Optical solitons in birefringent fibers with Biswas–Arshed model by generalized Jacobi elliptic function expansion method

Journal Pre-proof OPTICAL SOLUTIONS IN BIREFRINGENT FIBERS WITH BISWAS-ARSHED MODEL BY GENERALIZED JACOBI ELLIPTIC FUNCTION EXPANSIN METHOD Elsayed M.E. Zayed, Mohamed E.M. Alngar

PII:

S0030-4026(19)31820-0

DOI:

https://doi.org/10.1016/j.ijleo.2019.163922

Reference:

IJLEO 163922

To appear in:

Optik

Received Date:

28 November 2019

Accepted Date:

28 November 2019

Please cite this article as: Elsayed M.E. Zayed, Mohamed E.M. Alngar, OPTICAL SOLUTIONS IN BIREFRINGENT FIBERS WITH BISWAS-ARSHED MODEL BY GENERALIZED JACOBI ELLIPTIC FUNCTION EXPANSIN METHOD, (2019), doi: https://doi.org/10.1016/j.ijleo.2019.163922

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OPTICAL SOLUTIONS IN BIREFRINGENT FIBERS WITH BISWAS-ARSHED MODEL BY GENERALIZED JACOBI ELLIPTIC FUNCTION EXPANSIN METHOD

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Elsayed M. E. Zayed 1 , Mohamed E. M. Alngar

Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt

Abstract

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E-mail addresses: eme [email protected], [email protected]

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In this paper, we employ a generalized Jacobi elliptic function expansion method to extract optical solitons and other solutions to Biswas-Arshed model in birefringent fibers. Jacobi elliptic functions solutions, bright solitons, singular solitons, dark solitons, periodic wave solutions and other solutions have been found.

INTRODUCTION

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OCIS Codes: 060.2310, 060.4510, 060.5530, 190.3270.190.4370 Key words: Solitons; Biswas-Arshed model; Birefringent fibers.

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Optical solitons give inspiration to work in different areas of photonics, comprising spectroscopy and nonlinear optics [1-29]. There are various models that effectively describe the dynamics of soliton propagation such as Fokas-lenells equation, complex Ginzburg–Landau equation, Lakshmanan–Porsezian–Daniel model, Kaup– Newell model, resonant nonlinear Schrodinger’s equation, Radhakrishnan–Kundu–Lakshmanan equation, Kundu–Eckhaus equation, Gerdjikov–Ivanov equation [18-29]. The basic principal for the existence of solitons in optical fibers and metamaterials is to maintain a slight balance between nonlinearity and group velocity dispersion (GVD). In certain circumstances, there may happen situations which lead to low GVD and small nonlinearity. Recently, Biswas and Arshed anticipated a very inventive idea to handle the situations where both nonlinearity and GVD are small. This is simulated in their model and is being stated as Biswas–Arshed model [1-8]. The most important reason for studying this model is that this new model might be a precious one during a probable crisis in telecommunications industry. Thus, the proposed model is going to be of great asset in this industry by getting critical optical solitons of its. Because of this reason and making contributions to previous studies for this model, the Biswas-Arshed model without four-wave mixing terms (FWM) in birefringent fibers, which is first written in two component form for vector solitons by using the trial equation technique [9] and by using modified simple equation approach in [10].

1

1.1

GOVERNING MODEL

The (1+1)-dimensional Biswas-Arshed model [1–8] is given by: [ ( ) ( ) ] 2 2 2 iψt + a1 ψxx + a2 ψxt + i (b1 ψxxx + b2 ψxxt ) = i λ |ψ| ψ + µ |ψ| ψ + Θ |ψ| ψx . x

(1)

x

x

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The complex valued function ψ(x, t) is referred to the wave profile. The first term in this model is defined by the temporal evolution of pulses when the existence of group velocity dispersion with spatio-temporal dispersion sequentially are provided from the coefficients of a1 and a2 . Next, the coefficients of b1 and b2 are defined by the existence of third order dispersion and third order spatio-temporal dispersion sequentially. Finally, the self-steepening and nonlinear dispersions are applied by the coefficients of λ, µ and Θ sequentially. The Biswas-Arshed model without FWM in birefringent fibers is given as [9, 10]: [ ( ) ( ) ] 2 2 iψt + a1 ψxx + b1 ψxt + i (c1 ψxxx + d1 ψxxt ) = i λ1 |ψ| ψ + γ1 |ϕ| ϕ x ] x [ ( ) ( ) ] [ (2) 2 2 2 2 i µ1 |ψ| + α1 |ϕ| ψ + i Θ1 |ψ| + β1 |ϕ| ψx , x

[ ( ) ( ) ] 2 2 iϕt + a2 ϕxx + b2 ϕxt + i (c2 ϕxxx + d2 ϕxxt ) = i λ2 |ϕ| ϕ + γ2 |ψ| ψ x ] x ( ) ] [ ( ) [ 2 2 2 2 + α2 |ψ| i µ2 |ϕ| ϕ + i Θ2 |ϕ| + β2 |ψ| ϕx . x

x

(3)

2

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From Eqs. (2) and (3), the complex valued functions ψ(x, t) and ϕ(x, t) are referred to the wave profile. The first terms are defined by the temporal evolution of pulses when the existence of group velocity dispersion and spatio-temporal dispersion sequentially are supplied by the coefficients of aj and bj (j = 1, 2) in this equation. Next, the existence of third order dispersion and third order spatio-temporal dispersion are ensured with the coefficients of cj and dj sequentially. Finally, the nonlinear terms that signifies self-steepening and nonlinear dispersions are ensured by the coefficients of λj , γj , µj , αj , Θj and βj sequentially. The objective of this paper is to find many new Jacobi elliptic functions solutions, bright solitons, dark solitons, singular solitons, periodic wave solutions and other solutions for the Biswas-Arshed model in birefringent fibers (2) and (3). This article is organized as follows: In Sec. 2, the mathematical preliminaries are displayed. In Sec. 3 the generalized Jacobi elliptic function expansion approach are discussed. In Sec. 4, conclusions are given.

