F-Expansion method for the nonlinear generalized Ito System

F-Expansion method for the nonlinear

generalized Ito System

Eman Salem A. Alaidarous

Department of Mathematics, Faculty of Science, King Abdul Aziz University, P O Box 80203,

Jeddah 21589, Saudi Arabia.

Abstract

In this work, we derive new exact soliton solutions of the generalized Ito system. We will use the improved F-expansion method to explore periodic wave solutions expressed by various Jacobi elliptic functions, Weierstrass elliptic function, triangle functions, hyperbolic functions and other type of functions. That we can obtain a great variety of classes of solutions of generalized Ito system.

Introduction:

In recent years, the nonlinear partial deferential equations (NPDEs) are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry and biology, etc. With the development of soliton theory, many powerful methods have been presented, such as inverse scattering transform method [1], Hirota.s bilinear method [2,3], B¯acklund transform method [4], homogeneous balance method [5,6], truncated Painleve expansion method [7,8], similarity reduction method [9], hyperbolic function method [10],the tanh method and its extension [11-13], the sine.cosine method [14], the homotopy perturbation method [15,16], the adomian method [17], the variational approach [18-19], the algebraic method [20], the Jacobi elliptic function expansion method [21],and so on. F-expansion method is powerful to solve nonlinear partial deferential equations (NPDEs) and can help to get many new exact solutions which we have never seen before. Within our knowledge. In this work , we will apply the extended F-expansion method to explore the exact solutions for the nonlinear generalized Ito system

ut  vx,

vt  2vxxx6uvx 12wwx 6px,

wt  wxxx 3uwx,

pt  pxxx3upx.

1. 1

The paper is organized as follows:

In the next section, we will use improved F-expansion method to explore the

periodic wave solutions expressed by various Jacobi elliptic functions,

Weierstrass elliptic function, triangle functions, hyperbolic functions and

other type of functions for the nonlinear generalized Ito system. At last, some

conclusions are made.

Exact solutions for the nonlinear generalized Ito system:

Let

(2.1)

where k,λ are constants that would be determined later, and ξ0 is an arbitrary constant.

Substituting (2.1) into Eq. (1.1) yields ODEs for

(2.2)

It is easy to see that the rank of every term in Eq.(2.2) is even after integrating

once. According to the idea of the improved F-expansion method, we assume

that

can be expressed as follows:

Case 1:

_{Balancing the highest -order linear term with non-linear terms in Eq.(2.2), we get the}

following:

u  a0 a1Fξa2F2ξ,

v  b0 b1Fξb2F2ξ,

w  c0 c1Fξc2F2ξ,

p  e0 e1Fξe2F2ξ. _(2.3) 2.3

Where a0,a1,a2,b0,b1,b2,c0,c1,c2,e0,e1,e2 are constants to be determined later, Fξ

is a solution of the ODE

F2ξ  PF4 _QF2 _{R, P  0, 2.4}

λa2 kb2  0,

λa1 kb1  0,

2Pk3_b

2 ka2b2 kc2 2 _{ 0,}

2Pk3_b

1 3ka1b2 3ka2b1 6kc1c2  0,

2λb2 16Qk3b2 12ka1b1 12ka2b0 12ka0b2 24kc0c2 12kc12 12ke2  0,

λb1 2Qk3b1 6ka1b0 6ka0b1 12kc0c1 6ke1  0,

4Pk3_c

2 ka2c2  0,

2Pk3_c

1 2ka1c2 ka2c1  0,

2λc2 8Qk3c2 6ka0c2 3ka1c1  0,

λc1 Qk3c1 3ka0c1  0,

4Pk3_e

2 ka2e2  0,

2Pk3_e

1 2ka1e2 ka2e1  0,

2λe2 8Qk3e2 6ka0e2 3ka1e1  0,

λe1 Qk3e1 3ka0e1  0. 2.5

(2.5)

When solving the system of algebraic equations that is obtained above, we can get the solutions:

c2  e2  a1  b1  0,c0  const. ,e0  const. ,a0  const. ,c1  const. ,

b2  2Pk4Q6k2Pa0,e1  2c0c1,a2  2k2P,

b0  

3k6_Q2_P12Pk4_Qa

0 9Pk2a02 c12

2k2_P ,λ  k

3_Q3ka

0. 2.6

(2.6) Substituting 2. 6 into 2. 3, the general traveling wave solutions for Eq. 1. 1 can be obtained as follows:

u  a0 2k2PF2ξ,

v  3k

6_Q2_P12Pk4_Qa

0 9Pk2a0 2 _c

1 2

2k2_P 2k

4_PQ6k2_Pa

0F2ξ,

w  c0 c1Fξ,

p  e0 2c0c1Fξ,

ξ  kxk3_Q3ka

0tξ0. _(2.7) 2.7

Where k is an arbitrary constant, Fξ is a solution of ODE 2. 2 .

