# Finding General Solutions of Nonlinear Evolution

## Transcript Of Finding General Solutions of Nonlinear Evolution

Math. Sci. Lett. 3, No. 1, 1-8 (2014)

1

Mathematical Sciences Letters

An International Journal

http://dx.doi.org/10.12785/msl/030101

Finding General Solutions of Nonlinear Evolution ′

Equations by Improved (GG )-expansion Method

E. Osman1, M. Khalfallah2 and H. Sapoor1,∗ 1 Mathematics Department, Faculty of Science, Sohag University, Sohag, Egypt 2 Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt

Received: 6 Jun. 2013, Revised: 2 Oct. 2013, Accepted: 3 Oct. 2013 Published online: 1 Jan. 2014

Abstract: In this paper, an improved method named the improved ( GG′ )-expansion is introduced. Then we studied the modiﬁed equal width (MEW) and modiﬁed Benjamin-Bona-Mahony (MBBM) equations. The periodic and solitary wave solutions are constructed by using improved ( GG′ )-expansion method. Some new traveling wave solutions involving parameters, expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions. The solitary wave solutions are derived from the hyperbolic function solutions.

Keywords: Improved ( GG′ )-expansion method, modiﬁed equal width wave equation, modiﬁed Benjamin–Bona–Mahony equation,Solitary wave solution.

1 Introduction

Nonlinear wave phenomena play a major role in sciences such as plasma physics, optical ﬁbers, ﬂuid mechanics, chemical physics and geo-chemistry. Many powerful methods have been proposed to obtain exact solutions of nonlinear evolution equations, such as sine–cosine method [1, 2, 3, 4, 5], extended tanh method [6, 7, 8, 9], Hirota’s bilinear scheme [10], homogeneous balance method [11], Riccati equation rational expansion method [12, 13], and so on.

In recent years, with the development of symbolic

computation packages like Maple and Mathematica,

which enable us to perform the tedious and complex

′

computations

on

computer

The

(

G G

)-expansion

method

proposed by Wang et al. [14], is one of the most effective

direct methods to obtain travelling wave solutions of a

large number of NLEEs,such as the KdV equation,the

mKdV equation, the variant Boussinesq equations,the

Hirota–Satsuma equations,and so on .Later, the further

developed methods named the generalized

′

′

(

G G

)-expansion

method,the

modiﬁed

(

G G

)-expansion

∗ Corresponding author e-mail: hussien [email protected]

′

method

and

the

extended

(

G G

)-expansion

method

have

been proposed in Refs. [15, 16, 17], respectively. As we

know, when using the direct method, the choice of an

appropriate ansatz is great importance . In this paper, by

introducing a new general ansatze, we propose the ′

improved ( GG )-expansion method, which can be used to obtain travelling wave solutions of NLEEs.

2 Description of the improved ( GG′ )-expansion method

Suppose that a nonlinear evolution equation, say in two independent variables x and t, is given by

N(u, ut , ux, utt , uxx, uxt , ...) = 0,

(2.1)

where u = u(x,t) is an unknown function, N is a polynomial in u = u(x,t) and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved. To determine u explicitly, we take the following ﬁve steps:

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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...

Step 1: Use the travelling wave transformation:

u(x,t) = u(ξ ),

ξ = x − wt, (2.2)

where w is a constant to be determined latter. Then, the NLEE (2.1) is reduced to a nonlinear ordinary differential equation (NLODE) for u = u(ξ )

N(u, −wu′ , u′ , w2u′′ , u′′ , −wu′′ , ...) = 0. (2.3)

Step 2: We suppose that the NLODE (2.3) has the following solution:

∑ ∑ ′

′

−1

u(ξ ) =

ai(

G G

)i

m

+ a0 +

ai( GG )i , (2.4)

(1 + β ( G′ ))i

i=1 (1 + β ( G′ ))i

i=−m

G

G

where β , ai (i = −m, −m + 1, ..., m − 1, m) are constants to be determined later, m is a positive integer, and G =

G(ξ ) satisﬁes the following second order linear ordinary differential equation(LODE) :

G′′ + µG = 0, (2.5)

where µ is a real constant. The general solutions of Eq. (2.5) can be listed as follows.When µ < 0, we obtain the hyperbolic function solution of Eq. (2.5)

G(ξ ) = A1 cosh √−µξ + A2 sinh √−µξ , (2.6)

where A1 and A2 are arbitrary constants.When µ > 0, we obtain the trigonometric function solution of Eq. (2.5)

G(ξ ) = A1 sin √µξ + A2 cos √µξ ,

(2.7)

where A1 and A2 are arbitrary constants.When µ = 0, we obtain the rational function solution of Eq. (2.5)

G(ξ ) = A1 + A2ξ

(2.8)

where A1 and A2 are arbitrary constants.

Step 3:

Determine the positive integer m by balancing the highest

order derivative and nonlinear terms in Eq. (2.3).

