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Symbolic computation of exactsolutions for the compound KdV-Sawada–Kotera equationJiao Zhang a , Xiaoli Wei a & Jingchen Hou aa College of Science , Liaoning University of Petroleum andChemical Technology , Liaoning, People's Republic of ChinaPublished online: 22 Aug 2008.

To cite this article: Jiao Zhang , Xiaoli Wei & Jingchen Hou (2010) Symbolic computation of exactsolutions for the compound KdV-Sawada–Kotera equation, International Journal of ComputerMathematics, 87:1, 94-102, DOI: 10.1080/00207160801965289

To link to this article: http://dx.doi.org/10.1080/00207160801965289

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International Journal of Computer MathematicsVol. 87, No. 1, January 2010, 94–102

Symbolic computation of exact solutions for the compoundKdV-Sawada–Kotera equation

Jiao Zhang*, Xiaoli Wei and Jingchen Hou

College of Science, Liaoning University of Petroleum and Chemical Technology, Liaoning,People’s Republic of China

(Received 14 August 2007; revised version received 13 January 2008; accepted 27 January 2008)

The generalized F-expansion method is applied to construct the exact solutions of the compound KdV-Sawada–Kotera equation by the aid of the symbolic computation system Maple. Some new exact solutionswhich include Jacobi elliptic function solutions, soliton solutions and triangular periodic solutions areobtained via this method.

Keywords: exact solutions; compound KdV-Sawada–Kotera equation; generalized F-expansion method;soliton solution; symbolic computation

PACS Classification: 02.30Jr; 02.30Ik; 05.45Yv; 02.30Mv; 02.60Lj

1. Introduction

The study of exact solutions of nonlinear evolution equations plays an important role in solitontheory. Travelling wave solutions to nonlinear partial differential equations (NLPDEs) play anessential role in the nonlinear science, especially they may provide much physical informationand more inside to the physical aspect of the problem and help one to understand the mechanismthat governs these physical models or to better provide knowledge of the physical problem andthus lead to further applications. For this end, various methods have been developed, such as theinverse scattering transform [3], the Darboux transformation [11], Bäcklund transformation [13],the Hirota method [9], theWronskian technique [6,7], homogeneous balance method [5], truncatedPainlevé expansion method [14,15], symmetry method [12], F-expansion method [1,2,17], Jacobielliptic function method [10] and tan h-function method [4], and so on.

In reference [8], Hirota and Ito consider Sawada–Kotera equation

ut + b(15u3 + 15uuxx + uxxxx)x = 0, (1)

with a non-vanishing boundary condition

u|x=±∞ = c,

*Corresponding author. Email: zhangjiaofc@163.com

ISSN 0020-7160 print/ISSN 1029-0265 online© 2010 Taylor & FrancisDOI: 10.1080/00207160801965289http://www.informaworld.com

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where c is a constant. Let’s replace u by u + a/15b, and use a Galilei transformation to removeux , then Equation (1) becomes the KdV-Sawada–Kotera equation

ut + a(3u2 + uxx)x + b(15u3 + 15uuxx + uxxxx)x = 0, (2)

which reduces to the KdV equation for b = 0 and to the Sawada–Kotera equation for a = 0.In this paper, the generalized F-expansion method is applied to the compound KdV-Sawada–

Kotera equation. As a result, many new explicit exact solutions are obtained including Jacobielliptic function solutions, soliton solutions, trigonometric function solutions.

2. The generalized F-expansion method based on the symbolic computation

For a given NLPDE with independent variable x = (t, x1, x2, . . . , xm) and dependent variableuj (x),

Fj (uj , ujt , ujx1 , ujx2 , . . . , ujxn, ujtt , ujx1x1 , ujx1t , ujx2x2 , . . .) = 0, (3)

Equation (2) can be turned to an ODE by the travelling wave transformation uj (xi, t) = uj (ξ),ξ = k(x1 + l2x2 + · · · + lj xj + · · · + vt), where k is the wave number and λ is the wave speed,

Nj(uj , u′j , u

′′j , . . .) = 0, (4)

we assume that Equation (4) has solutions in the forms

uj (ξ) = aj0 +n∑

i=1

(ajiFi(ξ) + bjiF

−i (ξ ) + cjiFi−1(ξ)F ′(ξ) + djiF

−i (ξ )F ′(ξ)), (5)

where aj0 = aj0(x), aji = aji(x), bji = bji(x), cji = cji(x), dji = dji(x), (j = 1, 2, . . . , m),and F(ξ), F ′(ξ) satisfy

F ′2(ξ) = PF 4(ξ) + QF 2(ξ) + R, (6)

further, we can have

F ′′(ξ) = 2PF 3(ξ) + QF(ξ),

F ′′′(ξ) = (6PF 2(ξ) + Q)F ′(ξ), (7)

· · · · · · ,

where P , Q and R are all parameters and the prime denotes d/dξ . Given different values of P ,Q and R, the different Jacobi elliptic function solutions F(ξ) can be obtained from Equation (6)(see Appendix A).