MATHEMATICAL PRELIMINARIES

In order to solve the coupled system (2) and (3), we assume that the hypothesis as:

ϕ(x, t) = P2 (ξ) exp [iφ(x, t)] ,

(5)

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(4)

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and

ψ(x, t) = P1 (ξ) exp [iφ(x, t)] ,

ξ = x − vt,

φ(x, t) = −kx + ωt + θ0 ,

(6)

where v, k, ω and θ0 are all nonzero constants to be determined which represent soliton velocity, soliton frequency, wave number and phase constant, respectively. Next, φ(x, t) is a real function which represents the phase component of the soliton, while Pj (τ ) for j = 1, 2 are real functions which represent the shape of the pulse of the solitons. Substituting (4) and (5) along with (6) into the coupled system (2) and (3), separating the real and imaginary parts, we deduce that the real parts are [ ] [a1 − b1 v + 3c1 k − (ω + 2vk) d1 ] P1′′ + ωk (b1 + d1 k) − c1 k 3 − ω − k 2 a1 P1 (7) −k (λ1 + Θ1 ) P13 − kγ1 P23 − kβ1 P1 P22 = 0, 2

[ ] [a2 − b2 v + 3c2 k − (ω + 2vk) d2 ] P2′′ + ωk (b2 + d2 k) − c2 k 3 − ω − k 2 a2 P2 −k (λ2 + Θ2 ) P23 − kγ2 P13 − kβ2 P2 P12 = 0, while the imaginary parts are ( ) ′ (c1 − vd1 ) P1′′′ + vkb1 + ωb1 − v − 2ka1 − 3k 2 c1 + vk 2 d1 + 2ωkd1 P1 − (2µ1 + Θ1 + 3λ1 ) P12 P1′ − 2α1 P1 P2 P2′ − β1 P1′ P22 − 3γ1 P22 P2′ = 0, ( ) ′ (c2 − vd2 ) P2′′′ + vkb2 + ωb2 − v − 2ka2 − 3k 2 c2 + vk 2 d2 + 2ωkd2 P2 − (2µ2 + Θ2 + 3λ2 ) P22 P2′ − 2α2 P2 P1 P1′ − β2 P2′ P12 − 3γ2 P12 P1′ = 0

(8)

(9)

(10)

Setting (11)

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P2 = χ1 P1 , where χ1 is a nonzero constant, such that χ1 ̸= 1. Consequently, Eqs. (7) and (8) reduce to [ ] [a1 − b1 v + 3c1 k − (ω + 2vk) d1 ] P1′′ + ωk (b1 + d1 k) − c1 k 3 − ω − k 2 a1 P1 ] [ −k λ1 + Θ1 + γ1 χ31 + β1 χ21 P13 = 0,

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[ ] χ1 [a2 − b2 v + 3c2 k − (ω + 2vk) d2 ] P1′′ + χ1 ωk (b2 + d2 k) − c2 k 3 − ω − k 2 a2 P1 ] [ −k (λ2 + Θ2 ) χ31 + γ2 + β2 χ1 P13 = 0,

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while, Eqs. (9) and (10) reduce to ( ) ′ (c1 − vd1 ) P1′′′ + vkb1 + ωb1 − v − 2ka1 − 3k 2 c1 + vk 2 d1 + 2ωkd1 P1 ) ( − 2µ1 + Θ1 + 3λ1 + 2α1 χ21 + β1 χ21 + 3γ1 χ31 P12 P1′ = 0,

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( ) ′ χ1 (c2 − vd2 ) P1′′′ + χ1 vkb2 + ωb2 − v − 2ka2 − 3k 2 c2 + vk 2 d2 + 2ωkd2 P1 ] [ − (2µ2 + Θ2 + 3λ2 ) χ31 + 2α2 χ1 − β2 χ1 − 3γ2 P12 P1′ = 0.

(12)

(13)

(14)

(15)

The linearly independent principle is applied on (12) and (13) to get: (16)

ωk (b1 + d1 k) − c1 k 3 − ω − k 2 a1 = 0, [ ] k λ1 + Θ1 + γ1 χ31 + β1 χ21 = 0,

(17)

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a1 − b1 v + 3c1 k − (ω + 2vk) d1 = 0,

(18)

χ1 [a2 − b2 v + 3c2 k − (ω + 2vk) d2 ] = 0, [ ] χ1 ωk (b2 + d2 k) − c2 k 3 − ω − k 2 a2 = 0, [ ] k (λ2 + Θ2 ) χ31 + γ2 + β2 χ1 = 0.

(19)

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(20) (21)

and

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From (16) and (19), one can obtain the velocity of the soliton v as: v=

a1 + 3kc1 − ωd1 , b1 + 2kd1

(22)

v=

a2 + 3kc2 − ωd2 . b2 + 2kd2

(23)

From (22) and (23), we have the constraint condition: (a1 + 3kc1 − ωd1 ) (b2 + 2kd2 ) − (a2 + 3kc2 − ωd2 ) (b1 + 2kd1 ) = 0. 3

(24)

From (17) and (20), one can obtain the wave number ω as: ω=

k 2 (a1 + c1 k) , k (b1 + d1 k) − 1

(25)

ω=

k 2 (a2 + c2 k) . k (b2 + d2 k) − 1

(26)

and

From (25) and (26), we have the constraint condition: (a1 + c1 k) [k (b2 + d2 k) − 1] − (a2 + c2 k) [k (b1 + d1 k) − 1] = 0.

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(27)

By integrating (14) and (15) with zero constant of integrations, one gets: ( ) 3 (c1 − vd1 ) P1′′ + 3 vkb1 + ωb1 − v − 2ka1 − 3k 2 c1 + vk 2 d1 + 2ωkd1 P1 ( ) − 2µ1 + Θ1 + 3λ1 + 2α1 χ21 + β1 χ21 + 3γ1 χ31 P13 = 0,

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( ) 3χ1 (c2 − vd2 ) P1′′ + 3χ1 vkb2 + ωb2 − v − 2ka2 − 3k 2 c2 + vk 2 d2 + 2ωkd2 P1 ] [ − (2µ2 + Θ2 + 3λ2 ) χ31 + 2α2 χ1 − β2 χ1 − 3γ2 P13 = 0.

