2. Description of the Extended Mapping Method

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Volume 2012, Article ID 597983,14pages doi:10.1155/2012/597983

Research Article

Extended Mapping Method and Its Applications to Nonlinear Evolution Equations

J. F. Alzaidy

Mathematics Department, Faculty of Science, Taif University, Saudi Arabia

Correspondence should be addressed to J. F. Alzaidy,[email protected] Received 2 April 2012; Revised 15 July 2012; Accepted 31 July 2012 Academic Editor: Renat Zhdanov

Copyrightq2012 J. F. Alzaidy. 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.

We use extended mapping method and auxiliary equation method for finding new periodic wave solutions of nonlinear evolution equations in mathematical physics, and we obtain some new periodic wave solution for the Boussinesq system and the coupled KdV equations. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial diﬀerential equations arising in mathematical physics.

1. Introduction

The effort in finding exact solutions to nonlinear equations is important for the understanding of most nonlinear physical phenomena. For instance, the nonlinear wave phenomena observed in fluid dynamics, plasma, and optical fibers are often modeled by the bell- shaped sech solutions and the kink-shaped tanh solutions. Many effective methods have been presented, such as inverse scattering transform method1, Bäcklund transformation 2, Darboux transformation3, Hirota bilinear method 4, variable separation approach 5, various tanh methods6–9, homogeneous balance method10, similarity reductions method11,12,G/G-expansion method13, the reduction mKdV equation method14, the trifunction method15,16, the projective Riccati equation method17, the Weierstrass elliptic function method18, the Sine-Cosine method 19, 20, the Jacobi elliptic function expansion 21,22, the complex hyperbolic function method 23, the truncated Painlevé expansion 24, the F-expansion method 25, the rank analysis method 26, the ansatz method27,28, the exp-function expansion method29, and the sub-ODE method30.

The main objective of this paper is using the extended mapping method to construct the exact solutions for nonlinear evolution equations in the mathematical physics via the Boussinesq system and the coupled KdV equations.

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2. Description of the Extended Mapping Method

Suppose we have the following nonlinear PDE:

Fu, ut, ux, utt, uxx, uxt, . . . 0, 2.1

whereu ux, tis an unknown function,F is a polynomial inu ux, tand its various partial derivatives in which the highest order derivatives and nonlinear terms are involved.

In the following we give the main steps of a deformation method.

Step 1. The traveling wave variable

ux, t uξ, ξkx−ωt, 2.2

where k and ω are the wave number and the wave speed, respectively. Under the transformation2.2,2.1becomes an ordinary diﬀerential equationODEas

P

u, u, u, u, . . .

0. 2.3

Step 2. If all the terms of2.3contain derivatives inζ, then by integrating this equation and taking the constant of integration to be zero, we obtain a simplified ODE.

Step 3. Suppose that the solution2.3has the following form:

uξ a0ⁿ

i1

aifⁱξ bif⁻ⁱξ ⁿ

i2

cifⁱ⁻²ξfξ ⁿ

i−1

difⁱξfξ, 2.4

where a0, ai, bi, ci, and di are constants to be determined later, while fξ satisfies the nonlinear ODE:

fξ₂

pf⁴ξ qf²ξ r, 2.5

wherep,q, andrare constants.

Step 4. The positive integer “n” can be determined by considering the homogeneous balance between the highest derivative term and the nonlinear terms appearing in2.3. Therefore, we can get the value ofnin2.4.

Step 5. Substituting 2.4 into 2.3 with the condition 2.5, we obtain polynomial in fⁱξfξ^j,i . . . ,−2,−1,0,1,2, . . .;j 0,1. Setting each coeﬃcient of this polynomial to be zero yields a set of algebraic equations fora0,ai,bi,ci,di,ω, andk.

Step 6. Solving the algebraic equations by use of Maple or Mathematica, we havea0,ai,bi,ci, di, andkexpressed byp,q,r.

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Step 7. Since the general solutions of2.5have been well known for usseeAppendix A, then substituting the obtained coeﬃcients and the general solution of2.5into2.4, we have the travelling wave solutions of the nonlinear PDE2.1.

3. Applications of the Method

In this section, we apply the extended mapping method to construct the exact solutions for the Boussinesq system and the coupled KdV equations, which are very important nonlinear evolution equations in mathematical physics and have been paid attention by many researchers.