We can obtain the exact solutions expressed by Jacobi elliptic functions for Eq.1. 1 . When P  m2

u1  a0 2k2m2sn2ξ,

v1  

3k6_m2_1m2_2 _12m2_k4_1m2_a

0 9m2k2a0 2 _c

1 2

2k2_m2

2k4_m2_1m2_6k2_m2_a

0sn2ξ,

w1  c0 c1snξ,

p1  e0 2c0c1snξ. _(2.8) 2.8

When P  m2 Q 2m2 1 R  1m2

u2  a0 2k2m2cn2ξ,

v2  

3k6_2m2 _1m2 _12m2_k4_2m2 _1a

0 9m2k2a0 2 _c

1 2

2k2_m2

2k4_m2_2m2 _16k2_m2_a

0cn2ξ,

w2  c0 c1cnξ,

p2  e0 2c0c1cnξ. _(2.9) 2.9

When P  1 Q  2m2

R  m2 1

u3  a0 2k2dn2ξ,

v3  

3k6_2m2_2 _12k4_2m2_a

0 9k2a0 2 _c

1 2

2k2

2k4_2m2_6k2_a

0dn2ξ,

w3  c0 c1dnξ,

p3  e0 2c0c1dnξ. _(2.10) 2.10

When P  1 Q  1m2_

R  m2

u4  a0 2k2 ns2ξ,

v4  

3k6_1m2_2 _12k4_1m2_a

0 9k2a0 2 _c

1 2

2k2

2k4_1m2_6k2_a

0ns2ξ,

w4  c0 c1nsξ,

p4  e0 2c0c1nsξ. _(2.11) 2.11

u5  a0 2k21m2nc2ξ,

v5  

3k6_2m2 _12_1m2_{121m}2_k4_2m2 _1a

0 91m2k2a02 c12

2k2_P

2k4_1m2_2m2 _16k2_1m2_a

0nc2ξ,

w5  c0 c1ncξ,

p5  e0 2c0c1ncξ. 2.12

(2.12) When P  m2 1 Q  2m2

R  1

u6  a0 2k2m2 1nd2ξ,

v6  

3k6_2m2_2_m2 _{112m}2 _1k4_2m2_a

0 9m2 1k2a0 2 _c

1 2

2k2_m2 _1

2k4_m2 _{12m}2_6k2_m2 _1a

0nd2ξ,

w6  c0 c1ndξ,

p6  e0 2c0c1ndξ. 2.13

(2.13) When P 1m2 Q  2m2 R  1

u7  a0 2k21m2sc2ξ,

v7  

3k6_2m2_2_1m2_{121m}2_k4_2m2_a

0 91m2k2a0 2 _c

1 2

2k2_1m2_

2k4_1m2_2m2_6k2_1m2_a

0sc2ξ,

w7  c0 c1scξ,

p7  e0 2c0c1scξ. 2.14

(2.14) When P  m21m2 Q 2m2 1 R  1

u8  a0 2m21m2k2sd2ξ,

v8  

3k6_2m2 _12_m2_1m2_12m2_1m2_k4_2m2 _1a

0 9m21m2k2a02 c12

2k2_m2_1m2_

2k4_m2_1m2_2m2 _16k2_m2_1m2_a

0sd2ξ,

w8  c0 c1sdξ,

p8  e0 2c0c1sdξ. 2.15

(2.15) When P  1 Q  2m2

u9  a0 2k2 cs2ξ,

v9  

3k6_2m2_2 _12k4_2m2_a

0 9k2a0 2 _c

1 2

2k2

2k4_2m2_6k2_a

0cs2ξ,

w9  c0 c1csξ,

p9  e0 2c0c1csξ. _(2.16) 2.16

When P  1 Q  2m2 _{1 R  m}2_1m2_

u10  a0 2k2 ds2ξ,

v10  

3k6_2m2 _12 _12k4_2m2 _1a

0 9k2a0 2 _c

1 2

2k2

2k4_2m2 _16k2_a

0ds2ξ,

w10  c0 c1dsξ,

p10  e0 2c0c1dsξ. _(2.17) 2.17

where k is arbitrary constants. It is worthwhile to note that nsξ  snξ1_{,ncξ  cnξ}1_{,ndξ  dnξ}1_{,scξ } snξ

cnξ, sdξ 

snξ dnξ, csξ  cnξ

snξ,dsξ 

dnξ

snξ. Now ,when

c0  const. ,e0  const. ,b0  const. ,c2  const. ,c1  e1  a1  b1  0,

a0  

32k6_P2_Qc 2 2

24k4_P2 ,a2  4k 2_P,e

2  

c₂4 _128b

0k8P4 64c0c2P3k6

32P3_k6 ,

b2 

c₂2

2k2_P,λ  

c₂2

8k3_P2. 2.18

(2.18) Substituting 2. 6 into 2. 3, the general traveling wave solutions for Eq. 1. 1 can be obtained as follows

u  32k

6_P2_Qc 2 2

24k4_P2 4k

2_PF2_ξ,

v  b0 

c₂2

2k2_PF 2_ξ,

w  c0 c2F2ξ,

p  e0 

c₂4 _128b

0k8P4 64c0c2P3k6

32P3_k6 F

2_ξ,

ξ  kx c2 2

8k3_P2tξ0. 2.19

Where k is an arbitrary constant, Fξ is a solution of ODE 2. 2 .