(28)

(29)

Eqs. (28) and (29) have the same form under the constraint conditions: =

vkb1 +ωb1 −v−2ka1 −3k2 c1 +vk2 d1 +2ωkd1 χ1 (vkb2 +ωb2 −v−2ka2 −3k2 c2 +vk2 d2 +2ωkd2 )

=

2µ1 +Θ1 +3λ1 +2α1 χ21 +β1 χ21 +3γ1 χ31 . (2µ2 +Θ2 +3λ2 )χ31 +2α2 χ1 −β2 χ1 −3γ2

(30)

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c1 −vd1 χ1 (c2 −vd2 )

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where v is given by (22) and ω is given by (25). The task now is to solve Eq. (28) using the following proposed method:

GENERALIZED JACOBI ELLIPTIC FUNCTION EXPANSION METHOD

P1 (ξ) =

na

According to this method [30], we assume that Eq. (28) has the formal solution: N ∑ [ ] Aj U j (ξ) + Bj U ′ (ξ)U j−1 (ξ) ,

(31)

ur

j=0

where N is a positive integer, Aj , Bj (j = 0, 1, ..., N ) are constants to be determined and U (ξ) is the solutions of the Jacobian elliptic equation: U ′2 (ξ) = l0 + l2 U 2 (ξ) + l4 U 4 (ξ).

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(32)

The parameters l0 , l2 and l4 are constants to be determined later. We determine N in (31) by balancing the highest derivative term with the nonlinear term in Eq. (28). It is well known [30] that Eq. (32) has the following generalized Jacobian elliptic function solutions: w4 m2 (m2 − 1) l2 Type 1. If l0 = , l4 < 0, w2 = , 0 < m < 1, then 2 l4 2m − 1 wm U1 (ξ) = √ cn (wξ) . −l4

(33)

4

Type 2. If l0 =      U2 (ξ) =

w 4 m2 l2 , l4 > 0, w2 = − , l2 < 0, 0 < m < 1, then l4 1 + m2 wm √ sn (wξ) , l4 (34)

 wm    √ cd (wξ) . l4

l2 w4 (1 − m2 ) , l4 < 0, w2 = , 0 < m < 1, then l4 2 − m2 w U3 (ξ) = √ dn (wξ) . −l4

Type 3. If l0 =

w4 (1 − m2 ) l2 , l4 > 0, w2 = , 0 < m < 1, then l4 2 − m2

w U4 (ξ) = √ cs (wξ) . l4 Type 5. If l0 =

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Type 4. If l0 =

(35)

(36)

l2 w4 m2 (m2 − 1) , l4 > 0, w2 = , 0 < m < 1, then l4 2m2 − 1

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w U5 (ξ) = √ ds (wξ) . l4

w4 m2 (m2 − 1) l2 , l4 > 0, w2 = , 0 < m < 1, then l4 2m2 − 1 √ w 1 − m2 √ U6 (ξ) = nc (wξ) . l4

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Type 6. If l0 =

l2 w4 (1 − m2 ) , l4 < 0, w2 = , 0 < m < 1, then l4 2 − m2 √ w 1 − m2 √ U7 (ξ) = nd (wξ) . −l4

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Type 7. If l0 =

w4 (1 − m2 ) l2 , l4 > 0, w2 = , 0 < m < 1, then l4 2 − m2 √ w 1 − m2 √ sc (wξ) . U8 (ξ) = l4

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Type 8. If l0 =

Type 9. If l0 =

U9 (ξ) =

(38)

(39)

(40)

l2 w 4 m2 , l4 > 0, w2 = − , l2 < 0, 0 < m < 1, then l4 1 + m2 w √ dc (wξ) , l4

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    

(37)

(41)

 w    √ ns (wξ) . l4

w4 m2 (m2 − 1) l2 , l4 < 0, w2 = , 0 < m < 1, then l4 2m2 − 1 √ wm 1 − m2 √ sd (wξ) . U10 (ξ) = −l4

Type 10. If l0 =

5

(42)

w4 (1 − m2 )2 2l2 , l4 < 0, w2 = , 0 < m < 1, then 16l4 1 + m2 w U11 (ξ) = √ [mcn (wξ) ± dn (wξ)] . 2 −l4

Type 11. If l0 =

Type 12. If l0 =      

w4 (1 − m2 )2 2l2 , l4 > 0, w2 = , 0 < m < 1, then 16l4 1 + m2 √ w 1 − m2 √ [nc (wξ) ± sc (wξ)] , 2 l4 (44)

√ [ ] cn (wξ) w 1 − m2 √ . 1 ± sn (wξ) 2 l4

    

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U12 (ξ) =

(43)

w 4 m4 2l2 , l4 > 0, w2 = 2 , 0 < m < 1, then 16l4 m −2 ] w [√ U13 (ξ) = √ 1 − m2 nc (wξ) ± dc (wξ) . 2 l4 ( )2 w 4 1 − m2 2l2 Type 14. If l0 = , l4 < 0, w2 = , 0 < m < 1, then 16l4 1 + m2 √ w 1 − m2 √ U14 (ξ) = [msd (wξ) ± nd (wξ)] . 2 −l4

(45)

(46)

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Type 13. If l0 =

U15 (ξ) = √ and

3℘′ (ξ, g2 , g3 ) , l4 [6℘ (ξ, g2 , g3 ) + l2 ]

√

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Type 15. Also, Eq. (20) has the following Weierstrass elliptic function solutions: (a)

na

l0 [6℘ (ξ, g2 , g3 ) + l2 ] , 3℘′ (ξ, g2 , g3 ) ) ( l2 36l0 l4 − l22 l22 where g2 = l0 l4 + , g3 = . 12 216 (b) [

and

3℘ (ξ, g2 , g3 ) − l2 3l4

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U17 (ξ) =

[

U18 (ξ) =

where g2 = (c)

(48)

ur

U16 (ξ) =

3l0 3℘ (ξ, g2 , g3 ) − l2

(47)

] 12

,

(49)

,

(50)

] 12

) ) 4(2 4 ( l2 − 3l0 l4 , g3 = 9l0 l2 l4 − 2l23 . 3 27

U19 (ξ) = √

3℘′ (ξ, g2 , g3 ) , l4 [6℘ (ξ, g2 , g3 ) + l2 ]

(51)

6

and

√ U20 (ξ) =

l0 [6℘ (ξ, g2 , g3 ) + l2 ] , 3℘′ (ξ, g2 , g3 )

(52)

5l22 2l2 l3 , g2 = 2 and g3 = 2 . 36l4 9 54 Here ℘ (ξ, g2 , g3 ) is called a Weierstrass elliptic function which satisfies the Eq. ℘′2 = 4℘3 − g2 ℘ − g3 , such that g2 and g3 are called invariants of the Weierstrass elliptic function. where l0 =

Remark 1. The special cases of l0 , l2 and l4 in the above solutions (33) − (46) are well-known and given respectively as follows:

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(1): If l0 = 1 − m2 , l2 = 2m2 − 1, l4 = −m2 , w = 1, then U1 (ξ) = cn (ξ) . ( ) (2): If l0 = 1, l2 = − 1 + m2 , l4 = m2 , w = 1, then U2 (ξ) = sn (ξ) or cd (ξ) . (3): If l0 = m2 − 1, l2 = 2 − m2 , l4 = −1, w = 1, then U3 (ξ) = dn (ξ) . (4): If l0 = 1 − m2 , l2 = 2 − m2 , l4 = 1, w = 1, then U4 (ξ) = cs (ξ) . (5): If l0 = m2 (m2 − 1), l2 = 2m2 − 1, l4 = 1, w = 1, then U5 (ξ) = ds (ξ) . (6): If l0 = −m2 , l2 = 2m2 − 1, l4 = 1 − m2 , w = 1, then U6 (ξ) = nc (ξ) . ( ) (7): If l0 = −1, l2 = 2 − m2 , l4 = − 1 − m2 , w = 1, then U7 (ξ) = nd (ξ) . (8): If l0 = 1, l2 = 2 − m2 , l4 = 1 − m2 , w = 1, then U8 (ξ) = sc (ξ) . ( ) (9): If l0 = m2 , l2 = − 1 + m2 , l4 = 1, w = 1, then U9 (ξ) = dc (ξ) or ns (ξ) . ( ) (10): If l0 = 1, l2 = 2m2 − 1, l4 = −m2 1 − m2 , w = 1, then U10 (ξ) = sd (ξ) . (1 − m2 )2 1 + m2 1 (11): If l0 = − , l2 = , l4 = − , w = 1, then U11 (ξ) = mcn (ξ) ± dn (ξ) . 4 2 4 1 − m2 1 + m2 1 − m2 cn (ξ) (12): If l0 = , l2 = , l4 = , w = 1, then U12 (ξ) = nc (ξ) ± sc (ξ) or . 4 2 4 1 ± sn (ξ) √ m2 − 2 1 m4 , l2 = , l4 = , w = 1, then U13 (ξ) = 1 − m2 nc (ξ) ± dc (ξ) . (13): If l0 = 4 2 4 m2 − 1 1 + m2 m2 − 1 (14): If l0 = , l2 = , l4 = , w = 1, then U14 (ξ) = msd (ξ) ± nd (ξ) , 4 2 4 where 0 < m < 1 is the modulus of the Jacobi elliptic functions. If m → 1, then the Jacobi elliptic functions are reduced to the hyperbolic functions as follows: cn(ξ, 1) = sech(ξ), sn(ξ, 1) = tanh(ξ), dn(ξ, 1) = sech(ξ), cs(ξ, 1) = csch(ξ), ds(ξ, 1) = csch(ξ), ns(ξ, 1) = coth(ξ), nc(ξ, 1) = cosh(ξ), nd(ξ, 1) = cosh(ξ), sc(ξ, 1) = sinh(ξ), sd(ξ, 1) = sinh(ξ), while if m → 0, then the Jacobi elliptic functions are reduced to the trigonometric functions as follows: cn(ξ, 0) = cos(ξ), sn(ξ, 0) = sin(ξ), dn(ξ, 0) = 1, cs(ξ, 0) = cot(ξ), ds(ξ, 0) = csc(ξ), ns(ξ, 0) = csc(ξ), nc(ξ, 0) = sec(ξ), nd(ξ, 0) = 1, sc(ξ, 0) = tan(ξ), sd(ξ, 0) = sin(ξ).

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Remark 2. The derivatives of the Jacobi elliptic functions are given by: (1): (snξ)′ = (cnξ)(dnξ), (cnξ)′ = −(snξ)(dnξ), (dnξ)′ = −m2 (snξ)(cnξ), ′ ′ (2): (nsξ) = −(csξ)(dsξ), (csξ) = −(nsξ)(dsξ), (dsξ)′ = −(nsξ)(csξ), ′ ′ (3): (scξ) = (ncξ)(dcξ), (ncξ) = (scξ)(dcξ), (dcξ)′ = (1 − m2 )(ncξ)(scξ), (4): (sdξ)′ = (ndξ)(cdξ), (cdξ)′ = (m2 − 1)(sdξ)(ndξ), (ndξ)′ = m2 (cdξ)(sdξ).

3.1

ON SOLVING EQ. (28) USING THE PROPOSED METHOD

To this aim, we first balance P1′′ and P13 in Eq. (28), one gets: N + 2 = 3N =⇒ N = 1.

(53) 7

Now, Eq. (28) has the formal solution: P1 (ξ) = A0 + A1 U (ξ) +

U ′ (ξ) [B0 + B1 U (ξ)] , U (ξ)

(54)

where A0 , A1 , B0 and B1 are constants to be determined. Substituting (54) along with (32) into Eq. (28), s r collecting all the coefficients of [U (ξ)] [U ′ (ξ)] (r = 0, 1, ..., 8, s = 0, 1), we get the following set of algebraic equations: [U (ξ)] [U ′ (ξ)] : ∆3 B13 l4 = 0, 6 [U (ξ)] [U ′ (ξ)] : 3∆3 B12 B0 l4 = 0, 5 [U (ξ)] [U ′ (ξ)] : 3∆3 B1 B02 l4 + 6∆0 B1 l4 + 3∆3 A21 B1 + ∆3 B13 l2 = 0, 4 [U (ξ)] [U ′ (ξ)] : 3∆3 B12 B0 l2 + 3∆3 A21 B0 + 6∆3 A0 A1 B1 + 2∆0 B0 l4 + ∆3 B03 l4 = 0, 3 [U (ξ)] [U ′ (ξ)] : ∆3 B13 l0 + 3∆3 A30 B1 + B1 ∆1 + 6∆3 A0 A1 B0 + 3∆3 B1 B02 l2 + B1 l2 ∆0 = 0, 2 [U (ξ)] [U ′ (ξ)] : 3∆3 B12 B0 l0 + 3B0 ∆3 A20 + ∆3 B03 l2 + B0 ∆1 = 0, 1 [U (ξ)] [U ′ (ξ)] : 3∆3 B1 B02 l0 = 0, 0 [U (ξ)] [U ′ (ξ)] : 2∆0 B0 l0 + ∆3 B03 l0 = 0, 8 [U (ξ)] : 3∆3 B12 A1 l4 = 0, 7 [U (ξ)] : 3∆3 B12 A0 l4 + 6∆3 B1 B0 A1 l4 = 0, 6 [U (ξ)] : ∆3 A31 + 2∆0 A1 l4 + 6∆3 B1 B0 A0 l4 + 3∆3 B03 A1 l4 + 3∆3 B13 A1 l2 = 0, 5 [U (ξ)] : 3∆3 A0 A21 + 3∆3 B02 A0 l4 + 3∆3 B12 A0 l2 + 6∆3 B1 B0 A1 l2 ) = 0, 4 [U (ξ)] : A1 l2 ∆0 + 3A1 ∆3 A20 + A1 ∆1 + 6∆3 B1 B0 A0 l2 + 3∆3 B02 A1 l2 + 3∆3 B12 A1 l0 = 0, 3 [U (ξ)] : 3∆3 B12 A0 l0 + A0 ∆1 + ∆3 A30 + 6∆3 B1 B0 A1 l0 + 3∆3 B02 A0 l2 = 0, 2 [U (ξ)] : 6∆3 B1 B0 A0 l0 + 3∆3 B02 A1 l0 = 0, 1 [U (ξ)] : 3∆3 B02 A0 l0 = 0,