In order to make better use of the generalized F-expansion method, we take its main steps asfollows:

Step 1 Determine the integer n. Balancing the nonlinear term and the highest order derivativeterm in Equation (4), we can obtain the algebraic equation concerning n.

Step 2 Derive a system of nonlinear algebraic equations. Substituting Equations (5) and (6) intoEquation (4) with the value of n obtained in Step 1. Collecting the coefficients of F j (ξ)F ′p(ξ)(p =0, 1; j = 0, ±1, ±2, . . .), then setting each coefficient to zero, we can get a set of over-determinednon-linear algebraic equations for aj0, aji, bji, cji and dji(i = 1, 2, . . . , n).

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Step 3 Solve the systems of the algebraic equations. By using the Wu elimination method [16]with the aid of symbolic computation packages Maple, we can obtain the values of aj0, aji , bji ,cji , dji , k, lp and v, where p = 2, 3, . . . , m.

Step 4 Obtain the exact travelling wave solutions. Select P , Q, R and F(ξ) from Appendix Aand substitute them along with aj0, aji , bji , cji , dji , k, lp and υ into Equation (5) to obtain Jacobielliptic function solutions of Equation (3), from which exact soliton solutions and trigonometricfunction solutions can be obtained in the limit cases when m → 1 and m → 0 (see Appendix B).

3. The exact travelling wave solutions

We firstly make the following travelling wave transformation

ξ = kx + vt, (8)

where k and v are all constants to be determined.Substituting Equations (8) into (2), we get

vu′(ξ) + 6aku(ξ)u′(ξ) + ak3u(3)(ξ) + 45bku2(ξ)u′(ξ) + 15bk3u′(ξ)u(2)(ξ)

+ 15bk3u(ξ)u(3)(ξ) + bk5u(5)(ξ) = 0 (9)

considering the homogeneous balance between u(ξ)u(3)(ξ) and u(5)(ξ) in Equation (9), yieldsn = 2, we suppose that the solution of Equation (9) can be expressed by

u(ξ) = a0 + a1F(ξ) + b1

F(ξ)+ c1F

′(ξ) + d1F′(ξ)

F (ξ)+ a2F

2(ξ) + b2

F 2(ξ)

+ c2F(ξ)F ′(ξ) + d2F′(ξ)

F 2(ξ), (10)

where a0, a1, b1, c1, d1 a2, b2, c2, d2 are constants to be determined later, F(ξ) and F ′(ξ) satisfyEquations (6) and (7).

Substituting Equation (10) along with Equations (6) and (7) into Equation (9), the left-handside of Equation (9) is converted into a polynomial ofF j (ξ)F ′p(ξ)(p = 0, 1; j = 0, ±1, ±2, . . .),then setting each coefficient to zero, we get a set of over-determined algebraic system for a0, a1,b1, c1, d1 a2, b2, c2, d2, k and v. Solving the system of over-determined algebraic equations usingWu-elimination method [16] and Maple, we get the following solutions:

Case 1

c1 = c2 = b2 = d1 = d2 = b1 = a1 = 0, a2 = −2Pk2,

v = −45a20bk − 12bPk5R − 16bk5Q2 − 4ak3Q − 6a0ak − 60a0bk3Q. (11)

Case 2

c1 = c2 = b2 = d1 = d2 = b1 = a1 = 0, a2 = −4Pk2,

a0 = −a + 20k2bQ

15b, v = k(a2 − 80b2k4Q2 + 240b2k4PR)

5b. (12)

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Case 3

c1 = c2 = a1 = a2 = d1 = d2 = b1 = 0, b2 = −2Rk2,

v = −45a20bk − 12bPk5R − 16bk5Q2 − 4ak3Q − 6a0ak − 60a0bk3Q. (13)

Case 4

c1 = c2 = d1 = d2 = b1 = a1 = a2 = 0, a0 = −a + 20k2bQ

15b,

b2 = −4Rk2, v = k(a2 − 80b2k4Q2 + 240b2k4PR)

5b. (14)

Case 5

c1 = c2 = d1 = d2 = b1 = a1 = 0, a2 = −2Pk2, b2 = −2Rk2,

v = −45a20bk + 48bPk5R − 16bk5Q2 − 4ak3Q − 6a0ak − 60a0bk3Q. (15)

Case 6

c1 = c2 = d1 = d2 = b1 = a1 = 0, a0 = −a + 20k2bQ

15b, a2 = −4Pk2,

b2 = −4Rk2, v = −k(−a2 + 80b2k4Q2 + 960b2k4PR)