Step 4:

Substituting (2.4) along with Eq. (2.5) into Eq. (2.3) and ′

then setting all the coefﬁcients of ( GG ) of the resulting system’s numerator to zero, yields a set of

over-determined nonlinear algebraic equations for w, β

and ai(i = −m, −m + 1, ..., m − 1, m).

Step 5:

Assuming

that

the

constants

w, β ,

ai(i = −m, −m + 1, ..., m − 1, m) can be obtained by

solving the algebraic equations in Step 4, then

substituting these constants and the known general

solutions of Eq. (2.5) into (2.4), we can obtain the explicit

solutions of Eq. (2.1) immediately.

3 General traveling wave solutions of the MEW equation

We ﬁrst consider the modiﬁed equal width equation in its normalized form [18, 19]

ut + a(u3)x + buxxt = 0,

(3.1)

where a and b are arbitrary constants. Making the transformationu(x,t) = u(x − ct) = u(ξ ),

then substituting u(x − ct) into (3.1) we obtain the

following nonlinear ODE equation

−cu′ + a(u3)′ − bcu′′′ = 0,

(3.2)

where”′” is the derivative with respect to ξ (i.e. u′ = uξ ) and c is wave speed.

Integrating (3.2) once and setting the integral constant

as zero, we obtain wave equation

−cu + a(u3) − bcu′′ = 0.

(3.3)

By balancing the highest order derivative terms and nonlinear terms in Eq. (3.3), we ﬁnd that Eq. (3.3) own the solutions in the form

′

′

u(ξ ) = a0 +

a1(

G G

)

+

b1(1

+

β

(

G G

))

,

+ β ( G′ )

( G′ )

1

G

G

(3.4)

where G = G(ξ ) satisﬁes Eq. (2.5),β , a0, a1,b1, a and b are

constants to be determined latter.

Substituting (3.4) along with Eq. (2.5) into Eq. (3.3)

′

and

then

setting

all

the

coefﬁcients

of

(

G G

)

of

the

resulting system’s numerator to zero, yields a set of

over-determined nonlinear algebraic equations for

β , a0, a1,b1, a and b. Solving the over-determined

algebraic equations by Maple or Mathematica, we can

obtain the following sets of solutions

When µ < 0, we obtain the hyperbolic function solutions

of Eq. (3.3)

Case 1: a0 = 0, a1 = 0, b1 = ∓

2acb , µ = −2b1 , β = 0

u(ξ ) = ∓ c × 2ab A1 cosh √−µξ + A2 sinh √−µξ )

√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,

(3.5)

where A1, A2 are arbitrary constants In particular, when setting A1 = 0 , A2 = 0, the solutions

(3.5) can be written as

u(ξ ) = ∓

c tanh

1 ξ.

(3.6)

a

2b

Setting again A1 > 0, A21 > A22 the following kink-shaped solution of Eq. (3.3)

u(ξ ) = ± c tanh( 1 ξ + ξ0),

(3.7)

a

2b

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where ξ0 = tanh−1 AA12 .

Case 2: a0 = 0, a1 = ∓

2abc , b1 = 0, µ = −2b1 , β = 0

u(ξ ) = ∓ 2bc ×

√a

√

√

( −µ(A1 sinh −µξ + A2 cosh −µξ ))

A1 cosh √−µξ + A2 sinh √−µξ ,

(3.8)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.8) can be written as

u(ξ ) = ±

c coth

1 ξ.

(3.9)

a

2b

Setting again A1 > 0, A21 > A22 the following singular soliton solution of Eq. (3.3)

u(ξ ) = ± where ξ0 = tanh−1 AA12 .

c coth( a

1 ξ + ξ0), 2b

(3.10)

the 3D graphs of kink and anti-kink wave solutions are shown in Fig. 1. In the graphs, the abscissa axis is t, the cordinate axis is x and the vertical axis is u.

Case 3: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 41 2acb , µ = −8b1 , β =0

u(ξ ) = ∓ 2bc ×

√a

√

√

( −µ(A1 sinh −µξ + A2 cosh −µξ ))

A1 cosh √−µξ + A2 sinh √−µξ ∓

1 c× 4 2ab

(A1 cosh √−µξ + A2 sinh √−µξ ) √−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,

(3.11)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.11) can be written as

u(ξ ) = ∓

c tanh

1 ξ∓

c coth

1 ξ . (3.12)

4a

8b

4a

8b

Fig. 1: The 3D graphs of Eq. (3.6) and Eq.(3.9) as a=2.5, b=2, c=4 : (a)kink wave 1, (b) anti-kink wave 1, (c) kink wave 2, (d) anti-kink wave 2

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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...

Case 4: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 21 2acb , µ = 41b , β =0

u(ξ ) = ∓ 2bc ×

√a

√

√

( −µ(A1 sinh −µξ + A2 cosh −µξ ))

A1 cosh √−µξ + A2 sinh √−µξ ∓

1 c× 2 2ab

(A1 cosh √−µξ + A2 sinh √−µξ ) √−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,

(3.13)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.13) can be written as

u(ξ ) = ∓I

c tanh

−1 ξ ± I coth

−1 ξ ,

(3.14)

2a

4b

4b

√ where I = −1.