ro of

7

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(55)

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where

na

∆0 = 3 (c1 − vd1 ) , ( ) ∆1 = 3 vkb1 + ωb1 − v − 2ka1 − 3k 2 c1 + vk 2 d1 + 2ωkd1 , ) ( ∆3 = − 2µ1 + Θ1 + 3λ1 + 2α1 χ21 + β1 χ21 + 3γ1 χ31 .

(56)

By solving the above algebraic equations (55) by using the Maple, one gets the following results: Result 1.

√

2l4 ∆0 ∆1 , B0 = 0, B1 = 0, l0 = l0 , l2 = − , l4 = l4 , ∆3 ∆0

Jo

provided

−

ur

A0 = 0, A1 =

l4 ∆0 ∆3 < 0.

(57)

(58)

By the aid of the solutions (33) − (52), we have the following families of Jacobian elliptic solutions for the coupled system (2) and (3) as follows: ∆1 w4 m2 (m2 − 1) , l4 < 0, w2 = − , 0 < m < 1, then substituting (57) along Family 1. If l0 = l4 ∆0 (2m2 − 1) with (33) into Eq. (54), we have: √ [ (√ )] 2∆1 ∆1 ψ(x, t) = ±m − cn − (x − vt) ei(−kx+ωt+θ0 ) , (59) ∆3 (2m2 − 1) ∆0 (2m2 − 1) 8

and

√

[

2∆1 ±m − cn ∆3 (2m2 − 1)

ϕ(x, t) = χ1

(√

∆1 − (x − vt) ∆0 (2m2 − 1)

)] ei(−kx+ωt+θ0 ) ,

(60)

provided ∆1 ∆3 (2m2 − 1) < 0 and ∆0 ∆1 (2m2 − 1) < 0.

(61)

and

[ √

2∆1 ± − sech ∆3

ϕ(x, t) = χ1

(√

ro of

In particular, if m → 1, then we have the bright soliton solution of the coupled system (2) and (3) in the form: (√ )] [ √ 2∆1 ∆1 sech − (x − vt) ei(−kx+ωt+θ0 ) , (62) ψ(x, t) = ± − ∆3 ∆0 )] ∆1 − (x − vt) ei(−kx+ωt+θ0 ) , ∆0

(63)

-p

provided ∆1 ∆3 < 0 and ∆0 ∆1 < 0.

(64)

w4 m2 ∆1 , l4 > 0, w2 = , 0 < m < 1, then substituting (57) along with (34) l4 ∆0 (1 + m2 ) into Eq. (54), we have: √ [ (√ )] 2∆1 ∆1 ψ(x, t) = ±m − sn (x − vt) ei(−kx+ωt+θ0 ) , (65) ∆3 (1 + m2 ) ∆0 (1 + m2 ) √

[ ϕ(x, t) = χ1

or

2∆1 sn ±m − ∆3 (1 + m2 ) √

[

ur

2∆1 ψ(x, t) = ±m − cd ∆3 (1 + m2 ) and

[

(√

√

2∆1 ±m − cd ∆3 (1 + m2 )

Jo

ϕ(x, t) = χ1

(√

∆1 (x − vt) ∆0 (1 + m2 )

)] ei(−kx+ωt+θ0 ) ,

na

and

lP

re

Family 2. If l0 =

)] ∆1 (x − vt) ei(−kx+ωt+θ0 ) , ∆0 (1 + m2 )

(√

)] ∆1 (x − vt) ei(−kx+ωt+θ0 ) , ∆0 (1 + m2 )

(66)

(67)

(68)

provided

∆1 ∆3 < 0 and ∆0 ∆1 > 0.

(69)

In particular, if m → 1 in (65) and (66), then we have the dark soliton solution of the coupled system (2) and (3) in the form: (√ )] [ √ ∆1 ∆1 tanh (x − vt) ei(−kx+ωt+θ0 ) , (70) ψ(x, t) = ± − ∆3 2∆0 9

and

[ √ ϕ(x, t) = χ1

∆1 ± − tanh ∆3

(√

)] ∆1 ei(−kx+ωt+θ0 ) , (x − vt) 2∆0

(71)

and

[ √ ϕ(x, t) = χ1

2∆1 ± − dn ∆3 (2 − m2 )

ro of

provided the same constraint condition (69) is satisfied. w4 (1 − m2 ) ∆1 Family 3. If l0 = , l4 < 0, w2 = − , 0 < m < 1, then substituting (57) along with l4 ∆0 (2 − m2 ) (35) into Eq. (54), we have: [ √ (√ )] 2∆1 ∆1 ψ(x, t) = ± − dn − (x − vt) ei(−kx+ωt+θ0 ) , (72) ∆3 (2 − m2 ) ∆0 (2 − m2 ) (√

)] ∆1 − (x − vt) ei(−kx+ωt+θ0 ) , ∆0 (2 − m2 )

(73)

[ √ ϕ(x, t) = χ1 ±

2∆1 cs ∆3 (2 − m2 )

(√

)] ∆1 − ei(−kx+ωt+θ0 ) , (x − vt) ∆0 (2 − m2 )

na

provided

lP

and

re

-p

provided the same constraint condition (64) is satisfied. In particular, if m → 1, then we have the same bright soliton solution (62) and (63). w4 (1 − m2 ) ∆1 Family 4. If l0 = , l4 > 0, w2 = − , 0 < m < 1, then substituting (57) along with l4 ∆0 (2 − m2 ) (36) into Eq. (54), we have: (√ )] [ √ 2∆1 ∆1 cs − (x − vt) ei(−kx+ωt+θ0 ) , (74) ψ(x, t) = ± ∆3 (2 − m2 ) ∆0 (2 − m2 )

∆1 ∆3 > 0 and ∆0 ∆1 < 0.

