New Jacobi Elliptic Function Solutions for Coupled KdV-mKdV Equation

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New Jacobi Elliptic Function Solutions for

Coupled KdV-mKdV Equation

Yunjie Yang, Yan He, Aifang Feng

Abstract—A generalized (G0/G)-expansion method is used to search for the exact traveling wave solutions of the coupled KdV-mKdV equation. As a result, some new Jacobi elliptic function solutions are obtained. It is shown that the method is straightforward, concise, effective, and can be used for many other nonlinear evolution equations in mathematical physics.

Index Terms—generalized (G0/G)-expansion method; the coupled KdV-mKdV equation; Jacobi elliptic function solutions

I. INTRODUCTION

Seeking the traveling wave solutions of nonlinear evolution equations(NLEEs) has been an interesting and hot topic in mathematics physics for a long time. Many effective methods to construct traveling wave solutions of NLEEs have been established [1-11]. However, no method can be used for finding all solutions for all types of NLEEs. Recently, the

(G0/G)-expansion method [12] has become popular in the research community, and the initial idea has been refined by many studies [13-17]. It is shown that the(G0/G)-expansion method is very effective, and many nonlinear equations have been successfully solved.

In this paper, some exact solutions of the coupled KdV-mKdV equation which are expressed by the Jacobi ellip-tic function are obtained by using the (G0/G)-expansion method.

A. Description of the generalized(G0/G)-expansion method

Assume that the nonlinear partial differential equation

F(u, ux, ut, uxx, uxt, utt,· · ·) = 0, (1)

where F is a polynomial in its arguments. The main steps of the generalized(G0/G)-expansion method are descripted as follows.

Step 1. Seeking traveling wave solutions of (1) by taking

u(x, t) = u(ξ), ξ = x−ct, and transforming (1) to the ordinary differential equation(ODE)

F(u, u0,−cu0, c2u00,−cu00, u00,· · ·) = 0. (2)

Step 2.Looking for its solution (u(ξ)) in the polynomial form

u(ξ) =a0+

m

X

i=1

ai

_f0 f

i

, (3)

This work was supported by the Natural Science Foundations of Educa-tion Committee of Yunnan Province (2013C079, 2012Y452).

Yunjie Yang is with the Department of Mathematics, Kunming University, Kunming, Yunnan, 650214, PR China e-mail: ([email protected]).

Yan He is with the Department of Computer Science and Technology, Chuxiong Teachers College, Chuxiong, Yunnan, 675000, PR China.

Aifang Feng is with the Department of Mathematics, Kunming University, Kunming, Yunnan, 650214, PR China.

where a0, ai(i = 1,2,· · ·, m) are constants which will be determined later, f = f(ξ) is the solution of the auxiliary LODE

f02=P f4+Qf2+R, (4)

whereP, Q andR are constants.

Step 3. Determining the parameter m by balancing the highest order nonlinear term and the highest order partial derivative ofuin (2).

Step 4. Substituting (3) and (4) into (2), and setting all the coefficients of all terms with the same powers of

(f0/f)k(k= 1,2,· · ·) to zero. Then a system of nonlinear algebraic equations (NAEs) with respect to the parameters

c, a0, ai(i= 1,2,· · ·, m) is obtained. By solving the NAEs if available, those parameters can be determined explicitly. Step 5. Assuming that the constantsc, a0, ai(i= 1,2,· · ·, m) can be obtained by solving the algebraic equations in Step 4, and substituting these constants and the known general solutions into (3). Then the explicit solutions of (1) can be obtained immediately.

B. Applications of method

In this section, we apply the (G0/G)-expansion method to seek the exact solutions of the coupled KdV-mKdV equation [18,19] as follows:

ut+αuux+βu2ux+uxxx= 0, (5)

whereα andβ are two constant parameters. Let u(x, t) = u(ξ), ξ=x−ctin (5). Then

−cu0+αuu0 +βu2u0+u000 = 0, (6)

where c is a constant which will be determined later. By integrating both sides of (6) with respect toξ,

−cu+α 2u

2₊β

3u

3₊_u00_{= 0.} ₍₇₎

Thenm = 1 by balancing u3 _and_u00 _{in (7). According to}

(3) and (4) we have

u(ξ) =a0+a1

f0 f

!