Example 3.1the Boussinesq system. We start the Boussinesq system32in the following form:

vt 1

3uxxx8 3uux, utvx.

3.1

The traveling wave variable2.2permits us converting3.1into the following ODE:

ωv1

3k²u8

3uu0, ωuv0.

3.2

Integrating3.2with respect toξonce and taking the constant of integration to be zero, we obtain

ωv1

3k²u4

3u²0, 3.3

ωuv0. 3.4

Suppose that the solutions of3.3and3.4can be expressed by

uξ a0ⁿ

i1

i2

cifⁱ⁻²ξfξ ⁻ⁿ

i−1

difⁱξfξ,

vξ A0^m

i1

Aifⁱξ Bif⁻ⁱξ ^m

i2

Lifⁱ⁻²ξfξ ^−m

i−1

Hifⁱξfξ,

3.5

wherea0,ai,bi,ci,di,Ai,Bi,Li, andHiare constants to be determined later.

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Considering the homogeneous balance between the highest order derivativeu and the nonlinear termu² in3.3, the order ofuandvin3.4, then we can obtainn m 2, hence the exact solutions of3.5can be rewritten as,

uξ a0a1fξ b1

1

fξa2f²ξ b2

1

f²ξc2fξ d1fξ

fξ d2fξ f²ξ, vξ A0A1fξ B1 1

fξA2f²ξ B2 1

f²ξL2fξ H1fξ

fξ H2fξ f²ξ,

3.6

where a0, a1, a2, b1, b2, c2, d1, d2, A0, A1, B1, B2, L2, H1, and H2 are constants to be determined later. Substituting3.6with the condition2.5into3.3and3.4and collecting all terms with the same power offⁱξfξ^j,i. . . ,−2,−1,0,1,2, . . .;j0,1. Setting each coeﬃcients of this polynomial to be zero, we get a system of algebraic equations which can be solved by Maple or Mathematica to get the following solutions.

Case 1. Consider

a0a1a2b1c2d1d2A1A2B1L1H1H2 0, A0arbitrary constant, b2−9ω²r

8q , B2 9ω³r

8q , k±ω√ 3 2√

q. 3.7

Case 2. Consider

a0a1b2b1 c2d1d2A1B2B1L1H1H20, A0arbitrary constant, a2−9ω²p

8q , A2 9ω³p

8q , k±ω√ 3 2√

q.

3.8

Case 3. Consider

a0a1a2b1c2d1A2A1B1L1H10, b2−9ω²r

4q , d2∓9ω²√ r

4q , B2 9ω³r

4q , H2±9ω³√ r 4q , A0arbitrary constant, k ω√

√3 q.

3.9

Case 4. Consider

a0a1b1c2d1d2A1B1L1H1H20, a2−9ω²p

8q , b2−9ω²r

8q , A2 9ω³p

8q , B2 9ω³r 8q , A0arbitrary constant, k±ω√

3 2√

q.

3.10

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Note that there are other cases which are omitted here. Since the solutions obtained here are so many, we just list some of the exact solutions corresponding to Case4to illustrate the eﬀectiveness of the extended mapping method.

Substituting3.10into3.6yields

uξ −9ω²p

8q f²ξ− 9ω²r 8q

1 f²ξ, vξ 9ω³p

8q f²ξ 9ω³r 8q

1 f²ξ,

3.11

where

ξ±ω√ 3 2√

qx−ωt. 3.12

According toAppendix A, we have the following families of exact solutions.

Family 1. Ifr1, q−1m², pm², fξ snξ, then we get

uξ 9ω²

81m² m²sn²ξ ns²ξ ,

vξ − 9ω³

81m² m²sn²ξ ns²ξ ,

3.13

where

ξ± ω√ 3 2i√

1m²x−ωt. 3.14

Family 2. Ifr1−m²,q2m²−1,p−m²,fξ cnξ, then we get

uξ 9ω²

82m²−1 m²cn²ξ−

1−m² nc²ξ

,

vξ − 9ω³m²

82m²−1 cn²ξ−

1−m² nc²ξ

,

3.15

where

ξ± ω√ 3 2√

2m²−1x−ωt. 3.16

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Family 3. Ifrm²−1,q2−m²,p−1,fξ dnξ, then we get