(75)

(76)

and

Jo

ur

In particular, if m → 1, then we have the singular soliton solution of the coupled system (2) and (3) in the form: [ √ (√ )] 2∆1 ∆1 ψ(x, t) = ± csch − (x − vt) ei(−kx+ωt+θ0 ) , (77) ∆3 ∆0 [ √

ϕ(x, t) = χ1 ±

2∆1 csch ∆3

(√

)] ∆1 (x − vt) ei(−kx+ωt+θ0 ) , − ∆0

(78)

provided the same constraint condition (76) is satisfied. While, if m → 0, then we have the periodic wave solution of the coupled system (2) and (3) in the form: (√ )] [ √ ∆1 ∆1 cot − (x − vt) ei(−kx+ωt+θ0 ) , (79) ψ(x, t) = ± ∆3 2∆0 10

and

[ √ ϕ(x, t) = χ1 ±

∆1 cot ∆3

(√

)] ∆1 − (x − vt) ei(−kx+ωt+θ0 ) , 2∆0

(80)

and

[ √ ϕ(x, t) = χ1 ±

2∆1 ds ∆3 (2m2 − 1)

ro of

provided the same constraint condition (76) is satisfied. w4 m2 (m2 − 1) ∆1 Family 5. If l0 = , l4 > 0, w2 = − , 0 < m < 1, then substituting (57) along l4 ∆0 (2m2 − 1) with (37) into Eq. (54), we have: (√ )] [ √ ∆1 2∆1 ψ(x, t) = ± ds − (x − vt) ei(−kx+ωt+θ0 ) , (81) ∆3 (2m2 − 1) ∆0 (2m2 − 1) (√

)] ∆1 − (x − vt) ei(−kx+ωt+θ0 ) , ∆0 (2m2 − 1)

provided

-p

∆1 ∆3 (2m2 − 1) > 0 and ∆0 ∆1 (2m2 − 1) < 0.

(82)

(83)

and

[ √ ϕ(x, t) = χ1

2∆1 ± − csc ∆3

(√

lP

re

In particular, if m → 1, then we have the same singular soliton solution (77) and (78). While, if m → 0, then we have the periodic wave solution of the coupled system (2) and (3) in the form: [ √ (√ )] 2∆1 ∆1 ei(−kx+ωt+θ0 ) , (84) ψ(x, t) = ± − csc (x − vt) ∆3 ∆0 )] ∆1 (x − vt) ei(−kx+ωt+θ0 ) , ∆0

(85)

[ √

Jo

and

ur

na

provided the same constraint condition (69) is satisfied. w4 m2 (m2 − 1) ∆1 Family 6. If l0 = , l4 > 0, w2 = − , 0 < m < 1, then substituting (57) along l4 ∆0 (2m2 − 1) with (38) into Eq. (54), we have: [ √ (√ )] 2 (1 − m2 ) ∆1 ∆1 nc (x − vt) ei(−kx+ωt+θ0 ) , (86) ψ(x, t) = ± − ∆3 (2m2 − 1) ∆0 (2m2 − 1)

ϕ(x, t) = χ1 ±

2 (1 − m2 ) ∆1 nc ∆3 (2m2 − 1)

(√

)] ∆1 (x − vt) ei(−kx+ωt+θ0 ) , − ∆0 (2m2 − 1)

(87)

provided the same constraint condition (83) is satisfied. In particular, if m → 0, then we have the periodic wave solution of the coupled system (2) and (3) in the form: [ √ (√ )] 2∆1 ∆1 ψ(x, t) = ± − sec (x − vt) ei(−kx+ωt+θ0 ) , (88) ∆3 ∆0 11

and

[ √ ϕ(x, t) = χ1

2∆1 ± − sec ∆3

(√

)] ∆1 ei(−kx+ωt+θ0 ) , (x − vt) ∆0

(89)

ro of

provided the same constraint condition (69) is satisfied. ∆1 w4 (1 − m2 ) , l4 < 0, w2 = − , 0 < m < 1, then substituting (57) along with Family 7. If l0 = l4 ∆0 (2 − m2 ) (39) into Eq. (54), we have: [ √ (√ )] 2 (1 − m2 ) ∆1 ∆1 ψ(x, t) = ± − nd − (x − vt) ei(−kx+ωt+θ0 ) , (90) ∆3 (2 − m2 ) ∆0 (2 − m2 ) and [ √ ϕ(x, t) = χ1

2 (1 − m2 ) ∆1 ± − nd ∆3 (2 − m2 )

(√

)] ∆1 − (x − vt) ei(−kx+ωt+θ0 ) , ∆0 (2 − m2 )

(91)

and ϕ(x, t) = χ1 ±

2 (1 − m2 ) ∆1 sc ∆3 (2 − m2 )

(√

)] ∆1 − (x − vt) ei(−kx+ωt+θ0 ) , ∆0 (2 − m2 )

lP

[ √

re

-p

provided the same constraint condition (64) is satisfied. w4 (1 − m2 ) ∆1 Family 8. If l0 = , l4 > 0, w2 = − , 0 < m < 1, then substituting (57) along with l4 ∆0 (2 − m2 ) (40) into Eq. (54), we have: [ √ (√ )] 2 (1 − m2 ) ∆1 ∆1 ψ(x, t) = ± sc − (x − vt) ei(−kx+ωt+θ0 ) , (92) ∆3 (2 − m2 ) ∆0 (2 − m2 )

(93)

and

ur

na

provided the same constraint condition (76) is satisfied. In particular, if m → 0, then we have the periodic wave solution of the coupled system (2) and (3) in the form: [ √ (√ )] ∆1 ∆1 ψ(x, t) = ± tan − (x − vt) ei(−kx+ωt+θ0 ) , (94) ∆3 2∆0 [ √