Now, We can obtain the exact solutions expressed by Jacobi elliptic functions for Eq.1. 1 .

When P  m2 Q  1m2 R  1

u11  

32k6_m4_1m2_c 2 2

24k4_m4 4k

2_m2_sn2_ξ,

v11  b0 

c₂2 2k2_m2sn

2_ξ,

w11  c0 c1sn2ξ,

p11  e0 

c₂4 _128b

0k8m8 64c0c2m6k6

32m6_k6 sn

2_{ξ. 2.20}

(2.20) When P  m2 Q 2m2 1 R  1m2

u12  

32k6_m4_2m2 _1c 2 2

24k4_m4 4k

2_m2_cn2_ξ,

v12  b0 

c₂2

2k2_m2cn 2_ξ,

w12  c0 c1cn2ξ,

p12  e0 

c₂4 _128b

0k8m8 64c0c2m6k6

32m6_k6 cn

2_{ξ. 2.21}

(2.21) When P  1 Q  2m2

R  m2 1

u13  

32k6_2m2_c 2 2

24k4 4k

2_dn2_ξ,

v13  b0 

c₂2 2k2dn

2_ξ,

w13  c0 c1dn2ξ,

p13  e0 

c₂4 _128b

0k8 64c0c2k6

32k6 dn

2_{ξ. 2.22}

(2.22) When P  1 Q  1m2_

R  m2

u14 

32k6_1m2_c 2 2

24k4 4k

2_ns2_ξ,

v14 b0

c₂2

2k2 ns 2_ξ,

w14 c0c1ns2ξ,

p14 e0

c₂4 _128b

0k864c0c2k6

32k6 ns

2_{ξ. 2.23}

When P  1m2

Q  2m2 1 R  m2

u15  

32k6_1m2_2_2m2 _1c 2 2

24k4_1m2_2 4k

2_1m2_nc2_ξ,

v15  b0 

c₂2

2k2_1m2_nc 2_ξ,

w15  c0 c1nc2ξ,

p15  e0 

c₂4 _128b

0k81m24 64c0c21m23k6

321m2_3_k6 nc

2_{ξ. 2.24}

(2.24) When P  m2 _{1 Q}  ₂_m2

R  1

u16  

32k6_m2 _12_2m2_c 2 2

24k4_m2 _12 4k

2_m2 _1nd2_ξ,

v16  b0 

c₂2

2k2_m2 _1nd 2_ξ,

w16  c0 c1nd2ξ,

p16  e0 

c₂4 _128b

0k8 m2 14 64c0c₂3m2 13k6

32m2 _13_k6 nd

2_{ξ. 2.25}

(2.25) When P  1m2

Q  2m2 R  1

u17  

32k6_1m2_2_2m2_c 2 2

24k4_1m2_2 4k

2_1m2_sc2_ξ,

v17  b0 

c₂2

2k2_1m2_sc 2_ξ,

w17  c0 c1sc2ξ,

p17  e0 

c₂4 _128b

0k81m24 64c0c21m23k6

321m2_3_k6 sc

2_{ξ. 2.26}

(2.26) When P  m21m2 Q 2m2 1 R  1

u18  

32k6_m4_1m2_2_2m2 _1c 2 2

24k4_m4_1m2_2 4k

2_m2_1m2_sd2_ξ,

v18  b0 

c₂2

2k2_m2_1m2_sd 2_ξ,

w18  c0 c1sd2ξ,

p18  e0 

c₂4 _128b

0k8m81m24 64c0c2m61m23k6

32m6_1m2_3_k6 sd

2_{ξ. 2.27}

When P  1 Q  2m2

R  1m2

u19  

32k6_2m2_c 2 2

24k4 4k

2 _cs2_ξ,

v19  b0 

c₂2

2k2cs 2_ξ,

w19  c0 c1cs2ξ,

p19  e0 

c₂4 _128b

0k8 64c0c2k6

32k6 cs

2_{ξ. 2.28}

(2.28) When P  1 Q  2m2 _{1 R  m}2_1m2_

u20  

32k6_2m2 _1c 2 2

24k4 4k

2 _ds2_ξ,

v20  b0 

c₂2

2k2ds 2_ξ,

w20  c0 c1ds2ξ,

p20  e0 

c₂4 _128b

0k8 64c0c2k6

32k6 ds

2_{ξ. 2.29}

(2.29) Where k is arbitrary constants.