uξ 9ω²

82−m² dn²ξ−

m²−1 nd²ξ

,

vξ − 9ω³

82−m² dn²ξ−

m²−1 nd²ξ

,

3.17

where

ξ± ω√ 3 2√

m²−1x−ωt. 3.18

Family 4. Ifrm²,q−1m²,p1,fξ dcξ, then we get

uξ 9ω²

81m² dc²ξ m²cd²ξ ,

vξ − 9ω³

81m² dc²ξ m²cd²ξ ,

3.19

where

ξ± ω√ 3 2i√

1m²x−ωt. 3.20

Family 5. Ifr1,q2−m²,p1−m²,fξ scξ, then we get

uξ − 9ω²

82−m² 1−m²

sc²ξ cs²ξ ,

vξ 9ω³

82−m² 1−m²

sc²ξ cs²ξ ,

3.21

where

ξ± ω√ 3 2√

2−m²x−ωt. 3.22

Family 6. Ifr1/4,q 1/21−2m²,p1/4,fξ nsξ±csξ, then we get

uξ 9ω²

1−2ns²ξ 8

1−2m² , vξ −9ω³

1−2ns²ξ 8

1−2m² ,

3.23

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where

ξ± ω√ 3 2

1/2

1−2m²x−ωt. 3.24

Family 7. Ifr 1/41−m²,q 1/41m²,p 1/41−m²,fξ ncξ±scξ, then we get

uξ −9ω²

1−m²

sc²ξ nc²ξ

41m² ,

vξ 9ω³

1−m²

sc²ξ nc²ξ

41m² ,

3.25

where

ξ± ω√

√ 3

1m²x−ωt. 3.26

Similarly, we can write down the other families of exact solutions of3.1which are omitted for convenience.

Example 3.2 the coupled KdV equations. In this subsection, consider the coupled KdV equations32:

utuxxx6uux6vvx,

vtvxxx6uvx6vux. 3.27

ωuk²u3u²v²0, ωvk²v6uv0.

3.28

Integrating3.2with respect toξonce and taking the constant of integration to be zero, we obtain

ωuk²u3

u²v²

0, 3.29

ωvk²v6uv 0. 3.30

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Suppose that the solutions of3.27can be expressed by

uξ a0ⁿ

i1

i2

cifⁱ⁻²ξfξ ⁻ⁿ

i−1

difⁱξfξ,

vξ α0^m

i1

αifⁱξ βif⁻ⁱξ ^m

i2

γifⁱ⁻²ξfξ ^−m

i−1

eifⁱξfξ,

3.31

wherea0,ai,bi,ci,di,αi,βi,γi, andeiare constants to be determined later.

Balancing the order ofuandv²in3.29, the order ofvanduvin3.30, then we can obtainnm2, so3.31can be rewritten as

uξ a0a1fξ b1

1

fξa2f²ξ b2

1

f²ξc2fξ d1

fξ fξ d2

fξ f²ξ, vξ α0α1fξ β1 1

fξα2f²ξ β2 1

f²ξγ2fξ e1fξ

fξ e2fξ f²ξ,

3.32

wherea0,a1,a2,b1,b2,c2,d1,d2,α0,α1,β1,β2,γ2,e1, ande2are constants to be determined later. Substituting 3.31 with the condition 2.5 into 3.29 and 3.30 and collecting all terms with the same power offⁱξfξ^j,i . . . ,−2,−1,0,1,2, . . .;j 0,1. Setting each coeﬃcient of this polynomial to be zero, we get a system of algebraic equations which can be solved by Maple or Mathematica to get the following solutions.

Case 1. Consider

a1a2b1c2d1d2α1α2 β1e1 e2γ20,

a0−ω 12

⎛

⎜⎝1 q

q²−3pr

⎞

⎟⎠, b2 − ωr 4

q²−3pr ,

α0−ω 12

⎛

⎜⎝1 q

q²−3pr

⎞

⎟⎠, β2− ωr 4

q²−3pr

, k±

√ω 2⁴

q²−3pr .

3.33

Case 2. Consider

a1b1b2c2d1d2α1β1β2e1e2γ20,

a0−ω 12

⎛

⎜⎝1 q

q²−3pr

⎞

⎟⎠, a2− ωp 4

q²−3pr ,

α0 ω 12

⎛

⎜⎝1 q

q²−3pr

⎞

⎟⎠, α2 ωp 4

q²−3pr

, k±

√ω 2⁴

q²−3pr .