When µ > 0 ,b ⋖ 0 we get the trigonometric function

solutions of Eq. (3.3)

Case 1: a0 = 0, a1 = 0, b1 = ∓

2acb , µ = −2b1 , β = 0

u(ξ ) = ∓ c × 2ab

A1 sin √µξ + A2 cos √µξ ) √µ(A1 cos √µξ − A2 sin √µξ ) ,

(3.15)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.15) can be written as

u(ξ ) = ∓ −c cot −1 ξ .

a

2b

(3.16)

Case 2: a0 = 0, a1 = ∓

2abc , b1 = 0, µ = −2b1 , β = 0

u(ξ ) = ∓ 2bc ×

√a √

√

µ(A1 cos µξ − A2 sin µξ )

A1 sin √µξ + A2 cos √µξ ) ,

(3.17)

where A1, A2 are arbitrary constants. Setting A1 = 0 , A2 = 0, we get travelling wave

solutions of the type of tangent function

u(ξ ) = ± −c tan −1 ξ .

a

2b

(3.18)

The 3D graphs of travelling wave solutions of the type of tangent function are shown in Fig. 2. In the graphs, the abscissa axis is t, the ordinate axis is x and the vertical axis is u.

Fig. 2: The 3D graphs of (3.18) as a=-2,b=-3,c=4,,x∈(-0.5,0.5)

Case 3:

a0 = 0, a1 = ∓ 2abc , b1 = ∓ 41 2acb , β =0

u(ξ ) = ∓ 2bc ×

√a √

√

( µ(A1 cos µξ − A2 sin µξ ))

A1 sin √µξ + A2 cos √µξ ∓

1 c× 4 2ab

(A1 sin √µξ + A2 cos √µξ ) √µ ((A1 cos √µξ − A2 sin √µξ )) ,

µ = −8b1 , (3.19)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.19) can be written as

u(ξ ) = ±I

c tan

−1 ξ ∓ I

c cot

−1 ξ , (3.20)

4a

8b

4a

8b

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5

√ where I = −1.

Case 4: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 21 2acb , µ = 41b , β =0

u(ξ ) = ∓ 2bc ×

√a √

√

( µ(A1 cos µξ − A2 sin µξ ))

A1 sin √µξ + A2 cos √µξ ∓

1 c× 2 2ab

(A1 sin √µξ + A2 cos √µξ ) √µ ((A1 cos √µξ − A2 sin √µξ )) ,

(3.21)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.21) can be written as

u(ξ ) = ±

c tan

1 ξ±

c cot

1 ξ , (3.22)

2a

4b

2a

4b

When µ = 0 we get the rational function solutions of Eq. (3.3)

u(ξ ) = ∓ 2bc ( A2 ), a A1 + A2ξ

(3.23)

where A1, A2 are arbitrary constants.

4 General traveling wave solutions of the MBBM equation

′

We

now

employ

the

improved

(

G G

)

-expansion

to

the

modiﬁed Benjamin–Bona–Mahony equation [20]

ut + ux + u2ux + uxxt = 0

(4.1)

As described in Section 3, the wave variable

u(x,t) = u(x − ct) = u(ξ )

carries (4.1) into the ODE (1 − c)u′ + u2u′ − cu′′′ = 0,

(4.2)

Integrating (4.2) and setting the constant of integration to be zero we obtain.

(1 − c)u + u3 − cu′′ = 0, 3

(4.3)

According to equaion (3.4) in section 3 we have When µ < 0, we obtain the hyperbolic function

solutions of Eq. (4.3)

Case 1:

√

(c−1)

a0 = 0, a1 = ± 6c, b1 = 0, µ = − 2c , β = 0

√√

√

√

u(ξ ) = ±

6c

−µ(A1 sinh √

−µξ + A2 cosh √

−µξ ) ,

A1 cosh −µξ + A2 sinh −µξ )

(4.4)

where A1, A2 are arbitrary constants

In particular, when setting A2 = 0 , A1 = 0, the solutions

(4.4) can be written as

u(ξ ) = ± 3(c − 1) tanh (c − 1) ξ . (4.5) 2c

setting again A1 = 0 , A2 = 0, the solutions (4.4) can be written as

u(ξ ) = ± 3(c − 1) coth (c − 1) ξ . (4.6) 2c

Case 2: a0 = 0,

β =0

√ a1 = ± 6c,

b1 = ∓ (1−4 c)

6c ,

µ

=

(c−1)

4c ,

√√

√

√

u(ξ ) = ±

6c

−µ(A1 sinh √

−µξ + A2 cosh √

−µξ )

A1 cosh −µξ + A2 sinh −µξ )

∓ (1 − c) 6 ×

4 √c

√

A1 cosh −µξ + A2 sinh −µξ )

√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) .