When

e0  const. ,b0  const. ,c2  const. ,c0  const. ,c1  e1  a1  b1  e2  0,

a0  

32k6_P2_Qc 2 2

24k4_P2 ,a2  4k 2_P,b

2 

c₂2

2k2_P,λ  

c₂2

8k3_P2 . 2.30

Second: Substituting 2. 6 into 2. 3 , the general traveling wave solutions for Eq. 1. 1 _{can be obtained as follows:}

u 32k

6_P2_Qc 2 2

24k4_P2 4k

2_PF2_ξ,

v b0 

c₂2

2k2_PF 2_ξ,

w  c2

4 _128b 0k8P4

64k6_P3_c 2

c1F2ξ,

p  e0,

ξ  kx c2

2

8k3_P2tξ0. 2.31

We can obtain the exact solutions expressed by Jacobi elliptic functions for Eq.1. 1 . When P  m2

Q  1m2 R  1

u21  

32k6_m4_1m2_c 2 2

24k4_m4 4k

2_m2_sn2_ξ,

v21  b0 

c₂2

2k2_m2sn 2_ξ,

w21 

c₂4 _128b 0k8m8

64k6_m6_c 2

c1sn2ξ,

p21  e0. 2.32

(2.32) When P  m2 Q 2m2 1 R  1m2

u22  

32k6_m4_2m2 _1c 2 2

24k4_m4 4k

2_m2_cn2_ξ,

v22  b0 

c₂2

2k2_m2 cn 2_ξ,

w22  

c₂4 _128b 0mmk8

64k6_m6_c 2

c1cn2ξ,

p22  e0. _(2.33) 2.33

When P  1 Q 2m2

R  m2 1

u23  

32k6_2m2_c 2 2

24k4 4k

2_dn2_ξ,

v23  b0 

c₂2

2k2dn 2_ξ,

w23  

c₂4 _128b 0k8

64k6_c 2

c1dn2ξ,

p23  e0. _(2.34) 2.34

When P  1 Q  1m2_

R  m2

u24  

32k6_1m2_c 2 2

24k4 4k

2 _ns2_ξ,

v24  b0 

c₂2

2k2 ns 2_ξ,

w24 

c₂4 _128b 0k8

64k6_c 2

c1ns2ξ,

p24  e0. _(2.35) 2.35

When P  1m2

u25  

32k6_1m2_2_2m2 _1c 2 2

24k4_1m2_2 4k

2_1m2_nc2_ξ,

v25  b0 

c₂2

2k2_1m2_nc 2_ξ,

w25 

c₂4 _128b

0k81m24

64k6_1m2_3_c 2

c1nc2ξ,

p25  e0. _(2.36) 2.36

When P  m2 _1

Q  2m2

R  1

u26  

32k6_m2 _12_2m2_c 2 2

24k4_m2 _12 4k

2_m2 _1nd2_ξ,

v26  b0 

c₂2

2k2_m2 _1nd 2_ξ,

w26 

c₂4 _128b

0k8m2 14

64k6_m2 _13_c 2

c1nd2ξ,

p26  e0. _(2.37) 2.37

When P  1m2

Q  2m2 R  1

u27  

32k6_1m2_2_2m2_c 2 2

24k4_1m2_2 4k

2_1m2_sc2_ξ,

v27  b0 

c₂2

2k2_1m2_sc 2_ξ,

w27 

c₂4 _128b

0k81m24

64k6_1m2_3_c 2

c1sc2ξ,

p27  e0. _(2.38) 2.38

When P m21m2_

Q  2m2 1 R  1

u28  

32k6_m4_1m2_2_2m2 _1c 2 2

24k4_m4_1m2_2 4k

2_m2_1m2_sd2_ξ,

v28  b0 

c₂2

2k2_m2_1m2_sd 2_ξ,

w28 

c₂4 _128b

0k8m81m24

64k6_m6_1m2_3_c 2

c1sd2ξ,

p28  e0. _(2.39) 2.39

When P  1 Q  2m2

R  1m2

u29  

32k6_2m2_c 2 2

24k4 4k

2 _cs2_ξ,

v29  b0 

c₂2

2k2cs 2_ξ,

w29 

c₂4 _128b 0k8

64k6_c 2

c1cs2ξ,

p29  e0. _(2.40) 2.40

When P  1 Q 2m2 1 R  m2_1m2_

u30  

32k6_2m2 _1c 2 2

24k4 4k

2 _ds2_ξ,

v30  b0 

c₂2

2k2ds 2_ξ,

w30 

c₂4 _128b 0k8

64k6_c 2

c1ds2ξ,

p30  e0. _(2.41) 2.41

where k is arbitrary constants.

When

c0  const. ,e0  const. ,a0  const. ,c1  c2  e1  e2  a1  b1  0,

a2  2k2P,b0  

λλ8k3_Q6ka 0

6k2 ,b2  2λkP. 2.42

(2.42)

Substituting 2. 6 into 2. 3 , the general traveling wave solutions for Eq. 1. 1 can be obtained as follows:

u  a0 2k2PF2ξ,

v  λλ8k

3_Q6ka 0

6k2 2λkPF

2_ξ,

w  c0,

p  e0. _(2.43) 2.43

Where k,λ are arbitrary constants, Fξ is a solution of ODE 2. 2 .

