exp-function method with the

help of symbolic computation.

Letting ( ) ( ) where,

( ) ( ) (57)

Where , μ, and β are parameters that to be determined. Assume the transformation

, equation (56) turns to the following ordinary differential equation:

( ) (

) (58)

By integrating equation (58) twice with zero constant, we have

(

) (59)

assume

(60)

or

(61)

then

, (62)

and

.

/

(63)

Substitute Eq.(61-63) in Eq.(59), we get

,

.

/

- .

/ (64)

then

, ( ) - (

) (65)

3.3.1. Tan method

Applying Tan method, equation (65) becomes,

( )

{

(

( ) ( ) ( )

( ) ( ) )

( ) ( ( ) ( ))}

. ( ) ( )

( )/

(66)

Equating the exponents gives:

(67)

Equating the identical powers and the coefficients of each pair of the tan function, leads

us to the following algebraic system:

* ( ) +

(68)

Solving the algebraic system of (68), we obtain

( )

, - , √

, ( )- , - (69)

where:

√ , -

( )

(70)

the solution ( ) will be obtained in the following form:

( ) .( )

, - / ( ( √

, ( )- , - )) (71)

then

( ) [.( )

, - / ( ( √

, ( )- , - ))]

(72)

where , ( )- , - (73)

3.3.2. Sech method

Applying sech method to solve Eq.(65) then

.( ) ( ) ( )/

( ) ( ) ( ) , - ( )

. ( ) ( )/

(74)

Equating the exponents β , then

β (75)

Hence setting their respective coefficients to zero yields the following system of

algebraic equations:

, -

(76)

Solving system (76), to find

√

, - , √

, -

(77)

Then substitute Eq.(77) to get

( ) √

, -

, ( √

, - )- (78)

then

( ) * √

, -

, ( √

, - )-+

(79)

where: , - (80)

clearly and , and is unrestricted value.

4. Conclusions

In this paper, the tan and Sech function method has been successfully implemented

to establish new solitary wave solutions for Bretherton equation, the generalized

Rosenau-kdv equation, and the Generalized-Pochhammer-Chree equation (GPC). The

two methods can be extended to solve the problems of nonlinear partial differential

equations.

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