(4.7)

In particular, when setting A2 = 0 , A1 = 0, the solutions (4.7) can be written as

u(ξ ) = ± ∓

3(1 − c) tanh 2

3(1 − c) coth 2

(1 − c) ξ 4c

(1 − c) ξ . 4c

(4.8)

Case 3: a0 = 0,

µ = − (c8−c1) ,

√ a1 = ± 6c, β =0

b1 = ∓(1 − c) 332c ,

√√

√

√

u(ξ ) = ±

6c

−µ(A1 sinh √

−µξ + A2 cosh √

−µξ )

A1 cosh −µξ + A2 sinh −µξ )

∓(1 − c) 3

√32c

√

A1 cosh −µξ + A2 sinh −µξ )

√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) .

(4.9)

In particular, when setting A2 = 0 , A1 = 0, the solutions (4.9) can be written as

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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...

u(ξ ) = ± 3(c − 1) tanh (c − 1) ξ

4

8c

± 3(c − 1) coth (c − 1) ξ .

4

8c

(4.10)

When µ > 0 we get the trigonometric function solutions

of Eq. (4.3)

Case 1:

√

(c−1)

a0 = 0, a1 = ± 6c, b1 = 0, µ = − 2c , β = 0

√√

√

√

u(ξ ) = ±

6c

µ(A1 cos √

µξ − A2 sin √

µξ) .

A1 sin µξ + A2 cos µξ )

(4.11)

In particular, when setting A2 = 0 , A1 = 0, the solutions (4.11) can be written as

u(ξ ) = ± 3(1 − c) cot (1 − c) ξ . 2c

(4.12)

setting again A1 = 0 , A2 = 0, the solutions (4.11) can be written as

u(ξ ) = ± 3(1 − c) tan (1 − c) ξ . 2c

(4.13)

The 3D graphs of travelling wave solutions of the type of

tangent function are shown in Fig. 3. In the graphs, the

abscissa axis is t, the ordinate axis is x and the vertical

axis is u.

Case 2:

√

a0 = 0, a1 = ± 6c, b1 = ∓ (1−4 c)

6c ,

µ

=

(c−1)

4c ,

β =0

√√

√

√

u(ξ ) = ±

6c(

µ(A1 cos √

µξ − A2 sin √

µξ ))

A1 sin µξ + A2 cos µξ

∓ (1 − c) 6

4 √c

√

(A1 sin µξ + A2 cos µξ )

× √µ ((A1 cos √µξ − A2 sin √µξ )) .

(4.14)

setting again A1 = 0 , A2 = 0, the solutions (4.14) can be written as

u(ξ ) = ∓ ∓

3(c − 1) tan 2

3(c − 1) cot 2

(c − 1) ξ 4c

(c − 1) ξ . 4c

(4.15)

Fig. 3: The 3D graphs of (4.13) as c=1/2, x∈(-0.5,0.5)

Case 3: a0 = 0,

µ = − (c8−c1) ,

√ a1 = ± 6c, β =0

b1 = ∓(1 − c) 332c ,

√√

√

√

u(ξ ) = ±

6c(

µ(A1 cos √

µξ − A2 sin √

µξ ))

A1 sin µξ + A2 cos µξ

∓(1 − c) 3

3√2c

√

(A1 sin µξ + A2 cos µξ )

× √µ ((A1 cos √µξ − A2 sin √µξ )) .

(4.16)

setting A1 = 0 , A2 = 0, the solutions (4.16) can be written as

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7

u(ξ ) = ∓ ±

3(1 − c) tan 4

3(1 − c) cot 2

(1 − c) ξ 8c

(1 − c) ξ . 8c

(4.17)

When µ = 0 we get the rational function solutions of Eq.

(4.3)

√

A2

u(ξ ) = ± 6( A1 + A2ξ ),

(4.18)

where A1, A2 are arbitrary constants.

5 Conclusions

′

The

improved

(

G G

)-expansion

method

is

applied

successfully for solving the modiﬁed equal width (MEW)

and modiﬁed Benjamin–Bona–Mahony (MBBM)

equations.These exact solutions include the hyperbolic

function solutions, trigonometric function solutions and

rational function solutions. When the parameters are

taken as special values, the solitary wave solutions are

derived from the hyperbolic function solutions. This

method has more advantages: it is direct and concise.

References

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[4] A. M. Wazwaz, M. A. Helal, Nonlinear variants of the BBM equation with compact and noncompact physical structures. Chaos,Soli tons &Fractals, 26, 767–776 (2005).

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[18] L. R. Gardner, G. A. Gardner, F. A. Ayoub, N. K. Amein, Simulations of the EW undular bore. Commun Numer Methods Eng., 13, 583–592 1998.

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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...

El-sayed

Osman

received the PhD degree

in Mathematics at Clarkson

University

(USA).His

research interests are in the

areas of applied mathematics

including Soliton theory

and mathematical methods

and models. He has published

research articles in Egyptian

and international journals of mathematical sciences.

Mohamed Khalfallah received the PhD degree in applied mathematics in ”The theory of Soliton”, Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt. He has a number of papers published in many of scientiﬁc journals.

Hussien

Sapoor

received the Master degree

in applied mathematics. His

research interests are in the

areas of applied mathematics

”The theory of Soliton”,

Mathematics Department,

Faculty of Science,Sohage

University, Sohage, Egypt.

c 2014 NSP Natural Sciences Publishing Cor.