We can obtain the exact solutions expressed by Jacobi elliptic functions for Eq.1. 1 . When P  m2

u31  a0 2k2m2sn2ξ,

v31  

λλ_8k3_1m2_6ka 0

6k2 2λkm

2_sn2_ξ,

w31  c0,

p31  e0. _(2.44) 2.44

When P  m2

Q  2m2 1 R  1m2

u32  a0 2k2m2cn2ξ,

v32  

λλ8k3_2m2 _{16ka} 0

6k2 2λkm

2_cn2_ξ,

w32  c0,

p32  e0. _(2.45) 2.45

When P  1 Q 2m2

R  m2 1

u33  a0 2k2dn2ξ,

v33  

λλ8k3_2m2_6ka 0

6k2 2λkdn

2_ξ,

w33  c0,

p33  e0. _(2.46) 2.46

When P  1 Q  1m2_

R  m2

u34  a0 2k2 ns2ξ,

v34  

λλ8k3_1m2_6ka 0

6k2 2λk ns

2_ξ,

w34  c0,

p34  e0. _(2.47) 2.47

When P  1m2

Q  2m2 1 R  m2

u35  a0 2k21m2nc2ξ,

v35  

λλ8k3_2m2 _{16ka} 0

6k2 2λk1m

2_nc2_ξ,

w35  c0,

p35  e0. _(2.48) 2.48

When P  m2 _1

Q  2m2

u36  a0 2k2m2 1nd2ξ,

v36  

λλ8k3_2m2_6ka 0

6k2 2λkm

2 _1nd2_ξ,

w36  c0,

p36  e0. _(2.49) 2.49

When P  1m2

Q  2m2

R  1 u37  a0 2k21m2sc2ξ,

v37  

λλ8k3_2m2_6ka 0

6k2 2λk1m

2_sc2_ξ,

w37  c0,

p37  e0. _(2.50) 2.50

When P m21m2 Q  2m2 1 R  1

u38  a0 2k2m21m2sd2ξ,

v38  

λλ8k3_2m2 _{16ka} 0

6k2 2λkm

2_1m2_sd2_ξ,

w38  c0,

p38  e0. _(2.51) 2.51

When P  1 Q  2m2

R  1m2

u39  a0 2k2cs2ξ,

v39  

λλ8k3_2m2_6ka 0

6k2 2λkcs

2_ξ,

w39  c0,

p39  e0. _(2.52) 2.52

When P  1 Q 2m2 _1

R  m21m2

u40  a0 2k2 ds2ξ,

v40  

λλ8k3_2m2 _{16ka} 0

6k2 2λkds

2_ξ,

w40  c0,

p40  e0. _(2.35) 2.53

When m 1 , the Jacobic functions degenerate to the hyperbolic functions , i.e.

snξ  tanhξ,cnξ  sechξ,dnξ  sechξ. .

snξ  sinξ,cnξ  cosξ,dnξ  1

When R  0 in Eq. 2. 5 _{, we can obtain the exact solutions that can be expressed as} follows:

Fξ  csch Qξ iff Q  0,P  0,

Fξ  sech Qξ iff Q  0,P  0,

Fξ  csch Qξ iff Q  0,P  0.

Case2:

According to the idea of the improved F-expansion method, we can assume that uξ,vξ,wξ are of the form as follows:

u  a0 a1Fξa2F2ξ,

v  b0 b1Fξb2F2ξ,

w  c0 c1Fξc2F2ξ,

p  e0 e1Fξ. _(3.1) 3.1

Where a0,a1,a2,b0,b1,b2,c0,c1,c2,e0,e1 are constants to be determined later, Fξ is

a solution of ODE 2. 4. , by inserting 3. 1 into Eqs. 2. 2 , and considering Eq. 2. 4 simultaneously, The left-hand sides of the Eqs.2. 2 become polynomials in Fξ, if canceling F , setting the coefficients of the polynomial to zero yields a set of algebraic equations

λa2 kb2  0,

λa1 kb1  0,

2Pk3_b

2 ka2b2 kc22  0,

2Pk3_b

1 3ka1b2 3ka2b1 6kc1c2  0,

2λb2 16k3b2Q12ka1b1 12ka2b0 12ka0b2 24kc0c2 12kc12  0,

λb1 2k3b1Q6ka1b0 6ka0b1 12kc0c1 6e1k  0,

4Pk3_c

2 ka2c2  0,

2Pk3_c

1 2ka1c2 ka2c1  0,

2λc2 8k3c2Q6ka0c2 3ka1c1  0,

λc1 k3c1Q3ka0c1  0,

2e1Pk3 e1ka2  0,

3e1ka1  0,

λe1 e1k3Q3e1ka0  0. 3.2

Solving the algebraic equations obtained above, we can have the solutions:

c1  const. ,a0  const. ,e0  const. ,λ  k3Q3ka0,a1  b1  c0  c2  e1  0,

a2  2Pk2,b2  2Pk4Q6Pk2a0,b0  

3Pk6_Q2 _12Pk4_Qa

0 9Pk2a₀2 c₁2

2Pk2 3.3

(3.3)

Substituting (3.3) into (3.1), the traveling wave solutions for Eq. (1.1) are derived as follows:

u a0 2Pk2F2ξ,

v 3Pk

6_Q2 _12Pk4_Qa

0 9Pk2a₀2 c₁2

2Pk2 2Pk

4_Q6Pk2_a

0F2ξ,

w  c1Fξ,

p  e0,

ξ  kxk3_Q3ka

0tξ0. _(3.4) 3.4

Where k is an arbitrary constant, Fξ is a solution of ODE 2. 2 . Similar to Case1 , we can obtain the other solutions to Eq. 1. 1 in terms of various Jacobi elliptic functions, triangle functions, hyperbolic functions and other type of functions, but we omit them here for simplicity.

