Travelling Wave Solutions to the Generalized

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Volume 2011, Article ID 629760,13pages doi:10.1155/2011/629760

Research Article

Travelling Wave Solutions to the Generalized

Pochhammer-Chree (PC) Equations Using the First Integral Method

Shoukry Ibrahim Atia El-Ganaini

^{1, 2}

1Mathematics Department, Faculty of Science at Dawadmi, Shaqra University, Dawadmi 11911, Saudi Arabia

2Mathematics Department, Faculty of Science, Damanhour University, Bahira 22514, Egypt

Correspondence should be addressed to Shoukry Ibrahim Atia El-Ganaini,[email protected]

Received 11 August 2011; Accepted 26 September 2011 Academic Editor: Anuar Ishak

Copyrightq2011 Shoukry Ibrahim Atia El-Ganaini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By using the first integral method, the traveling wave solutions for the generalized Pochhammer- ChreePCequations are constructed. The obtained results include complex exponential function solutions, complex traveling solitary wave solutions, complex periodic wave solutions, and complex rational function solutions. The power of this manageable method is confirmed.

1. Introduction

In this paper, we study the generalized Pochhammer-ChreePCequations:

u_tt−u_ttxx−

αuβuⁿ¹νu²ⁿ¹

xx 0, n≥1, 1.1

whereα, β, andνare constants. Equation1.1represents a nonlinear model of longitudinal wave propagation of elastic rods1–14.The model forα 1, β 1/n1, andν 0 was studied in4,7,8where solitary wave solutions for this model were obtained forn 1,2, and 4. A second model forα0, β−1/2, andν0 was studied by9, and solitary wave solutions were obtained as well.

However, a third model was investigated in10–13forn1,2 where explicit solitary wave solutions and kinks solutions were derived.

It is the objective of this work to further complement studies on a generalized PC equations in1–14.

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The first integral method, which is based on the ring theory of commutative algebra, was first proposed by Feng 15. This method was further developed by the same author in16–21and some other mathematicians22–26. Our first interest in the present work is to implement the first integral method to stress its power in handling nonlinear equations, so that one can apply it for solving various types of nonlinearity. The next interest is in the determination of exact traveling wave solutions for the generalized PC equations. The remaining structure of this paper is organized as follows: Section 2 is a brief introduction to the first integral method. In Section3, by implementing the first integral method, new exact traveling wave solutions to the generalized PC equations are reported with the aid of mathematical software Mathematica 8.0. This describes the ability and reliability of the method. A conclusion is given in Section4.

2. The First Integral Method

Consider a general nonlinear partial diﬀerential equation in the form

Pu, ut, u_x, u_xx, u_tt, u_xt, u_xxx, . . . 0. 2.1 Using the wave variable ξ x−ct carries 2.1 into the following ordinary diﬀerential equationODE:

Q

U, U, U, U, . . .

0, 2.2

where prime denotes the derivative with respect to the same variableξ.

Next, we introduce new independent variablesxu, yuξwhich change2.2to a system of ODEs:

xy, yf

x, y

. 2.3

According to the qualitative theory of diﬀerential equations 27, if one can find the first integrals to System 2.3under the same conditions, the analytic solutions to 2.3 can be solved directly. However, in general, it is diﬃcult to realize this even for a single first integral, because for a given plane autonomous system, there is no general theory telling us how to find its first integrals in a systematic way. A key idea of this approach here to find the first integral is to utilize the Division Theorem. For convenience, first let us recall the Division Theorem for two variables in the complex domainC15.

Division Theorem

Suppose thatPx, yand Qx, yare polynomials of two variablesxandy inCx, yand Px, yis irreducible inCx, y. If Qx, yvanishes at all zero points ofPx, y, then there exists a polynomialGx, yinCx, ysuch that

Q x, y

P x, y

G x, y

. 2.4

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3. The Generalized PC Equations

We conduct our analysis by examining all possible cases ofνfor the generalized PC equations 1.1.

Case 1.