5b. (16)

Case 7

a1 = b1 = d1 = b2 = c2 = d2 = 0, a0 = a − 5k2bQ

15b, a2 = −2k2P,

c1 = 2k2√

P , v = −k(60Pk4b2R + 5Qk4b2 − a2)

5b. (17)

Substituting Equations (11) to (16), into Equation (10), respectively, we obtain the followingnew exact travelling wave solutions

Family 1

u = a0 − 2Pk2F 2(ξ). (18)

Family 2

u = −a + 20k2bQ + 60bk2PF 2(ξ)

15b. (19)

Family 3

u = a0 − 2Rk2

F 2(ξ). (20)

Family 4

u = − a

15b− 4k2Q

3− 4k2R

F 2(ξ). (21)

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Family 5

u = a0 − 2Pk2F 2(ξ) − 2k2R

F 2(ξ). (22)

Family 6

u = − a

15b− 4k2Q

3− 4Pk2F 2(ξ) − 4k2R

F 2(ξ). (23)

Family 7

u = −5k2bQ + a − 30k2b√

PF ′(ξ) + 30k2bPF 2(ξ)

15b. (24)

We would like to select Family 1, Family 3, Family 5 and Family 7 to obtain new and generalexact solutions to the compound KdV-Sawada–Kotera equation in the following.

According to Appendix A, if we select

P = 1

4, Q = 1

2− m2, R = 1

4, F (ξ) = ns(ξ) + cs(ξ), (25)

from Family 1, we obtain combined non-degenerate Jacobi elliptic function solution of thecompound KdV-Sawada–Kotera equation

u = a0 − 1

2k2(ns(ξ) + cs(ξ))2, (26)

when m → 1, Equation (26) admits to the new solitary wave solution

u = a0 − 1

2k2(coth(ξ) + csch(ξ))2, (27)

when m → 0, from Equation (26), we obtain the trigonometric function solution

u = a0 − 1

2k2(csc(ξ) + cot(ξ))2. (28)

From Family 3, we obtain

u = −1

2

(−2a0ns2(ξ) − 4a0ns(ξ)cs(ξ) − 2a0cs2(ξ) + k2)

(ns(ξ) + cs(ξ))2, (29)

when m → 1, we get the new solitary wave solution

u = 4a0 + 4a0sech(ξ) − (2a0 + k2) tanh2(ξ)

2(1 + sech(ξ))2, (30)

as long as m → 0, Equation (29) admits to the trigonometric function solution

u = 4a0 + 4a0 cos(ξ) − (2a0 + k2) sin2(ξ)

2(1 + cos(ξ))2. (31)

From Equations (25) and (23), we can obtain

u = − 1

15

(15bk2ns4(ξ) + 60bk2cs(ξ)ns3(ξ)

+(a + 10bk2 − 20bm2k2 + 90bk2cs2(ξ))ns2(ξ) + (60bk2cs3(ξ)

+(2a + 20bk2 − 40bm2k2)cs(ξ))ns(ξ) + 15bk2cs4(ξ)

+(a + 10bk2 − 20bm2k2)cs2(ξ) + 15bk2)

(b(ns(ξ) + cs(ξ))2). (32)

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Equations (25) and (24) yield

u = ((a − 5bk2 − 5bm2k2)sn2(ξ) + 15bk2(1 + cn(ξ)dn(ξ) + dn(ξ) + cn(ξ)))

(−15bsn2(ξ)). (33)

From Appendix A, choosing

P = 1 − m2

4, Q = 1 + m2

2, R = 1 − m2

2, F (ξ) = nc(ξ) + sc(ξ), (34)

inserting Equation (34) into Equation (18), we obtain

u = a0 +(

−2k2 + 1

2m2k2

)(sc2(ξ) + nc2(ξ)) + (−4k2 + m2k2)nc(ξ)sc(ξ). (35)

In the limit case when m → 1, from Equation (35) we obtain solitary wave solution in the followingform:

u = a0 − 3

2k2(cosh(ξ) + sinh(ξ))2, (36)

when m → 0, from Equation (35) we obtain the trigonometric function solution

u = a0 − 2k2(sec(ξ) + tan(ξ))2. (37)

Inserting Equation (34) into Equation (20), we obtain combined Jacobi elliptic function solution

u = 1

2

(2a0nc2(ξ) + 4a0nc(ξ)sc(ξ) + 2a0sc2(ξ) + m2k2 − k2)

(nc(ξ) + sc(ξ))2, (38)

when m → 0, we get

u = 2a0 − k2 + (2a0 + k2) sin(ξ)

2(1 + sin(ξ)). (39)