Case 3:

If Fξ is a solution for other type of elliptic

F2ξ  AF3 _BF2 _{CFD , A  0, 4.1}

(4.1) where A,B,C,D are real constants, assume that uξ can be expressed by a finite power series of Fξ

uiξ 



djFjξ, 4.2

(4.2) where i 1, 2, 3, 4 define the independent variables ,integer N in 4. 2 can be determined by considering homogeneous balance between the nonlinear terms with the highest-order derivatives of uξ in Eq. 2. 2 which admits the following:

u  a0 a1Fξ,

v  b0 b1Fξ,

w  c0 c1Fξ,

p  e0 e1Fξ. _(4.3) 4.3

solution of ODE 4. 1 Eq. 4. 1 , D  0 that admits periodic wave solution expressed by Jacobi elliptic function F  sn2ξ,m , as C  4,B 4m2 1,A 4m2 , and periodic wave solution expressed by Jacobi elliptic function F  cn2ξ,m , as

C  41m2_{,B  42m}2 _{1,A 4m}2

and admits periodic wave solution expressed by Jacobi elliptic function F  dn2ξ,m , as C  41m2,B42m2,A 4 . Inserting 4. 3 into Eq. 2. 2 , and considering Eq. 4. 1 , D  0 Simultaneously, the left-hand sides of the Eq. 2. 2 become polynomials in Fξ if canceling F ,setting the coefficients of the polynomial to zero yields a set of algebraic equations:

a1λb1k  0,

6b1k3A12a1kb1 12c12k  0,

b1λ2b1k3B6a1kb0 6b1ka0 12c1kc0 6e1k  0,

3c1k3A3c1ka1  0,

c1λc1k3B3c1ka0  0,

3e1k3A3e1ka1  0,

e1λe1k3B3e1ka0  0. 4.4

(4.4) Solving the algebraic equations obtained above, we have the solutions:

c0  const. ,e0  const. ,b0  const. ,c1  const. ,a1  k2A,λ  

2c₁2

k3_A2,

a0  

k6_A2_B2c 1 2

3k4_A2 ,b1 

2c₁2

k2_A,e1  

2c₁4 _k8_A4_b

0 2c1c0A3k6

A3_k6 . 4.5

(4.5) where k is arbitrary constant.

With the aid of the solutions for Eq. 4. 1,D  0 , the exact solutions for Eq. 1. 1 are obtained as follows:

u k

6_A2_B2c 1 2

3k4_A2 k

2_AFξ,

v b0 

2c₁2

k2_AFξ,

w  c0 c1Fξ,

p  e0 

2c₁4 _k8_A4_b

0 2c1c0A3k6

A3_k6 Fξ,

ξ  kx 2c1 2

k3_A2tξ0. 4.6

We can obtain the exact solutions expressed by Jacobi elliptic functions for Eq.1. 1 .

When C  4, B  4m2 _{1, A 4m}2

u41  

2c₁2 _64m4_k6_m2 _1

48m4_k4 k

2_Asn2_ξ,m,

v41  b0 

c₁2 2k2_m2sn

2_ξ,m,

w41  c0 c1sn2ξ,m,

p41  e0 

2c₁4 _256k8_m8_b

0 128c1c0m6k6

64m6_k6 sn

2_{ξ,m. 4.7}

(4.7) When C  41m2, B  42m2 1, A  4m2

u42  

2c₁2 _64m4_k6_2m2 _1

48m4_k4 4k

2_m2_cn2_ξ,m,

v42  b0 

c₁2

2k2_m2cn

2_ξ,m,

w42  c0 c1cn2ξ,m,

p42  e0 

2c₁4 _k8_A4_b

0 2c1c0A3k6

A3_k6 cn

2_{ξ,m. 4.8}

When C  41m2_,

B  42m2_,

A  4 _(4.8)

u43  

4k6_2m2_2c 1 2

3k4 4k

2_dn2_ξ,m,

v43  b0 

c₁2

2k2dn

2_ξ,m,

w43  c0 c1dn2ξ,m,

p43  e0 

c₁4 _128k8_b

0 64c1c0k6

32k6 dn

2_{ξ,m. 4.9}

(4.9) When

λ  0,a0  k 2_B

3 ,a1  k

2_A,b

1  c1  0,e1  k2Ab0,

c0  const. ,e0  const. ,b0  const. _(4.10) 4.10

Where k is arbitrary constant.

u  k2B 3 k

2_AFξ,

v  b0,

w  c0,

p  e0 k2Ab0Fξ,

ξ  kxξ0. 4.11

(4.11)

We can obtain the exact solutions expressed by Jacobi elliptic functions for Eq.1. 1 .