1

Mathematical Sciences Letters

An International Journal

http://dx.doi.org/10.12785/msl/030101

Finding General Solutions of Nonlinear Evolution ′

Equations by Improved (GG )-expansion Method

E. Osman1, M. Khalfallah2 and H. Sapoor1,∗ 1 Mathematics Department, Faculty of Science, Sohag University, Sohag, Egypt 2 Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt

Received: 6 Jun. 2013, Revised: 2 Oct. 2013, Accepted: 3 Oct. 2013 Published online: 1 Jan. 2014

Abstract: In this paper, an improved method named the improved ( GG′ )-expansion is introduced. Then we studied the modiﬁed equal width (MEW) and modiﬁed Benjamin-Bona-Mahony (MBBM) equations. The periodic and solitary wave solutions are constructed by using improved ( GG′ )-expansion method. Some new traveling wave solutions involving parameters, expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions. The solitary wave solutions are derived from the hyperbolic function solutions.

Keywords: Improved ( GG′ )-expansion method, modiﬁed equal width wave equation, modiﬁed Benjamin–Bona–Mahony equation,Solitary wave solution.

1 Introduction

Nonlinear wave phenomena play a major role in sciences such as plasma physics, optical ﬁbers, ﬂuid mechanics, chemical physics and geo-chemistry. Many powerful methods have been proposed to obtain exact solutions of nonlinear evolution equations, such as sine–cosine method [1, 2, 3, 4, 5], extended tanh method [6, 7, 8, 9], Hirota’s bilinear scheme [10], homogeneous balance method [11], Riccati equation rational expansion method [12, 13], and so on.

In recent years, with the development of symbolic

computation packages like Maple and Mathematica,

which enable us to perform the tedious and complex

′

computations

on

computer

The

(

G G

)-expansion

method

proposed by Wang et al. [14], is one of the most effective

direct methods to obtain travelling wave solutions of a

large number of NLEEs,such as the KdV equation,the

mKdV equation, the variant Boussinesq equations,the

Hirota–Satsuma equations,and so on .Later, the further

developed methods named the generalized

′

′

(

G G

)-expansion

method,the

modiﬁed

(

G G

)-expansion

∗ Corresponding author e-mail: hussien [email protected]

′

method

and

the

extended

(

G G

)-expansion

method

have

been proposed in Refs. [15, 16, 17], respectively. As we

know, when using the direct method, the choice of an

appropriate ansatz is great importance . In this paper, by

introducing a new general ansatze, we propose the ′

improved ( GG )-expansion method, which can be used to obtain travelling wave solutions of NLEEs.

2 Description of the improved ( GG′ )-expansion method

Suppose that a nonlinear evolution equation, say in two independent variables x and t, is given by

N(u, ut , ux, utt , uxx, uxt , ...) = 0,

(2.1)

where u = u(x,t) is an unknown function, N is a polynomial in u = u(x,t) and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved. To determine u explicitly, we take the following ﬁve steps:

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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...

Step 1: Use the travelling wave transformation:

u(x,t) = u(ξ ),

ξ = x − wt, (2.2)

where w is a constant to be determined latter. Then, the NLEE (2.1) is reduced to a nonlinear ordinary differential equation (NLODE) for u = u(ξ )

N(u, −wu′ , u′ , w2u′′ , u′′ , −wu′′ , ...) = 0. (2.3)

Step 2: We suppose that the NLODE (2.3) has the following solution:

∑ ∑ ′

′

−1

u(ξ ) =

ai(

G G

)i

m

+ a0 +

ai( GG )i , (2.4)

(1 + β ( G′ ))i

i=1 (1 + β ( G′ ))i

i=−m

G

G

where β , ai (i = −m, −m + 1, ..., m − 1, m) are constants to be determined later, m is a positive integer, and G =

G(ξ ) satisﬁes the following second order linear ordinary differential equation(LODE) :

G′′ + µG = 0, (2.5)

where µ is a real constant. The general solutions of Eq. (2.5) can be listed as follows.When µ < 0, we obtain the hyperbolic function solution of Eq. (2.5)

G(ξ ) = A1 cosh √−µξ + A2 sinh √−µξ , (2.6)

where A1 and A2 are arbitrary constants.When µ > 0, we obtain the trigonometric function solution of Eq. (2.5)

G(ξ ) = A1 sin √µξ + A2 cos √µξ ,

(2.7)

where A1 and A2 are arbitrary constants.When µ = 0, we obtain the rational function solution of Eq. (2.5)

G(ξ ) = A1 + A2ξ

(2.8)

where A1 and A2 are arbitrary constants.

Step 3:

Determine the positive integer m by balancing the highest

order derivative and nonlinear terms in Eq. (2.3).

Step 4:

Substituting (2.4) along with Eq. (2.5) into Eq. (2.3) and ′

then setting all the coefﬁcients of ( GG ) of the resulting system’s numerator to zero, yields a set of

over-determined nonlinear algebraic equations for w, β

and ai(i = −m, −m + 1, ..., m − 1, m).