3.34

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Case 3. Consider

a1b1c2d1d2 α1 β2e1 e2γ20,

a0−ω 12

⎛

⎜⎝3 q

q²12pr

⎞

⎟⎠, a2− ωp 4

q²12pr

, b2 − ωr 4

q²12pr ,

α0−ω 12

⎛

⎜⎝1− q

q²12pr

⎞

⎟⎠, α2 ωp 4

q²12pr

, β2 ωr

4

q²12pr ,

k±

√ω 2⁴

q²12pr .

3.35

Case 4. Consider

a1b1b2d1d2α1β1β2e1 e20,

a0−ω 12

⎛

⎜⎝1 q

q²12pr

⎞

⎟⎠, a2− ωp 2

q²12pr

, c2− ω√ p 2

q²12pr ,

α0 ω 12

⎛

⎜⎝1 q

q²12pr

⎞

⎟⎠, α2 ωp 2

q²12pr

, γ2 ω√ p 2

q²12pr ,

k±

√ω

4

q²12pr .

3.36

Note that there are other cases which are omitted here. Since the solutions obtained here are so many, we just list some of the exact solutions corresponding to Case4 to illustrate the eﬀectiveness of the extended mapping method.

uξ −ω 12

⎛

⎜⎝1 q

q²12pr

⎞

⎟⎠− ωp 2

q²12pr

f²ξ− ω√ p 2

q²12pr fξ,

vξ ω 12

⎛

⎜⎝1 q

q²12pr

⎞

⎟⎠ ωp 2

q²12pr

f²ξ ω√ p 2

q²12pr fξ,

3.37

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where

ξ±

√ω

4

q²12pr

x−ωt. 3.38

According toAppendix A, we have the following families of exact solutions.