When C  4, B  4m2 _{1, A 4m}2

u44 

4k2_m2 _1

3 4m

2_k2_sn2_ξ,m,

v44  b0,

w44  c0,

p44  e0 4m2k2b0sn2ξ,m. _(4.12) 4.12

When C  41m2_,

B  42m2 _1,

A  4m2

u45 

4k2_2m2 _1

3 4m

2_k2_cn2_ξ,m,

v45  b0,

w45  c0,

p45  e0 4m2k2b0cn2ξ,m. _(4.13) 4.13

When C  41m2, B  42m2, A  4

u46  

4k2_2m2_

3 4k

2_dn2_ξ,m,

v46  b0,

w46  c0,

p46  e0 4k2Ab0dn2ξ,m. _(4.14) 4.14

When

e1  c1  0,a1  k 2_A

2 ,b0  

λλ2k3_B6ka 0

6k2 ,b1  

λkA 2 ,

a0  const. ,c0  const. ,e0  e0. _(4.15) 4.15

where k,λ are arbitrary constants.

u  a0  k 2_A

2 Fξ,

v  λλ2k

3_B6ka 0

6k2 

λkA 2 Fξ, w  c0,

p  e0. _(4.16) 4.16

We can obtain the exact solutions expressed by Jacobi elliptic functions for Eq. 1. 1 . When C  4, B  4m2 _{1, A 4m}2

u47  a0 2m2k2sn2ξ,m,

v47  

λλ_8k3_m2 _{16ka} 0

6k2 2λm

2_ksn2_ξ,m,

w47  c0,

p47  e0. _(4.17) 4.17

When C  41m2, B 42m2 1, A  4m2

u48  a0 2m2k2cn2ξ,m,

v48  

λλ8k3_2m2 _{16ka} 0

6k2 2λm

2_kcn2_ξ,m,

w48  c0,

p48  e0. _(4.18) 4.18

When C  41m2, B  42m2, A  4

u49  a0 2k2dn2ξ,m,

v49  

λλ_8k3_2m2_6ka 0

6k2 2λkdn

2_ξ,m,

w49  c0,

p49  e0. _(4.19) 4.19

In the limit case when m 1 , the Jacobic functions degenerate to the hyperbolic functions , i.e.

snξ  tanhξ,cnξ  sechξ,dnξ  sechξ.

and when m 0 , the Jacobic functions degenerate to the hyperbolic functions , i.e. snξ  sinξ,cnξ  cosξ,dnξ  1

If Eq. 4. 1 is rewritten as follows:

We can derive the exact solution for Eq. 1. 1 :

When B3  F  B2,

m B2 B3 B1 B3

u50 

k6_A3_B

3 B2 B12c12

3k4_A2 k

2_{A B}

3 B2 B3sn2 A

4 B1 B2,m ,

v50  b0 

2c₁2 _B

3 B2 B3sn2 A₄B1 B2,m

k2_A ,

w50  c0 c1 B3 B2 B3sn2 A

4B1 B2,m ,

p50  e0 

2c₁4 _k8_A4_b

0 2c1c0A3k6

A3_k6 B3 B2 B3sn 2_ A

4B1 B2,m . 4.21 (4.21)

or

u51 

k6_A3_B

3 B2 B12c1 2

3k4_A2 k

2_{A B}

3 B2 B3cn2 A

4B1 B2,m ,

v51  b0 

2c₁2 _B

3 B2 B3cn2 A₄B1 B2,m

k2_A ,

w51  c0 c1 B3 B2 B3cn2 A

4B1 B2,m ,

p51  e0 

2c₁4 _k8_A4_b

0 2c1c0A3k6

A3_k6 B3 B2 B3cn 2_ A

4B1 B2,m . 4.22 (4.22)

Where    2c₁2

k3_A2,k _{are arbitrary constants}

or

u52  1

3k

2_{AB3B2B1k}2_{A B}

3 B2 B3sn2 A

4B1 B2,m ,

v52  b0,

w52  c0,

p52  e0 k2Ab0 B3 B2 B3sn2 A

u53  1

3k

2_{AB3B2B1k}2_{A B}

3 B2 B3cn2 A

4B1 B2,m ,

v53  b0,

w53  c0,

p53  e0 k2Ab0 B3 B2 B3cn2 A

4B1 B2,m . 4.24 (4.24) where   0,k are arbitrary constants.

or

u54  a0  1

2k

2_{A B}

3 B2 B3sn2 A

4B1 B2,m ,

v54  

2k3_AB

3 2k3AB2 2k3AB1 6ka0

6k2

 1

2kA B3 B2 B3sn

2_ A

4B1 B2,m ,

w54  c0,

p54  e0. _(4.25) 4.25

or

u55  a0  1

2k

2_{A B}

3 B2 B3cn2 A

4B1 B2,m ,

v55  

2k3_AB

3 2k3AB2 2k3AB1 6ka0

6k2

 1₂kA B3 B2 B3cn2 A

4B1 B2,m ,

w55  c0,

p55  e0. _(4.26) 4.26

where ,k are arbitrary constants.