Step 5:

Assuming

that

the

constants

w, β ,

ai(i = −m, −m + 1, ..., m − 1, m) can be obtained by

solving the algebraic equations in Step 4, then

substituting these constants and the known general

solutions of Eq. (2.5) into (2.4), we can obtain the explicit

solutions of Eq. (2.1) immediately.

3 General traveling wave solutions of the MEW equation

We ﬁrst consider the modiﬁed equal width equation in its normalized form [18, 19]

ut + a(u3)x + buxxt = 0,

(3.1)

where a and b are arbitrary constants. Making the transformationu(x,t) = u(x − ct) = u(ξ ),

then substituting u(x − ct) into (3.1) we obtain the

following nonlinear ODE equation

−cu′ + a(u3)′ − bcu′′′ = 0,

(3.2)

where”′” is the derivative with respect to ξ (i.e. u′ = uξ ) and c is wave speed.

Integrating (3.2) once and setting the integral constant

as zero, we obtain wave equation

−cu + a(u3) − bcu′′ = 0.

(3.3)

By balancing the highest order derivative terms and nonlinear terms in Eq. (3.3), we ﬁnd that Eq. (3.3) own the solutions in the form

′

′

u(ξ ) = a0 +

a1(

G G

)

+

b1(1

+

β

(

G G

))

,

+ β ( G′ )

( G′ )

1

G

G

(3.4)

where G = G(ξ ) satisﬁes Eq. (2.5),β , a0, a1,b1, a and b are

constants to be determined latter.

Substituting (3.4) along with Eq. (2.5) into Eq. (3.3)

′

and

then

setting

all

the

coefﬁcients

of

(

G G

)

of

the

resulting system’s numerator to zero, yields a set of

over-determined nonlinear algebraic equations for

β , a0, a1,b1, a and b. Solving the over-determined

algebraic equations by Maple or Mathematica, we can

obtain the following sets of solutions

When µ < 0, we obtain the hyperbolic function solutions

of Eq. (3.3)

Case 1: a0 = 0, a1 = 0, b1 = ∓

2acb , µ = −2b1 , β = 0

u(ξ ) = ∓ c × 2ab A1 cosh √−µξ + A2 sinh √−µξ )

√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,

(3.5)

where A1, A2 are arbitrary constants In particular, when setting A1 = 0 , A2 = 0, the solutions

(3.5) can be written as

u(ξ ) = ∓

c tanh

1 ξ.

(3.6)

a

2b

Setting again A1 > 0, A21 > A22 the following kink-shaped solution of Eq. (3.3)

u(ξ ) = ± c tanh( 1 ξ + ξ0),

(3.7)

a

2b

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3

where ξ0 = tanh−1 AA12 .

Case 2: a0 = 0, a1 = ∓

2abc , b1 = 0, µ = −2b1 , β = 0

u(ξ ) = ∓ 2bc ×

√a

√

√

( −µ(A1 sinh −µξ + A2 cosh −µξ ))

A1 cosh √−µξ + A2 sinh √−µξ ,

(3.8)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.8) can be written as

u(ξ ) = ±

c coth

1 ξ.

(3.9)

a

2b

Setting again A1 > 0, A21 > A22 the following singular soliton solution of Eq. (3.3)

u(ξ ) = ± where ξ0 = tanh−1 AA12 .

c coth( a

1 ξ + ξ0), 2b

(3.10)

the 3D graphs of kink and anti-kink wave solutions are shown in Fig. 1. In the graphs, the abscissa axis is t, the cordinate axis is x and the vertical axis is u.

Case 3: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 41 2acb , µ = −8b1 , β =0

u(ξ ) = ∓ 2bc ×

√a

√

√

( −µ(A1 sinh −µξ + A2 cosh −µξ ))

A1 cosh √−µξ + A2 sinh √−µξ ∓

1 c× 4 2ab

(A1 cosh √−µξ + A2 sinh √−µξ ) √−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,

(3.11)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.11) can be written as

u(ξ ) = ∓

c tanh

1 ξ∓

c coth

1 ξ . (3.12)

4a

8b

4a

8b

Fig. 1: The 3D graphs of Eq. (3.6) and Eq.(3.9) as a=2.5, b=2, c=4 : (a)kink wave 1, (b) anti-kink wave 1, (c) kink wave 2, (d) anti-kink wave 2

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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...

Case 4: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 21 2acb , µ = 41b , β =0

u(ξ ) = ∓ 2bc ×

√a

√

√

( −µ(A1 sinh −µξ + A2 cosh −µξ ))

A1 cosh √−µξ + A2 sinh √−µξ ∓

1 c× 2 2ab

(A1 cosh √−µξ + A2 sinh √−µξ ) √−µ(A1 sinh √−µξ + A2 cosh √−µξ ) ,

(3.13)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.13) can be written as

u(ξ ) = ∓I

c tanh

−1 ξ ± I coth

−1 ξ ,

(3.14)

2a

4b

4b

√ where I = −1.