, (8)

u00(ξ) = 2a1

f0 f

!

 f0

f !2

−Q



. (9)

With the aid of Maple, substituting (8) and (9) into (7), the left-hand side of (7) becomes a polynomial in (f0/f)

and ξ. Setting their coefficients to zero yields a system of algebraic equations in a0, a1, c, β. Solving these

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a0=

12Q α , a1=

12√Q

α , c= 4Q, β=− α2

24Q. (10)

With the aid of the appendix [20] and from the formal solution (10), we get the following set of exact solutions of (5).

Case 1. Choosing P = m2_{, Q} ₌ ₋_{(1 +}_m2_{), R} _{= 1,} _and

f(ξ) =sn(ξ), we obtain the Jacobi elliptic function solution of (5)

u1=−12_α (1 +m2)− √

−1−m2_cs(ξ)dn(ξ) , ξ=x+ 4(1 +m2_)t. (11)

Case 2. Choosing P = m2_{, Q} ₌ ₋_{(1 +}_m2_{), R} _{= 1,} _and

f(ξ) =cd(ξ), we obtain the Jacobi elliptic function solution of (5)

u2= 12_α (1−m2) √

−1−m2_sd(ξ)nc(ξ)

−12

α(1 +m

2_),

ξ=x+ 4(1 +m2)t.

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Case 3. Choosing P =−m2, Q= 2m2−1, R= 1−m2,

and f(ξ) = cn(ξ) , we obtain the Jacobi elliptic function solution of (5)

u3= 12_α (2m2−1)− √

2m2₋_{1dc(ξ)sn(ξ)} ,

ξ=x−4(2m2−1)t. (13)

Case 4. Choosing P = −1, Q = 2−m2_{, R} ₌ _m2₋_1,

and f(ξ) = dn(ξ) , we obtain the Jacobi elliptic function solution of (5)

u4=12_α (2−m2)−m2 √

2−m2_cd(ξ)sn(ξ) ,

ξ=x−4(2−m2_)t. (14)

Case 5. Choosing P = 1, Q = −(1 +m2_{), R} ₌ _m2_, _and

f(ξ) =ns(ξ), we obtain the Jacobi elliptic function solution of (5)

u5=−12_α (1 +m2) + √

−1−m2_cs(ξ)dn(ξ) , ξ=x+ 4(1 +m2_)t. (15)

Case 6. Choosing P = 1, Q = −(1 +m2_{), R} ₌ _m2_, _and

f(ξ) =dc(ξ), we obtain the Jacobi elliptic function solution of (5)

u6= 12_α (1−m2) √

−1−m2_sc(ξ)nd(ξ)

−12

α(1 +m

2_),

ξ=x+ 4(1 +m2_)t.

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Case 7. Choosing P = 1−m2_{, Q}_{= 2m}2₋_{1, R}₌₋_m2_,

and f(ξ) = nc(ξ) , we obtain the Jacobi elliptic function solution of (5)

u7= 12_α (2m2−1) + √

2m2₋_{1dc(ξ)sn(ξ)} ,

ξ=x−4(2m2−1)t. (17)

Case 8. Choosing P = m2−1, Q = 2−m2, R = −1,

and f(ξ) = nd(ξ) , we obtain the Jacobi elliptic function solution of (5)

u8= 12_α (2−m2) +m2 √

2−m2_cd(ξ)sn(ξ) ,

ξ=x−4(2−m2_)t. (18)

Case 9. Choosing P = 1−m2_{, Q} _{= 2}₋_m2_{, R} _{= 1,} _and

f(ξ) =sc(ξ), we obtain the Jacobi elliptic function solution of (5)

u9= 12_α (2−m2) + √

2−m2_dc(ξ)ns(ξ) ,

ξ=x−4(2−m2)t. (19)

Case 10.ChoosingP=−m2₍₁₋_m2_{), Q}_{= 2m}2₋_{1, R}_{= 1,}

and f(ξ) = sd(ξ) , we obtain the Jacobi elliptic function solution of (5)

u10=12_α (2m2−1) + √

2m2₋_{1cd(ξ)ns(ξ)} ,

ξ=x−4(2m2₋_1)t. (20)

Case 11. Choosing P = 1, Q = 2−m2_{, R}_{= 1}₋_m2_, _and

f(ξ) =cs(ξ), we obtain the Jacobi elliptic function solution of (5)

u11= 12_α (2−m2) + √

2−m2_ds(ξ)nc(ξ) ,

ξ=x−4(2−m2_)t. (21)