When B2  F  B1,

m B1 B2

u56 

k6_A3_B

3 B2 B12c₁2

3k4_A2 k

2_{A B}

2 B1 B2cn2 A

4B1 B3,m ,

v56  b0 

2c₁2 _B

2 B1 B2cn2 A₄B1 B3,m

k2_A ,

w56  c0 c1 B2 B1 B2cn2 A

4B1 B3,m ,

p56  e0 

2c₁4 _k8_A4_b

0 2c1c0A3k6

A3_k6 B2 B1 B2cn 2_ A

4B1 B3,m . 4.27 (4.27)

where    2c₁2

k3_A2,k _{are arbitrary constants.}

or

u57  1

3k

2_{AB3B2B1k}2_{A B}

2 B1 B2cn2 A

4B1 B3,m ,

v57  b0,

w57  c0,

p57  e0 k2Ab0 B2 B1 B2cn2 A

4B1 B3,m . 4.28 (4.28) where   0,k is arbitrary constant.

or

u58  a0  1

2k

2_{A B}

2 B1 B2cn2 A

4B1 B3,m ,

v58  

2k3_AB

3 2k3AB2 2k3AB1 6ka0

6k2

 1₂kA B2 B1 B2cn2 A

4B1 B3,m ,

w58  c0,

p58  e0. _(4.29) 4.29

where ,k are arbitrary constant. If Eq. 4. 1 is rewritten as follows:

F 2ξ  g3 g2F4F3, and g2,g3  const., _(4.30) 4.30

u59  

c₁2

24k4 4k

2_{ξ;g2,g3,}

v59  b0 

c₁2_{ξ;g2,g3}

2k2 ,

w59  c0 c1ξ;g2,g3,

p59  e0 

c₁4 _128b

0k8 64c1c0k6ξ;g2,g3

32k6 . 4.31

(4.31)

where    c₁2

8k3 ,k _{are arbitrary constants.}

or

u60  

c₁2

24k4 4k

2_{ξ;g2,g3,}

v60  b0 

c₁2_{ξ;g2,g3}

2k2 ,

w60 

c₁4 _128b 0k8

64k6_c 1

c1ξ;g2,g3,

p60  e0. _(4.32) 4.32

where    c₁2

8k3 ,k _{is arbitrary constant.}

or

u61  a0 2k2ξ;g2,g3,

v61  

6ka0

6k2 2kξ;g2,g3,

w61  c0,

p61  e0. _(4.33) 4.33

where ,k are arbitrary constants ,where ξ;g2,g3 _{is Weierstrass elliptic function.} If C  D  0 in 4. 1 , we have

F2ξ  AF3 _BF2 _{, A  0, 4.34}

u

k6A2B2c12

3k4_A2 k

2_Bsech2 B

2 ξ (iffB  0,AF  0)

k6A2B2c₁2 3k4_A2 k

2_Bcsch2 B

2 ξ (iffB  0,AF  0)

k6A2B2c12

3k4_A2 k

2_Bsech2 B

2 ξ (iffB  0)

,

v

b0  2c₁2_B

k2_A2 sech

2 B

2 ξ (iffB  0,AF  0)

b0  2c12B

k2_A2 csch

2 B

2 ξ (iffB  0,AF  0)

b0  2c₁2B k2_A2 sech

2 B

2 ξ (iffB  0)

,

w 

c0  Bc_A1 sech2 B

2 ξ (iffB  0,AF  0)

c0  Bc1

A csch 2 B

2 ξ (iffB  0,AF  0)

c0  Bc1

A sech 2 B

2 ξ (iffB  0)

,

p 

e0 

B2c₁4k8_A4_b

02c1c0A3k6

A4_k6 sech

2 B

2 ξ (iffB  0,AF  0)

e0 

B2c₁4_k8_A4_b

02c1c0A3k6

A4_k6 csch

2 B

2 ξ (iffB  0,AF  0)

e0 

B2c₁4k8_A4_b

02c1c0A3k6

A4_k6 sech

2 B

2 ξ (iffB  0)

. 4.35

(4.35)

where    2c₁2

or

u 

1 3k

2_BBk2_sech2 B

2 ξ (iffB  0,AF  0)

1 3k

2_BBk2_csch2 B

2 ξ (iffB  0,AF  0)

1 3k

2_BBk2_sech2 B

2 ξ (iffB  0)

,

v  b0,

w  c0,

p 

e0 Bk2sech2 B

2 ξ (iffB  0,AF  0)

e0 Bk2csch2 B

2 ξ (iffB  0,AF  0)

e0 Bk2sech2  B

2 ξ (iffB  0)

. 4.36

(4.36) where   0,k are arbitrary constants.

or

u 

a0  1₂Bk2sech2 B

2 ξ (iffB  0,AF  0)

a0  1₂Bk2csch2 B

2 ξ (iffB  0,AF  0)

a0  1₂Bk2sech2  B

2 ξ (iffB  0)

,

v 

2k3_B6ka 0

6k2 

kB 2 sech

2 B

2 ξ (iffB  0,AF  0)

2k3_B6ka 0

6k2 

kB 2 csch

2 B

2 ξ (iffB  0,AF  0)

2k3_B6ka 0

6k2 

kB 2 sech

2 B

2 ξ (iffB  0)

,

w  c0,

p  e0. 4.37

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