When µ > 0 ,b ⋖ 0 we get the trigonometric function

solutions of Eq. (3.3)

Case 1: a0 = 0, a1 = 0, b1 = ∓

2acb , µ = −2b1 , β = 0

u(ξ ) = ∓ c × 2ab

A1 sin √µξ + A2 cos √µξ ) √µ(A1 cos √µξ − A2 sin √µξ ) ,

(3.15)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.15) can be written as

u(ξ ) = ∓ −c cot −1 ξ .

a

2b

(3.16)

Case 2: a0 = 0, a1 = ∓

2abc , b1 = 0, µ = −2b1 , β = 0

u(ξ ) = ∓ 2bc ×

√a √

√

µ(A1 cos µξ − A2 sin µξ )

A1 sin √µξ + A2 cos √µξ ) ,

(3.17)

where A1, A2 are arbitrary constants. Setting A1 = 0 , A2 = 0, we get travelling wave

solutions of the type of tangent function

u(ξ ) = ± −c tan −1 ξ .

a

2b

(3.18)

The 3D graphs of travelling wave solutions of the type of tangent function are shown in Fig. 2. In the graphs, the abscissa axis is t, the ordinate axis is x and the vertical axis is u.

Fig. 2: The 3D graphs of (3.18) as a=-2,b=-3,c=4,,x∈(-0.5,0.5)

Case 3:

a0 = 0, a1 = ∓ 2abc , b1 = ∓ 41 2acb , β =0

u(ξ ) = ∓ 2bc ×

√a √

√

( µ(A1 cos µξ − A2 sin µξ ))

A1 sin √µξ + A2 cos √µξ ∓

1 c× 4 2ab

(A1 sin √µξ + A2 cos √µξ ) √µ ((A1 cos √µξ − A2 sin √µξ )) ,

µ = −8b1 , (3.19)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.19) can be written as

u(ξ ) = ±I

c tan

−1 ξ ∓ I

c cot

−1 ξ , (3.20)

4a

8b

4a

8b

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5

√ where I = −1.

Case 4: a0 = 0, a1 = ∓ 2abc , b1 = ∓ 21 2acb , µ = 41b , β =0

u(ξ ) = ∓ 2bc ×

√a √

√

( µ(A1 cos µξ − A2 sin µξ ))

A1 sin √µξ + A2 cos √µξ ∓

1 c× 2 2ab

(A1 sin √µξ + A2 cos √µξ ) √µ ((A1 cos √µξ − A2 sin √µξ )) ,

(3.21)

In particular, when setting A1 = 0 , A2 = 0, the solutions (3.21) can be written as

u(ξ ) = ±

c tan

1 ξ±

c cot

1 ξ , (3.22)

2a

4b

2a

4b

When µ = 0 we get the rational function solutions of Eq. (3.3)

u(ξ ) = ∓ 2bc ( A2 ), a A1 + A2ξ

(3.23)

where A1, A2 are arbitrary constants.

4 General traveling wave solutions of the MBBM equation

′

We

now

employ

the

improved

(

G G

)

-expansion

to

the

modiﬁed Benjamin–Bona–Mahony equation [20]

ut + ux + u2ux + uxxt = 0

(4.1)

As described in Section 3, the wave variable

u(x,t) = u(x − ct) = u(ξ )

carries (4.1) into the ODE (1 − c)u′ + u2u′ − cu′′′ = 0,

(4.2)

Integrating (4.2) and setting the constant of integration to be zero we obtain.

(1 − c)u + u3 − cu′′ = 0, 3

(4.3)

According to equaion (3.4) in section 3 we have When µ < 0, we obtain the hyperbolic function

solutions of Eq. (4.3)

Case 1:

√

(c−1)

a0 = 0, a1 = ± 6c, b1 = 0, µ = − 2c , β = 0

√√

√

√

u(ξ ) = ±

6c

−µ(A1 sinh √

−µξ + A2 cosh √

−µξ ) ,

A1 cosh −µξ + A2 sinh −µξ )

(4.4)

where A1, A2 are arbitrary constants

In particular, when setting A2 = 0 , A1 = 0, the solutions

(4.4) can be written as

u(ξ ) = ± 3(c − 1) tanh (c − 1) ξ . (4.5) 2c

setting again A1 = 0 , A2 = 0, the solutions (4.4) can be written as

u(ξ ) = ± 3(c − 1) coth (c − 1) ξ . (4.6) 2c

Case 2: a0 = 0,

β =0

√ a1 = ± 6c,

b1 = ∓ (1−4 c)

6c ,

µ

=

(c−1)

4c ,

√√

√

√

u(ξ ) = ±

6c

−µ(A1 sinh √

−µξ + A2 cosh √

−µξ )

A1 cosh −µξ + A2 sinh −µξ )

∓ (1 − c) 6 ×

4 √c

√

A1 cosh −µξ + A2 sinh −µξ )

√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) .