Case 12.ChoosingP= 1, Q= 2m2₋_{1, R}₌₋_m2₍₁₋_m2_),

and f(ξ) = ds(ξ) , we obtain the Jacobi elliptic function solution of (5)

u12=12α (2m

2₋₁₎₋√_2m2₋_{1cs(ξ)nd(ξ)} ,

ξ=x−4(2m2₋_1)t. (22)

Case 13.Choosing P = 1/4, Q = (1−2m2)/2, R= 1/4,

and f(ξ) = ns(ξ)±cs(ξ) , we obtain the Jacobi elliptic function solution of (5)

u13=12_α

1

2−m

2

−q1

2−m2ds(ξ)

,

ξ=x−2(1−2m2_)t. (23)

Case 14.Choosing P= (1−m2)/4, Q= (1 +m2)/2, R= (1−m2)/4,andf(ξ) =nc(ξ)±sc(ξ), we obtain the Jacobi elliptic function solution of (5)

u14= 12_α

1

2 1 +m

2

+ q

1

2(1 +m2)dc(ξ)

,

ξ=x−2(1 +m2_)t. (24)

Case 15.Choosing P = 1/4, Q= (m2−2)/2, R=m2/4,

and f(ξ) = ns(ξ)±ds(ξ) , we obtain the Jacobi elliptic function solution of (5)

u15= 12_α

1

2m

2₋₁

−q1

2m2−1cs(ξ)

,

ξ=x−2(m2₋_2)t. (25)

Case 16.ChoosingP =m2/4, Q= (m2−2)/2, R=m2/4,

and f(ξ) =sn(ξ)±icn(ξ) , we obtain the Jacobi elliptic function solution of (5)

u16= 12_α

q

1

2m2−1

dn(ξ)(cn(ξ)−isn(ξ))

sn(ξ)+icn(ξ)

+12_α 1₂m2₋₁ , ξ=x−2(m2₋_2)t.

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Case 17.ChoosingP =m2_{/4, Q}_{= (m}2₋_{2)/2, R}₌_m2_/4,

and f(ξ) = √m2₋_1sd(ξ)_±_cd(ξ), _{we obtain the Jacobi}

elliptic function solution of (5)

u17= 12_α 1₂m2−1

+12 α q 1 2m

2₋₁

. √

m2₋₁_cn₍_ξ₎₊₍_m2₋₁₎_sn₍_ξ₎ dn(ξ)(√m2₋₁_sn₍_ξ₎₊_cn₍_ξ₎₎

,

ξ=x−2(m2−2)t.

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Case 18. Choosing P = 1/4, Q= (1−2m2_{)/2, R} _{= 1/4,}

andf(ξ) =mcd(ξ)±i√1−m2_nd(ξ),_{we obtain the Jacobi}

u18= 12α

1

2−m

2

+12_αq1₂−m2

.

msn(ξ)(−1+m2+i√1−m2_mcn₍_ξ₎₎ dn(ξ)(mcn(ξ)+i√1−m2₎

,

ξ=x−2(1−2m2)t.

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Case 19. Choosing P = 1/4, Q= (1−2m2_{)/2, R} _{= 1/4,}

and f(ξ) = msn(ξ)±idn(ξ)nd(ξ), we obtain the Jacobi elliptic function solution of (5)

u19= 12_α 1₂−m2+12_α

q

1

2−m2

.mcn(_msnξ)(dn₍_ξ(₎₊ξ)−_idnimsn₍_ξ₎(ξ)), ξ=x−2(1−2m2_)t.

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Case 20. Choosing P = 1/4, Q= (1−2m2)/2, R = 1/4,

and f(ξ) = √m2₋_1sc(ξ)_±_idc(ξ), _{we obtain the Jacobi}

u20= 12_α 1₂−m2

+12_α

q

1

2 −m2

. √

m2₋₁_dn₍_ξ₎₋_isn₍_ξ₎₍_m2₋₁₎ cn(ξ)(√m2₋₁_sn₍_ξ₎₊_idn₍_ξ₎₎

,

ξ=x−2(1−2m2_)t.