Jo

ϕ(x, t) = χ1 ±

∆1 tan ∆3

(√

)] ∆1 − (x − vt) ei(−kx+ωt+θ0 ) , 2∆0

(95)

provided the same constraint condition (76) is satisfied. ∆1 w4 m2 , l4 > 0, w2 = , 0 < m < 1, then substituting (57) along with (41) Family 9. If l0 = l4 ∆0 (m2 + 1) into Eq. (54), we have: [ √ (√ )] ∆1 2∆1 ψ(x, t) = ± − dc (x − vt) ei(−kx+ωt+θ0 ) , (96) ∆3 (m2 + 1) ∆0 (m2 + 1)

12

[ √ ϕ(x, t) = χ1

or

2∆1 ± − dc ∆3 (m2 + 1)

[ √

2∆1 ns ψ(x, t) = ± − ∆3 (m2 + 1) and

(√

[ √ ϕ(x, t) = χ1

(√

2∆1 ± − ns ∆3 (m2 + 1)

∆1 (x − vt) ∆0 (m2 + 1)

∆1 (x − vt) ∆0 (m2 + 1)

(√

)] ei(−kx+ωt+θ0 ) ,

(97)

)] ei(−kx+ωt+θ0 ) ,

∆1 (x − vt) ∆0 (m2 + 1)

(98)

)] ei(−kx+ωt+θ0 ) .

(99)

ro of

and

[ √ ϕ(x, t) = χ1

∆1 ± − coth ∆3

(√

∆1 (x − vt) 2∆0

)]

ei(−kx+ωt+θ0 ) ,

re

and

-p

provided the same constraint condition (69) is satisfied. In particular, if m → 1 in (98) and (99), then we have the singular soliton solution of the coupled system (2) and (3) in the form: [ √ (√ )] ∆1 ∆1 ψ(x, t) = ± − coth (x − vt) ei(−kx+ωt+θ0 ) , (100) ∆3 2∆0

(101)

and

[

2∆1 (1 − m2 ) ±m − sd ∆3 (2m2 − 1)

ur

ϕ(x, t) = χ1

√

na

lP

provided the same constraint condition (69) is satisfied. While, if m → 0 in (96) − (99), then we have the same periodic wave solution (88), (89), (84) and (85), respectively. w4 m2 (m2 − 1) ∆1 Family 10. If l0 = , l4 < 0, w2 = − , 0 < m < 1, then substituting (57) along l4 ∆0 (2m2 − 1) with (42) into Eq. (54), we have: √ [ (√ )] 2∆1 (1 − m2 ) ∆1 ψ(x, t) = ±m − sd − (x − vt) ei(−kx+ωt+θ0 ) , (102) ∆3 (2m2 − 1) ∆0 (2m2 − 1) (√

∆1 − (x − vt) ∆0 (2m2 − 1)

)] ei(−kx+ωt+θ0 ) .

(103)

Jo

provided the same constraint condition (61) is satisfied. 2∆1 w4 (1 − m2 )2 , l4 < 0, w2 = − , 0 < m < 1, then substituting (57) along Family 11. If l0 = 16l4 ∆0 (m2 + 1) with (43) into Eq. (54), we have: [ √ { (√ ) (√ )}] ∆1 2∆1 2∆1 ψ(x, t) = ± − ∆3 (m mcn ± dn ei(−kx+ωt+θ0 ) , − ∆0 (m − ∆0 (m 2 +1) 2 +1) (x − vt) 2 +1) (x − vt) (104) and [ √ { (√ ) (√ )}] ∆1 2∆1 2∆1 ϕ(x, t) = χ1 ± − ∆3 (m mcn − ∆0 (m − ∆0 (m ± dn ei(−kx+ωt+θ0 ) . 2 +1) 2 +1) (x − vt) 2 +1) (x − vt) 13

(105) provided the same constraint condition (64) is satisfied. In particular, if m → 1, then we have the same bright soliton solution (62) and (63). 2∆1 w4 (1 − m2 )2 Family 12. If l0 = , l4 > 0, w2 = − , 0 < m < 1, then substituting (57) along 16l4 ∆0 (m2 + 1) with (44) into Eq. (54), we have: [ √ ] { (√ ) (√ )} ∆1 (1−m2 ) 2∆1 2∆1 ψ(x, t) = ± ∆3 (m2 +1) nc − ∆0 (m2 +1) (x − vt) ± sc − ∆0 (m2 +1) (x − vt) ei(−kx+ωt+θ0 ) , (106) and ∆1 (1−m2 ) ∆3 (m2 +1)

ϕ(x, t) = χ1 ±

] { (√ ) (√ )} 2∆1 2∆1 nc − ∆0 (m2 +1) (x − vt) ± sc − ∆0 (m2 +1) (x − vt) ei(−kx+ωt+θ0 ) ,

ro of

[ √

(107)

or

∆0

(m2 +1)

re

and

-p

(√ )    √ 2∆1   cn − (x − vt) 2 2 ∆0 (m +1) 1 (1−m ) (√ )  ei(−kx+ωt+θ0 ) , ψ(x, t) = ± ∆ ∆3 (m2 +1)  1 ± sn − 2∆1 (x − vt) 

(√ )    √ 2∆1   cn − (x − vt) 2 2 ∆0 (m +1) 1 (1−m ) )  ei(−kx+ωt+θ0 ) , (√ ϕ(x, t) = χ1 ± ∆ ∆3 (m2 +1)  1 ± sn − 2∆1 (x − vt) 

(108)

(109)

lP

∆0 (m2 +1)

[ √ { (√ ) (√ )}] 2∆1 2∆1 1 − − ϕ(x, t) = χ1 ± ∆ sec (x − vt) ± tan (x − vt) ei(−kx+ωt+θ0 ) , ∆3 ∆0 ∆0

ur

and

na

provided the same constraint condition (76) is satisfied. In particular, if m → 0, then from (106) and (107) we have the periodic wave solution of the coupled system (2) and (3) in the form: [ √ { (√ ) (√ )}] 2∆1 2∆1 1 ψ(x, t) = ± ∆ sec − (x − vt) ± tan − (x − vt) ei(−kx+ωt+θ0 ) , (110) ∆3 ∆0 ∆0

(111)

Jo

and from (108) and (109) we have the periodic wave solution of the coupled system (2) and (3) in the form: (√ )    1 √  cos  − 2∆ (x − vt) ∆ 0 1  ei(−kx+ωt+θ0 ) , (√ ) ψ(x, t) = ± ∆ (112) ∆3   2∆1 1 ± sin − ∆0 (x − vt) and  ϕ(x, t) = χ1 ±