Family 1. Ifr1,q2m²−1,pm²m²−1,fξ sdξ, then we get

uξ −ω 12

1 2m²−1

16m⁴−16m²1

− ωm² m²−1

sd²ξ 2

16m⁴−16m²1−ωm√

m²−1ndξcdξ 2

16m⁴−16m²1 , vξ ω

12

1 2m²−1 16m⁴−16m²1

ωm² m²−1

sd²ξ 2

16m⁴−16m²1 ωm√

m²−1ndξcdξ 2

16m⁴−16m²1 , 3.39

where

ξ±

√ω 4

16m⁴−16m²1x−ωt. 3.40

Family 2. Ifrm²m²−1,q2m²−1,p1,fξ dsξ, then we get

uξ −ω 12

1 2m²−1

16m⁴−16m²1

− ωds²ξ 2

16m⁴−16m²1 ωcsξnsξ 2

16m⁴−16m²1, vξ ω

12

1 2m²−1

16m⁴−16m²1

ωds²ξ 2

16m⁴−16m²1 − ωcsξnsξ 2

16m⁴−16m²1, 3.41

where

ξ±

√ω 4

16m⁴−16m²1x−ωt. 3.42

Family 3. Ifrm²/4,q 1/2m²−2,pm²/4,fξ snξ±icnξ, then we get

uξ−ω 12

1 m²−2 2√

m⁴−m²1

−ωm²snξ±icnξ² 8√

m⁴−m²1 −ωmcnξdnξ∓isnξdnξ 4√

m⁴−m²1 , vξω

12

1 m²−2 2√

m⁴−m²1

ωm²snξ±icnξ² 8√

m⁴−m²1 ωmcnξdnξ∓isnξdnξ 4√

m⁴−m²1 , 3.43

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where

ξ±

√ω

√4

m⁴−m²1x−ωt. 3.44

Family 4. Ifr1,q−1m²,pm²,fξ snξ, then we get

uξ −ω 12

1− 1m²

√m⁴14m²1

− ωm²sn²ξ 2√

m⁴14m²1− ωmcnξdnξ 2√

m⁴14m²1, vξ ω

12

1− 1m²

√m⁴14m²1

ωm²sn²ξ 2√

m⁴14m²1 ωmcnξdnξ 2√

m⁴14m²1,

3.45

where

ξ±

√ω

√4

m⁴14m²1x−ωt. 3.46

Family 5. Ifr1−m²,q2m²−1,p−m²,fξ cnξ, then we get

uξ −ω 12

1 2m²−1 16m⁴−16m²1

ωm²cn²ξ 2

16m⁴−16m²1 iωmsnξdnξ 2

16m⁴−16m²1, vξ ω

12

1 2m²−1

16m⁴−16m²1

− ωm²cn²ξ 2

16m⁴−16m²1 − iωmsnξdnξ 2

16m⁴−16m²1, 3.47

where

ξ±

√ω 4

16m⁴−16m²1x−ωt. 3.48

Family 6. Ifr1−m², q2−m²,p1,fξ csξ, then we get

uξ −ω 12

1√ 2−m² m⁴−16m²16

− ωcs²ξ 2√

m⁴−16m²16 ωnsξdsξ 2√

m⁴−16m²16, vξ ω

12

1√ 2−m² m⁴−16m²16

ωcs²ξ 2√

m⁴−16m²16− ωnsξdsξ 2√

m⁴−16m²16,

3.49

where

ξ±

√ω

√4

m⁴−16m²16x−ωt. 3.50

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Table 1

p q r fξ fξ

m² −1m² 1 snξ cnξdnξ

−m² 2m²−1 1−m² cnξ −snξdnξ

−1 2−m² m²−1 dnξ −m²snξcnξ

1 −1m² m² nsξ −dsξcsξ

m²−1 2−m² −1 ndξ m²sdξcdξ

1 2−m² 1−m² csξ −nsξdsξ

1−m² 2−m² 1 scξ ncξdcξ

m²m²−1 2m²−1 1 sdξ ndξcdξ

1 2m²−1 m²m²−1 dsξ −csξnsξ

1 4

1

21−2m² 1

4 nsξ±csξ −dsξcsξ∓nsξdsξ

1

41−m² 1

41m² 1

41−m² ncξ±scξ scξdcξ±ncξdcξ m²

4

1

2m²−2 m²

4 snξ±icnξ cnξdnξ∓isnξdnξ

Table 2

snξ → tanhξ cnξ → sechξ dnξ → sechξ nsξ → cothξ

csξ → cschξ dsξ → cschξ scξ → sinhξ sdξ → sinhξ

Family 7. Ifr−1,q2−m²,pm²−1,fξ ndξ, then we get

uξ −ω 12

1 2−m²

√m⁴−16m²16

− ω m²−1

nd²ξ 2√

m⁴−16m²16−ωm²√

m²−1 sdξcdξ 2√

m⁴−16m²16 , vξ ω

12

1√ 2−m² m⁴−16m²16

ω

m²−1 nd²ξ 2√

m⁴−16m²16ωm²√

m²−1 sdξcdξ 2√

m⁴−16m²16 , 3.51

where

ξ±

√ω

√4

m⁴−16m²16x−ωt. 3.52

4. Conclusion

The main objective of this paper is that we have found new exact solutions for the Boussinesq system and the coupled KdV equations by using the extended mapping method with the auxiliary equation method. Also, we conclude according to Appendix B that our results in terms of Jacobi elliptic functions generate into hyperbolic functions whenm → 1 and generate into trigonometric functions when m → 0. This method provides a powerful mathematical tool to obtain more general exact solutions of a great many nonlinear PDEs in mathematical physics.

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Table 3

ncξ 1

cnξ ndξ 1

dnξ cdξ cnξ

dnξ dcξ dnξ

cnξ csξ cnξ

snξ scξ snξ

cnξ sdξ snξ

dnξ dsξ dnξ

snξ

Appendices

A. The Jacobi Elliptic Functions

The general solutions to the Jacobi elliptic equation2.3and its derivatives31are listed in Table 1, where 0< m <1 is the modulus of the Jacobi elliptic functions andi√

−1.

B. Hyperbolic Functions

The Jacobi elliptic functions snξ, cnξ, dnξ, nsξ, csξ, dsξ, scξ, sdξgenerate into hyperbolic functions whenm → 1 as inTable 2.

C. Relations between the Jacobi Elliptic Functions

SeeTable 3.

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Mathematics

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Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

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Volume 2014

Applied Mathematics^{Journal of}

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

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Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

^{Journal of}

2. Description of the Extended Mapping Method

Research Article

Extended Mapping Method and Its Applications to Nonlinear Evolution Equations

J. F. Alzaidy

1. Introduction

2. Description of the Extended Mapping Method

3. Applications of the Method

4. Conclusion

Appendices

A. The Jacobi Elliptic Functions

B. Hyperbolic Functions

C. Relations between the Jacobi Elliptic Functions

References

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Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Discrete Mathematics

Stochastic Analysis