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Case 21.ChoosingP = (m2−1)/4, Q= (m2+ 1)/2, R= (m2₋_1)/4, _and _f_{(ξ) =} _msd(ξ)_±_nd(ξ), _{we obtain the}

Jacobi elliptic function solution of (5)

u21= 12_α

1

2 1 +m

2

+m q

1

2(1 +m

2_)cd(ξ)_,

ξ=x−2(1 +m2_)t. (31)

Case 22. Choosing P =m2_{/4, Q}_{= (m}2₋_{2)/2, R}_{= 1/4,}

andf(ξ) =sn(ξ)/(1±dn(ξ)),we obtain the Jacobi elliptic function solution of (5)

u22= 12_α

1

2m

2₋_{1 +}q1

2m

2₋_1cs(ξ)_,

ξ=x−2(m2−2)t.

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Case 23. Choosing P = −1/4, Q = (m2 _{+ 1)/2, R} ₌

1−m22

/4, andf(ξ) =mcn(ξ)±dn(ξ), we obtain the Jacobi elliptic function solution of (5)

u23=12_α

1

2(m

2_{+ 1)}₋_mq1

2(m

2_{+ 1)sn(ξ)}_,

ξ=x−2(m2+ 1)t.

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Case 24. Choosing P = 1−m22/4, Q = (m2 + 1)/2, R = 1/4, and f(ξ) = ds(ξ)±cs(ξ), we obtain the Jacobi elliptic function solution of (5)

u24=12_α

1

2(m

2_{+ 1)}₋q1

2(m2+ 1)ns(ξ)

,

ξ=x−2(m2_{+ 1)t.} (34)

Case 25.Choosing P = 1/4, Q= (m2₋_{2)/2, R}₌_m2_/4,

and f(ξ) = dc(ξ)±√1−m2_nc(ξ), _{we obtain the Jacobi}

u25= 12_α 1₂m2−1

+12_α

q

1

2m

2₋₁

.

sn(ξ)(1−m2₊√₁₋_m2_dn₍_ξ₎₎ cn(ξ)(dn(ξ)+√1−m2₎

,

ξ=x−2(m2₋_2)t.

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Case 26.Choosing R =m2_Q2_{/ m}2_{+ 1}2

P, Q < 0, P > 0,andf(ξ) =p−m2_Q/(m2_{+ 1)P sn(}p

−Q/(m2_{+ 1)ξ)}

, we obtain the Jacobi elliptic function solution of (5)

u26=12_αQ+12_α √

Qq₁₊−Q_m2cs

q −Q

1+m2ξ

.dnq₁₊−_mQ2ξ

,

ξ=x−4Qt.

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Case 27. Choosing R = (1−m2_)Q2_{/ m}2₋₂2 P, Q > 0, P < 0, and f(ξ) = p−Q/(2−m2_)P

dnpQ/(2−m2_)ξ_, _we _obtain _the _Jacobi _elliptic

function solution of (5)

u27= 12_αQ−12_α √

Q q

−Q m2₋₂m

2_cdq −Q

m2₋₂ξ

.snq_m−2Q₋₂ξ

,

ξ=x−4Qt.

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Case 28. Choosing R = m2_(m2 ₋

1)Q2_/ _2m2₋₁2

P, Q > 0, P < 0, and

f(ξ) = p−m2_Q/(2m2₋_{1)P cn}p

Q/(2m2₋_1)ξ_,

we obtain the Jacobi elliptic function solution of (5)

u28= 12_αQ−12_α √

Q q

Q

2m2₋₁dc

q _Q

2m2₋₁ξ

.snq₂_mQ2₋₁ξ

,

ξ=x−4Qt.

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Case 29. Choosing P = 1, Q = 2−4m2, R = 1, and

f(ξ) = sn(ξ)dn(ξ)/cn(ξ), we obtain the Jacobi elliptic function solution of (5)

u29= 12_α

√

2−4m2₍₋2m2_sn2₍_ξ₎₊_m2_sn4₍_ξ₎₊₁₎ cn(ξ)sn(ξ)dn(ξ)

,

+12_α(2−4m2) ξ=x−8(1−2m2)t.

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Case 30. Choosing P = m4_{, Q} _{= 2m}2₋_{4, R} _{= 1,} _and

f(ξ) = sn(ξ)cn(ξ)/dn(ξ), we obtain the Jacobi elliptic function solution of (5)

u30=12_α

√

2m2₋₄₍_m2_sn4₍_ξ₎₋₂_sn2₍_ξ₎₊₁₎ dn(ξ)sn(ξ)cn(ξ)

,

+12_α(2m2₋₄₎

ξ=x−8(m2₋_2)t.