β /0, ν /0. 3.1

Using the wave variableξx−ctand integrating twice, we obtain

c²−α

u−c²u−βuⁿ¹−νu²ⁿ¹0, 3.2 where prime denotes the derivative with respect to the same variableξ. Making the following transformation:

vuⁿ, 3.3

then3.2becomes

c²−α

n²v²−nc²vv−c²1−n

v2−n²βv³−n²νv⁴0, 3.4

wherevandvdenotedv/dξandd²v/dξ², respectively. Equation3.4is a nonlinear ODE, and we can rewrite it as

v−avbv²

v dv²fv³0, 3.5

where

a

1− α c²

n, b 1−n

n , d nβ

c², f nν

c². 3.6

Let x v, lety dv/dξ, and let 3.5 be equivalent to the following two-dimensional autonomous system

dx dξ y, dy

dξ ax−by²

x −dx²−fx³.

3.7

Assume that

dτ dξ

x, 3.8

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thus system3.7becomes

dx dτ xy, dy

dτ ax²−by²−dx³−fx⁴.

3.9

Now, we are applying the Division Theorem to seek the first integral to system3.9. Suppose thatx xτ, y yτare the nontrivial solutions to3.9, andpx, y _m

i0aixyⁱ is an irreducible polynomial inCx, y, such that

p xτ, yτ ^m

i0

a_ixτyτⁱ0, 3.10

wherea_ix i0,1, . . . , mare polynomials ofxanda_mx/0. We call3.10the first integral of polynomial form to system3.9. We start our study by assumingm1 in3.10. Note that dp/dτ is a polynomial inxandy, andpxτ, yτ 0 impliesdp/dτ|_3.9 0. According to the Division Theorem, there exists a polynomialHx, y hx gxyinCx, ysuch that

dp dτ

_3.9 ∂p

∂x

∂τ ∂p

∂y

∂τ

3.9

¹

i0

a_ixyⁱ·xy ¹

i0

iaixyⁱ⁻¹·

ax²−by²−dx³−fx⁴

hx gxy₁

i0

a_ixyⁱ

,

3.11

where prime denotes diﬀerentiation with respect to the variable x. On equating the coeﬃcients ofyⁱ i2,1,0on both sides of3.11, we have

xa₁x−ba1x gxa1x, 3.12

xa₀x hxa1x gxa0x, 3.13

a₁x

ax²−dx³−fx⁴

hxa0x. 3.14

Since,a₁xis a polynomial ofx, from3.12we conclude thata₁xis a constant andgx

−b. For simplicity, we takea1x 1, and balancing the degrees ofhxanda0xwe conclude that deghx 2 and dega0x 2 only. Now suppose that

hx A₂x²A₁xA₀, a₀x B₂x²B₁xB₀ A2/0, B₂/0, 3.15

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whereAi, Bi, i 0,1,2are all constants to be determined. Substituting3.15into3.13, we obtain

hx b2B2x² b1B1xbB0. 3.16

Substitutinga0x, a1x,andhxin3.14and setting all the coeﬃcients of powersxto be zero, we obtain a system of nonlinear algebraic equations, and by, solving it, we obtain the following solutions:

d−

√a32b

√ f

−2−b√

1b, B00, B1−

√a

√1b, B2− f

√−2−b, 3.17

d−

√a32b

√ f

−2−b√

1b, B00, B1

√a

√1b, B2 f

√−2−b, 3.18

d

√a32b

√ f

−2−b√

1b, B₀0, B₁−

√a

√1b, B₂ f

√−2−b, 3.19

d

√a32b

√ f

−2−b√

1b, B₀0, B₁

√a

√1b, B₂− f

√−2−b. 3.20

Setting3.17and3.18in3.10, we obtain that System3.9has one first integral

y∓ f

√−2−bx²

√a

√1bx

0, 3.21

respectively. Combining this first integral with3.9, the second-order diﬀerential equation 3.5can be reduced to