Substituting Equation (34) into Equation (24), we obtain

u =(−2a + 10k2bm2 − 65k2b + (2a − 55k2b + 20k2bm2)sn2(ξ)

+30k2b(m2 − 4)sn(ξ) + 60k2b√

4 − m2dn(ξ))

(30bcn2(ξ)). (40)

If we choose

P = m2

4, Q = m2 − 2

2, R = m2

4, F (ξ) = sn(ξ) + ics(ξ), (41)

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substituting Equation (41) into Equation (18), we have

u = a0 − 1

2m2k2sn2ξ − im2k2sn(ξ)cn(ξ) + 1

2m2k2cn2(ξ), (42)

when m tends to 1, we obtain the following new solitary wave solution

u = a0 − 1

2k2(tanh(ξ) + isech(ξ)

)2. (43)

From Equations (41) and (20), we have

u = −1

2

(−2a0sn2(ξ) − 4ia0sn(ξ)cn(ξ) + 2a0cn

2(ξ) + m2k2)

(sn(ξ) + icn(ξ))2, (44)

as long as m → 1, Equation (44) admits to the new solution

u = −2a0 − k2 + 4a0tanh2(ξ) + 4ia0tanh(ξ)sech(ξ)

2(tanh(ξ) + isech(ξ))2, (45)

by means of Equations (41) and (24), we get

u =2a + 5k2b(m2 − 2 − 6mcn(ξ)dn(ξ) + 6im dn(ξ)sn(ξ) + 3m2sn2(ξ)

+6im2sn(ξ)cn(ξ) − 3m2cn2(ξ))

−30b, (46)

for m → 1, Equation (46) admits to the new exact solution

u = 5k2b − 2a

30b− 1

2k2(tanh2(ξ) + 4isech(ξ) tanh(ξ) + 3sech2(ξ)). (47)

It is worth noting that we can obtain other Jacobi elliptic function solutions, solitary wave solutionsand trigonometric function solutions of the compound KdV-Sawada–Kotera equation with the aidof Appendix A and Appendix B choosing different Jacobi elliptic functions, but we omit themhere for simplicity.

4. Conclusion

In summary, the generalized F-expansion method for seeking more types of exact travellingwave solutions of the compound KdV-Sawada–Kotera equation is implemented. By using thisscheme, rich new families of exact Jacobi elliptic function solutions. When the modulus m → 1and m → 0 some of the obtained solutions degenerate as solitary wave solutions and triangularperiodic solutions. The method proposed in this paper can also be extended to solve some nonlinearevolution equations with variable coefficients in physics and it can be generalized further byintroducing new generalized ansatz equation. This is our task in the future.

Acknowledgements

The authors would like to express their sincere thanks to the referees for their helpful suggestion. This work is supportedby the Natural Science Foundation of Educational Committee of Liaoning Province of China.

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Appendix A

Relations between values of (P, Q, R) and corresponding F(ξ) in Equation (6)

P Q R F(ξ)

m2 −(1 + m2) 1 sn(ξ), cd(ξ)

−m2 2m2 − 1 1 − m2 cn(ξ)

−1 2 − m2 m2 − 1 dn(ξ)

1 −(1 − m2) m2 ns(ξ), dc(ξ)

1 − m2 2m2 − 1 −m2 nc(ξ)

m2 − 1 2 − m2 −1 nd(ξ)

1 − m2 2 − m2 1 sc(ξ)

−m2(1 − m2) 2m2 − 1 1 sd(ξ)

1 2 − m2 1 − m2 cs(ξ)

1 2m2 − 1 −m2(1 − m2) ds(ξ)

1

4

1 − 2m2

2

1

4ns(ξ) ± cs(ξ)

1 − m2

4

1 + m2

2

1 − m2

2nc(ξ) ± sc(ξ)

1

4

m2 − 2

2

m2

4ns(ξ) ± ds(ξ)

m2

4

m2 − 2

2

m2

4sn(ξ) ± ics(ξ)

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Appendix B

Jacobi elliptic functions degenerate as hyperbolic functions when m → 1 and m → 0

Jacobi ellipticfunctions m → 1 m → 0

sn(ξ) tanh(ξ) sin(ξ)

cn(ξ) sech(ξ) cos(ξ)

dn(ξ) sech(ξ) 1sc(ξ) sinh(ξ) tan(ξ)

sd(ξ) sinh(ξ) sin(ξ)

cd(ξ) 1 cos(ξ)

ns(ξ) coth(ξ) csc(ξ)

nc(ξ) cosh(ξ) sec(ξ)

nd(ξ) cosh(ξ) 1cs(ξ) csch(ξ) cot(ξ)

ds(ξ) csch(ξ) csc(ξ)

dc(ξ) 1 sec(ξ)

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