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Case 31.ChoosingP = 1, Q= 2m2_{+2, R}_{= 1}₋_2m2_+m4_,

andf(ξ) =cn(ξ)dn(ξ)/sn(ξ), we obtain the Jacobi elliptic function solution of (5)

u31= 12_α

2(m2_{+ 1) +} √

2(m2₊₁₎₍_m2_sn4₍_ξ₎₋₁₎ dn(ξ)sn(ξ)cn(ξ)

,

ξ=x−8(m2_{+ 1)t.}

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Case 32. Choosing P = A2_(m ₋ ₁₎2_{/4, Q} ₌

(m2 _{+ 1)/2 + 3m, R} _{= (m} ₋ ₁₎2_/4A2_, _and _f_{(ξ) =}

cn(ξ)dn(ξ)/A(1 +sn(ξ)) (1 +msn(ξ)),we obtain the Ja-cobi elliptic function solution of (5)

u32= 12_α

_m2_sn2₍_ξ₎₊_msn2₍_ξ₎₋_m₋₁ dn(ξ)cn(ξ)

.q1₂m2₊1

2+ 3m+

12

α

1

2m

2₊1

2+ 3m

,

ξ=x−2(m2+ 1 + 6m)t.

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Case 33. Choosing P = A2_(m _{+ 1)}2_{/4, Q} ₌

(m2 _{+ 1)/2} ₋ _{3m, R} _{= (m} _{+ 1)}2_/4A2_, _and _f_{(ξ) =}

cn(ξ)dn(ξ)/A(1 +sn(ξ)) (1−msn(ξ)), we obtain the Ja-cobi elliptic function solution of (5)

u33= 12_α

_m2_sn2₍_ξ₎₋_msn2₍_ξ₎₊_m₋₁ dn(ξ)cn(ξ)

. q

1

2m

2₊1

2−3m+

12

α

1

2m

2₊1

2−3m

, ξ=x−2(m2_{+ 1}₋_6m)t.

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Case 34. Choosing P = −4/m, Q = 6m −

m2 ₋ _{1, R} ₌ ₋_2m3 ₊ _m4 ₊ _m2_, _and _f(ξ) ₌

mcn(ξ)dn(ξ)/ msn2_{(ξ) + 1}

,we obtain the Jacobi elliptic function solution of (5)

u34= 12_αsn(ξ)

_msn2₍_ξ₎₍_m₊₁₎2₋_m₍_m₊₂₎₋₁ dn(ξ)cn(ξ)(msn2₍_ξ₎₊₁₎

.√6m−m2₋_{1 +} 12

α 6m−m

2₋₁

, ξ=x−4(6m−m2₋_1)t.

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Case 35. Choosing P = 4/m, Q = −6m−m2−1, R = 2m3+m4+m2,andf(ξ) =mcn(ξ)dn(ξ)/ msn2(ξ)−1

, we obtain the Jacobi elliptic function solution of (5)

u35= 12_αsn(ξ)

_msn2₍_ξ₎₍_m₋₁₎2₊_m₍_m₋₂₎₊₁ dn(ξ)cn(ξ)(msn2₍_ξ₎₋₁₎

.√−6m−m2₋_{1 +} 12

α −6m−m

2₋₁

, ξ=x+ 4(6m+m2_{+ 1)t.}

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Case 36. Choosing P = −(m2 + 2m + 1)B2, Q = 2m2 + 2, R = (2m − m2 − 1)/B2, and f(ξ) = msn2(ξ)−1/B msn2(ξ) + 1, we obtain the Jacobi elliptic function solution of (5)

u36=12_α

4m√2(m2₊₁₎_sn₍_ξ₎_cn₍_ξ₎_dn₍_ξ₎ m2_sn4₍_ξ₎₋₁

+12_α 2(m2+ 1), ξ=x−8(m2_{+ 1)t.}

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Case 37. Choosing P = −(m2 − 2m + 1)B2, Q = 2m2 + 2, R = −(2m + m2 + 1)/B2, and f(ξ) = msn2(ξ) + 1/B msn2(ξ)−1, we obtain the Jacobi elliptic function solution of (5)

u37=−12_α

4m√2(m2₊₁₎_sn₍_ξ₎_cn₍_ξ₎_dn₍_ξ₎ m2_sn4₍_ξ₎₋₁

+12_α 2(m2+ 1), ξ=x−8(m2+ 1)t.

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Remark 1. The validity of all the solutions which are obtained are verified.