dv

dξ ± f

√−2−bv²

√a

√1bv

. 3.22

Solving 3.22 directly and changing to the original variables, we obtain the following complex exponential function solutions to1.1:

u₁x, t

⎛

⎜⎝ iR exp

−n

1−α/c²x−ct−iRc₁

−√ ν/c

⎞

⎟⎠

1/n

, 3.23

u₂x, t

⎛

⎜⎝ iRexpiRc1

exp n

1−α/c²x−ct

−√ ν/c

expiRc1

⎞

⎟⎠

1/n

. 3.24

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Similarly, for the cases of3.19and3.20, we have another complex exponential function solutions:

u3x, t

iR

−exp−n

1−α/c² x−ct−iRc1

√ν/c _1/n

, 3.25

u4x, t

⎛

⎜⎝ iRexpiRc1

−exp n

1−α/c²x−ct √

ν/cexpiRc1

⎞

⎟⎠

1/n

, 3.26

where,R

1−α/c²√

1n,c1 is an arbitrary constant. These solutions are all new exact solutions. Now we assume thatm 2 in 3.10. By the Division Theorem, there exists a polynomialHx, y hx gxyinCx, ysuch that

dp dτ

3.9

∂p

∂x

∂τ ∂p

∂y

∂τ

3.9

²

m1

a_ixyⁱ·xy ²

m1

ia_ixyⁱ⁻¹·

ax²−by²−dx³−fx⁴

hx gxy₂

m1

aixyⁱ

,

3.27

On equating the coeﬃcients ofyⁱ i3,2,1,0on both sides of3.11, we have

xa₂x−2ba2x gxa2x, 3.28

xa₁x−ba1x hxa2x gxa1x, 3.29

xa₀x 2a2x

ax²−dx³−fx⁴

hxa1x gxa0x, 3.30

a1x

ax²−dx³−fx⁴

hxa0x. 3.31

Sincea2xis a polynomial ofx, from3.28we conclude thata2xis a constant andgx

−2b. For simplicity, we takea₂x 1, and balancing the degrees ofhx, a0x, anda₁xwe conclude that deghx 2 and dega1x 2. In this case, we assume that

hx A₂x²A₁xA₀, a₁x B₂x²B₁xB₀ A2/0, B₂/0, 3.32

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whereAi, Bi i 0,1,2are constants to be determined. Substituting3.32into3.29and 3.30, we have

hx 2bB2x² 1bB1xbB0, a0x

2f 2bB²₂ 22b

x⁴

2d 32bB1B2

32b

x³

−2a 1bB²₁21bB0B₂ 21b

x²B0B1xB₀²

2 Fx^−2b,

3.33

whereFis an arbitrary integration constant. Substitutinga0x,a1x, andhxin3.31and setting all the coeﬃcients of powersxto be zero, we obtain a system of nonlinear algebraic equations, and by solving it we obtain

F0, a 41bd²

32b²B²₂, f−1

42bB₂², B0 0, B1− 4d 32bB2.

3.34 Setting3.34in3.10, we obtain

y 4dx−32bB₂²x²

232bB2 . 3.35

Using this first integral, the second-order ODE3.5reduces to dv

dξ 4dv−32bB²₂v²

232bB2 . 3.36

Similarly, solving 3.36and changing to the original variables, we obtain the exponential function solutions:

u5x, t

2β2nB2S nexp

β2B212/nc1−xctS

2nB₂² _1/n

, 3.37

where S 2n²/2nc²B₂,c₁ is an arbitrary constant. These solutions are all new exact solutions.

Case 2.

β0, ν /0. 3.38

We now investigate the generalized PC equation1.1forβ0, then, we obtain c²−α

u−c²u−νu²ⁿ¹0, 3.39

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where prime denotes the derivative with respect toξ. Similarly as in Case1, making then the following transformation:

vuⁿ, 3.40

then3.39becomes

c²−α

n²v²−nc²vv−c²1−n v₂

−n²νv⁴0, 3.41

wherevandvdenotedv/dξandd²v/dξ², respectively. Let us rewrite3.41as

v−avbv²

v fv³0, 3.42

wherea, b, f are as given in3.6. Letxv, letydv/dξ, and3.42become the following two-dimensional autonomous system:

dx dξ y, dy

dξ ax−by² x −fx³.