(4.7)

In particular, when setting A2 = 0 , A1 = 0, the solutions (4.7) can be written as

u(ξ ) = ± ∓

3(1 − c) tanh 2

3(1 − c) coth 2

(1 − c) ξ 4c

(1 − c) ξ . 4c

(4.8)

Case 3: a0 = 0,

µ = − (c8−c1) ,

√ a1 = ± 6c, β =0

b1 = ∓(1 − c) 332c ,

√√

√

√

u(ξ ) = ±

6c

−µ(A1 sinh √

−µξ + A2 cosh √

−µξ )

A1 cosh −µξ + A2 sinh −µξ )

∓(1 − c) 3

√32c

√

A1 cosh −µξ + A2 sinh −µξ )

√−µ(A1 sinh √−µξ + A2 cosh √−µξ ) .

(4.9)

In particular, when setting A2 = 0 , A1 = 0, the solutions (4.9) can be written as

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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...

u(ξ ) = ± 3(c − 1) tanh (c − 1) ξ

4

8c

± 3(c − 1) coth (c − 1) ξ .

4

8c

(4.10)

When µ > 0 we get the trigonometric function solutions

of Eq. (4.3)

Case 1:

√

(c−1)

a0 = 0, a1 = ± 6c, b1 = 0, µ = − 2c , β = 0

√√

√

√

u(ξ ) = ±

6c

µ(A1 cos √

µξ − A2 sin √

µξ) .

A1 sin µξ + A2 cos µξ )

(4.11)

In particular, when setting A2 = 0 , A1 = 0, the solutions (4.11) can be written as

u(ξ ) = ± 3(1 − c) cot (1 − c) ξ . 2c

(4.12)

setting again A1 = 0 , A2 = 0, the solutions (4.11) can be written as

u(ξ ) = ± 3(1 − c) tan (1 − c) ξ . 2c

(4.13)

The 3D graphs of travelling wave solutions of the type of

tangent function are shown in Fig. 3. In the graphs, the

abscissa axis is t, the ordinate axis is x and the vertical

axis is u.

Case 2:

√

a0 = 0, a1 = ± 6c, b1 = ∓ (1−4 c)

6c ,

µ

=

(c−1)

4c ,

β =0

√√

√

√

u(ξ ) = ±

6c(

µ(A1 cos √

µξ − A2 sin √

µξ ))

A1 sin µξ + A2 cos µξ

∓ (1 − c) 6

4 √c

√

(A1 sin µξ + A2 cos µξ )

× √µ ((A1 cos √µξ − A2 sin √µξ )) .

(4.14)

setting again A1 = 0 , A2 = 0, the solutions (4.14) can be written as

u(ξ ) = ∓ ∓

3(c − 1) tan 2

3(c − 1) cot 2

(c − 1) ξ 4c

(c − 1) ξ . 4c

(4.15)

Fig. 3: The 3D graphs of (4.13) as c=1/2, x∈(-0.5,0.5)

Case 3: a0 = 0,

µ = − (c8−c1) ,

√ a1 = ± 6c, β =0

b1 = ∓(1 − c) 332c ,

√√

√

√

u(ξ ) = ±

6c(

µ(A1 cos √

µξ − A2 sin √

µξ ))

A1 sin µξ + A2 cos µξ

∓(1 − c) 3

3√2c

√

(A1 sin µξ + A2 cos µξ )

× √µ ((A1 cos √µξ − A2 sin √µξ )) .

(4.16)

setting A1 = 0 , A2 = 0, the solutions (4.16) can be written as

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7

u(ξ ) = ∓ ±

3(1 − c) tan 4

3(1 − c) cot 2

(1 − c) ξ 8c

(1 − c) ξ . 8c

(4.17)

When µ = 0 we get the rational function solutions of Eq.

(4.3)

√

A2

u(ξ ) = ± 6( A1 + A2ξ ),

(4.18)

where A1, A2 are arbitrary constants.

5 Conclusions

′

The

improved

(

G G

)-expansion

method

is

applied

successfully for solving the modiﬁed equal width (MEW)

and modiﬁed Benjamin–Bona–Mahony (MBBM)

equations.These exact solutions include the hyperbolic

function solutions, trigonometric function solutions and

rational function solutions. When the parameters are

taken as special values, the solitary wave solutions are

derived from the hyperbolic function solutions. This

method has more advantages: it is direct and concise.

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E. Osman et al: Finding General Solutions of Nonlinear Evolution Equations...

El-sayed

Osman

received the PhD degree

in Mathematics at Clarkson

University

(USA).His

research interests are in the

areas of applied mathematics

including Soliton theory

and mathematical methods

and models. He has published

research articles in Egyptian

and international journals of mathematical sciences.

Mohamed Khalfallah received the PhD degree in applied mathematics in ”The theory of Soliton”, Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt. He has a number of papers published in many of scientiﬁc journals.

Hussien

Sapoor

received the Master degree

in applied mathematics. His

research interests are in the

areas of applied mathematics

”The theory of Soliton”,

Mathematics Department,

Faculty of Science,Sohage

University, Sohage, Egypt.

c 2014 NSP Natural Sciences Publishing Cor.