Remark 2. In fact, there are more than three solutions compared to the latest related works [21] .

II. CONCLUSION

In this paper, some exact solutions of Jacobi elliptic function form from the coupled KdV-mKdV equation are derived. When the modulus of the Jacobi elliptic function

m → 0 or 1 , the corresponding solitary wave solutions and trigonometric function solutions are also obtained. It is shown that the (G0/G)-expansion method provides a very effective and powerful tool for solving nonlinear equations in mathematical physics.

REFERENCES

[1] C.T. Yan, A simple transformation for nonlinear waves, Phys. Lett. A, 224 (1996) 77-84.

[2] M.L. wang, Y.B. Zhou, Z.B. Li, Application of a homogeneous balance method to exact solutions of nonlinear evolution equations in mathe-matical physics, Phys. Lett. A, 216 (1996) 67-75.

[3] S. Zhang, A generalized auxiliary equation method and its application to (2+1)-dimensional Korteweg-de Vries equations, Comput. Math. Appl., 54 (2007) 1028-1038.

[4] E.G. Fan, Extended tanh-function method and its applications to non-linear equations, Phys. Lett. A, 277 (2000) 212-218.

[5] S.K. Liu, Z.T. Fu, S.D. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001) 69-74.

[6] Z.T. Fu, S.K. Liu, S.D. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A, 290 (2001) 72-76.

[7] K. Mohammed, New Exact traveling wave solutions of the (3+1)-dimensional Kadomtsev-Petviaghvili (KP) equation, IAENG Interna-tional Journal of Applied Mathematics, Vol. 37, no. 1, pp. 17-19, Aug. 2007.

[8] T. Hasuike, Exact and Explicit Solution Algorithm for Linear Program-ming Problem with a Second-Order Cone, IAENG International Journal of Applied Mathematics, Vol. 41, no. 3, pp. 1-5, Aug. 2011. [9] Y. Warnapala, R. Siegel and J. Pleskunas, The Numerical Solution of the

Exterior Boundary Value Problems for the Helmholtz’s Equation for the Pseudosphere, IAENG International Journal of Applied Mathematics, Vol. 41, no. 2, pp. 1-6, May. 2011.

[10] J. V. Lambers, Solution of Time-Dependent PDF Through Component-wise Approximation of Matrix Functions, IAENG International Journal of Applied Mathematics, Vol. 41, no. 1, pp. 1-10, Feb. 2011. [11] M.L. Wang, X.Z. Li, Application of F-expansion to periodic wave

solutions for a new Hamiltonian amplitude equation, Chaos, Solitons and Fractals, 24 (2005) 1257-1268.

[12] M.L. Wang, X.Z. Li, and J.L. Zhang, The(G0/G)-expansion method and traveling wave solutions of nonlinear evolution equations in math-ematical physics, Phys. Lett. A, 372 (2008) 417-423.

[13] E.M.E. Zayed, New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized(G0/G)-expansion method, Journal of Physics A, Vol. 42, no. 19, Article ID 195202, p. 13, 2009.

[14] S.M. Guo and Y.B. Zhou, The extended(G0/G)-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations, Applied Mathematics and Computation, Vol. 215, no. 9, pp. 3214-3221, 2010.

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[16] Y.M. Zhao, New Jacobi Elliptic Function Solutions for the Zakharov Equations, Journal of Applied Mathematics, Vol. 2012, Article ID 854619, 16 pages, 2012.

[17] N. Shang, B, Zheng, Exact Solutions for Three Fractional Partial Differential Equations by the(G0/G)Method, IAENG International Journal of Applied Mathematics, Vol. 43, no. 3, pp. 114-119, Aug. 2013.

[18] L.Y. Yang, G.T. Liu, Quasi-rational function solutions of an elliptic equation and its application to solve some nonlinear evolution equations, Applied Mathematics and Computation, Vol. 217, pp. 7377-7384, 2011. [19] H.L. L¨u, X.Q. Liu, L. Niu, A generalized(G0/G)-expansion method and its applications to nonlinear evolution equations, Applied Mathe-matics and Computation, 215 (2010) 3811-3816.

[20] A. Ebaid and E.H. Aly, Exact solutions for the transformed reduced Ostrovsky equation via the F-expansion method in terms of Weierstrass-elliptic and Jacobian-Weierstrass-elliptic functions, Wave Motion, Vol. 49, no. 2, pp. 296-308, 2012.