3.43

Assume that

dτ dξ

x, 3.44

thus system3.43becomes

dx dτ xy, dy

dτ ax²−by²−fx⁴.

3.45

Following the same procedures as in Case1, so we are applying the Division Theorem to seek the first integral to system3.45. Suppose thatxxτand yyτare the nontrivial solutions to3.45, andpx, y _m

i0a_ixyⁱ is an irreducible polynomial inCx, y, such that

p xτ, yτ ^m

i0

aixτyτⁱ0, 3.46

whereaix i0,1, . . . , mare polynomials ofxandamx/0. We call3.46the first integral of polynomial form to system3.45. We start by assumingm1 in3.46. Note thatdp/dτ

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is a polynomial inxandy, andpxτ, yτ 0 impliesdp/dτ|_3.44 0. According to the Division Theorem, there exists a polynomialHx, y hx gxyinCx, ysuch that

dp dτ

3.45

∂p

∂x

∂τ ∂p

∂y

∂τ

3.45

¹

i0

a_ixyⁱ·xy ¹

i0

ia_ixyⁱ⁻¹·

ax²−by²−fx⁴

hx gxy ₁

i0

aixyⁱ

,

3.47

where prime denotes diﬀerentiation with respect to the variable x. On equating the coeﬃcients ofyⁱ i2,1,0on both sides of3.47, we have

xa₁x−ba1x gxa1x, 3.48

xa₀x hxa1x gxa0x, 3.49

a1x

ax²−fx⁴

hxa0x. 3.50

Since,a₁xis a polynomial ofx, from3.48we conclude thata₁xis a constant andgx

−b. For simplicity, we takea1x 1, and balancing the degrees ofhxanda0xwe conclude that deghx 2 and dega0x 2 only. Now suppose that

hx A₂x²A₁xA₀, a₀x B₂x²B₁xB₀ A2/0, B₂/0, 3.51

whereA_i, B_i,i 0,1,2are constants to be determined. Substituting3.51into3.49, we have

hx 2bB2x² 1bB1xbB0. 3.52

Substitutinga₀x, a1x, andhxin3.50and setting all the coeﬃcients of powersxto be zero, we obtain a system of nonlinear algebraic equations, and, by solving it, we obtain the following solutions:

a−2 fB0

√−2−b, B2− f

√−2−b, B10,

a 2 fB₀

√−2−b, B₂ f

√−2−b, B₁0.

3.53

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Thus, by the similar procedure explained above in Case 1, the complex traveling solitary wave and the complex periodic wave solutions to the generalized PC equations in this Case 2are given, respectively, by

u₁x, t

⎛

⎜⎝−q B₀√

ctanh p

x−ct−i

11/nc₁ B₀ /

q√ c p

⎞

⎟⎠

1/n

,

u₂x, t

⎛

⎜⎝−q B₀√

ctan p

x−ct−i

11/nc₁ B₀ /

q√ c p

⎞

⎟⎠

1/n

,

3.54

wherep n^1/4ν^1/4, q i^1/411/n^1/4, c1 is an arbitrary constant. These solutions are all new exact solutions. Now we assume thatm 2 in3.46. By the Division Theorem, there exists a polynomialHx, y hx gxyinCx, ysuch that

dp dτ

_3.45 ∂p

∂x

∂τ ∂p

∂y

∂τ

3.45

²

i0

a_ixyⁱ·xy ²

i0

ia_ixyⁱ⁻¹·

ax²−by²−fx⁴

hx gxy ₂

i0

aixyⁱ

3.55

On equating the coeﬃcients ofyⁱ i3,2,1,0on both sides of3.55, we have

xa₂x−2ba2x gxa2x, 3.56

xa₁x−ba1x hxa2x gxa1x, 3.57 xa₀x 2a2x

ax²−fx⁴

hxa1x gxa0x, 3.58

a₁x

ax²−fx⁴

hxa0x. 3.59

Sincea2xis a polynomial ofx, from3.56we conclude thata2xis a constant andgx

−2b. For simplicity, we takea2x 1, and balancing the degrees ofhx, a0xanda1xwe conclude that deghx 1, dega1x 1 and deghx 2, dega1x 2.

Subcase 2.1. deghx 1 and dega1x 1. In this case, we assume that

hx A₁xA₀, a₁x B₁xB₀ A1/0, B₁/0, 3.60

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whereAi, Bi i0,1are constants to be determined. Inserting3.60into3.57and3.58, we deduce that

hx 1bB1xbB₀ a0x

f 2b

x⁴

−2a 1bB₁² 21b

x²B0B1xB²₀

2 Fx^−2b,

3.61

whereFis an arbitrary integration constant. Substitutinga₀x,a₁x, andhxin3.59and setting all the coeﬃcients of powersxto be zero, we obtain a system of nonlinear algebraic equations, and by solving it we obtain

a 1

4B²₁1b, F0, B00. 3.62

Then, by the similar procedure explained above, we get the complex exponential function solutions which can be expressed as

u₃x, t

iKexpKB1c₁

−expB1/2x−ct 2K/c√ ν√

n expKB1c₁ 1/n

,

u4x, t

− iKexpKB1c1

−expB1/2x−ct 2K/c√ ν√

n expKB1c1 _1/n

,

3.63

whereK

11/n. These solutions are all new exact solutions.

Subcase 2.2. deghx 2 and dega1x 2. Now suppose that

hx A₂x²A₁xA₀, a₁x B₂x²B₁xB₀ A2/0, B₂/0, 3.64

where,A_i, B_i,i 0,1,2are constants to be determined. Substituting3.64into3.57and 3.58, we have

hx 2bB2x² 1bB1xbB0 3.65

a0x

2f 2bB²₂ 2 2b

x⁴B1B2x³

−2a 1bB₁² 21bB0B₂ 21b

x²B0B1x B₀²

2 Fx⁻^2b,

3.66

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whereFis an arbitrary integration constant. Substitutinga0x, a1x, andhxin3.59and setting all the coeﬃcients of powersxto be zero, we obtain a system of nonlinear algebraic equations, and, by solving it, we obtain the following solutions:

F0, a0, B00, B10, B2− 2

√ f

−2−b, F0, a0, B₀0, B₁0, B₂ 2

√ f

−2−b.

3.67

Thus, as above, we obtain the complex rational function solutions which can be written as

u5x, t

iK

√n√ ν

∓x/√ αt

−iKc₁ _1/n

,

u₆x, t

− iK

√n√ ν

∓x/√ αt

iKc1

_1/n ,

3.68

whereKas defined above. These solutions are all new exact solutions.

Notice that the results in this paper are based on the assumption ofm 1,2 for the generalized PC equations. For the cases of m 3,4 for these equations, the discussions become more complicated and involves the irregular singular point theory and the elliptic integrals of the second kind and the hyperelliptic integrals. Some solutions in the functional form cannot be expressed explicitly. One does not need to consider the casesm≥5 because it is well known that an algebraic equation with the degree greater than or equal to 5 is generally not solvable.

4. Conclusion

In this work, we are concerned with the generalized PC equations for seeking their traveling wave solutions. We first transform each equation into an equivalent two-dimensional planar autonomous system then use the first integral method to find one first integral which enables us to reduce the generalized PC equations to a first-order integrable ordinary diﬀerential equations. Finally, a class of traveling wave solutions for the considered equations are obtained. These solutions include complex exponential function solutions, complex traveling solitary wave solutions, complex periodic wave solutions, and complex rational function solutions. We believe that this method can be applied widely to many other nonlinear evolution equations, and this will be done in a future work.

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Travelling Wave Solutions to the Generalized

Research Article