√

∆1 ∆3

(√ )   1  cos  − 2∆ (x − vt) ∆0 (√ )  ei(−kx+ωt+θ0 ) ,  1 ± sin  1 − 2∆ ∆0 (x − vt) 14

(113)

provided the same constraint condition (76) is satisfied. w 4 m4 2∆1 Family 13. If l0 = , l4 > 0, w2 = − , 0 < m < 1, then substituting (57) along with (45) 16l4 ∆0 (m2 − 2) into Eq. (54), we have: ) (√ )}] [ √ {√ (√ ∆1 2∆1 2∆1 ei(−kx+ωt+θ0 ) , ψ(x, t) = ± − ∆3 (2−m 1 − m2 nc 2) ∆0 (2−m2 ) (x − vt) ± dc ∆0 (2−m2 ) (x − vt) (114) and [ √ {√ (√ ) (√ )}] ∆1 2∆1 2∆1 ϕ(x, t) = χ1 ± − ∆3 (2−m 1 − m2 nc (x − vt) ± dc (x − vt) ei(−kx+ωt+θ0 ) , 2) 2 2 ∆0 (2−m ) ∆0 (2−m )

ro of

(115)

provided ∆1 ∆3 < 0 and ∆0 ∆1 > 0.

(116)

ψ(x, t) = ±

] { (√ ) (√ )} 2∆1 2∆1 msd − ∆0 (1+m2 ) (x − vt) ± nd − ∆0 (1+m2 ) (x − vt) ei(−kx+ωt+θ0 ) ,

re

∆1 (1−m2 ) − ∆3 (1+m2 )

and [ √

] { (√ ) (√ )} 2∆1 2∆1 msd − ∆0 (1+m2 ) (x − vt) ± nd − ∆0 (1+m2 ) (x − vt) ei(−kx+ωt+θ0 ) ,

na

ϕ(x, t) = χ1 ±

∆1 (1−m2 ) − ∆3 (1+m2 )

(117)

lP

[ √

-p

In particular, if m → 0, then we have the same periodic wave solution (88) and (89). ( )2 w 4 1 − m2 2∆1 Family 14. If l0 = , l4 < 0, w2 = − , 0 < m < 1, then substituting (57) along 16l4 ∆0 (1 + m2 ) with (46) into Eq. (54), we have:

(118)

and

Jo

ur

provided the same constraint condition (69) is satisfied. Family 15. Substituting (57) along with (47) into Eq. (54), one gets the Weierstrass elliptic function solutions of the coupled system (2) and (3) in the form: ( )    2 2 3 2 0 l0 l4 ∆1 −36∆0 ∆1 l0 l4 √  3∆0 ℘′ (x − vt), ∆1 +12∆  , 2 3 12∆0 216∆0 0  ei(−kx+ωt+θ0 ) , ( ) ψ(x, t) = ± − 2∆ (119) 2 2 3 2 ∆3  ∆ +12∆ l l ∆ −36∆0 ∆1 l0 l4 6∆0 ℘ (x − vt), 1 12∆20 0 4 , 1 216∆ − ∆1  3 0

0

( )   2 2 3 2 0 l0 l4 ∆1 −36∆0 ∆1 l0 l4 √   3∆0 ℘′ (x − vt), ∆1 +12∆ , 2 3 12∆0 216∆0 0  ei(−kx+ωt+θ0 ) , ) ( ϕ(x, t) = χ1 ± − 2∆ ∆3  ∆21 +12∆20 l0 l4 ∆31 −36∆20 ∆1 l0 l4  −∆ 6∆ ℘ (x − vt), , 

0

12∆20

216∆30

(120)

1

provided ∆0 ∆3 < 0.

(121) 15

Family 16. Substituting (57) along with (48) into Eq. (54), one gets the Weierstrass elliptic function solutions of the coupled system (2) and (3) in the form: ) (    2 2 3 2 0 l0 l4 ∆1 −36∆0 ∆1 l0 l4 √  6∆0 ℘ (x − vt), ∆1 +12∆  , − ∆ 2 3 1 12∆0 216∆0  ei(−kx+ωt+θ0 ) , (122) ( ) ψ(x, t) = ± − 2∆∆0 l30 l4 2 2 3 2   3∆ ℘′ (x − vt), ∆1 +12∆0 l0 l4 , ∆1 −36∆0 ∆1 l0 l4 0 12∆2 216∆3 0

and

0

( )   2 2 3 2 0 l0 l4 ∆1 −36∆0 ∆1 l0 l4 √  6∆0 ℘ (x − vt), ∆1 +12∆ , − ∆1  2 3 12∆ 216∆ 0 0  ei(−kx+ωt+θ0 ) , (123) ( ) ϕ(x, t) = χ1 ± − 2∆∆0 l30 l4  3∆ ℘′ (x − vt), ∆21 +12∆20 l0 l4 , ∆31 −36∆20 ∆1 l0 l4  

0

12∆20

216∆30

ro of

provided ∆0 ∆3 l0 l4 < 0.

(124)

-p

Family 17. Substituting (57) along with (49) into Eq. (54), one gets the Weierstrass elliptic function solutions of the coupled system (2) and (3) in the form: [ √ ) }] { ( 4∆21 − 12∆20 l0 l4 8∆31 − 36∆20 ∆1 l0 l4 2 , + ∆1 ei(−kx+ωt+θ0 ) , (125) ψ(x, t) = ± − 3∆3 3∆0 ℘ (x − vt), 3∆20 27∆30 and ϕ(x, t) = χ1 ±

{

2 − 3∆ 3

) }] ( 4∆21 − 12∆20 l0 l4 8∆31 − 36∆20 ∆1 l0 l4 , + ∆1 ei(−kx+ωt+θ0 ) , 3∆0 ℘ (x − vt), 3∆20 27∆30

re

[ √

lP

provided

(126)

∆3 < 0.

(127)

and

ur



na

Family 18. Substituting (57) along with (50) into Eq. (54), one gets the Weierstrass elliptic function solutions of the coupled system (2) and (3) in the form:   v u 2 6∆0 l0 l4  u i(−kx+ωt+θ0 ) ) ] [ ( ψ(x, t) = ±t− , (128) e 4∆21 −12∆20 l0 l4 8∆31 −36∆20 ∆1 l0 l4 , + ∆ ∆3 3∆0 ℘ (x − vt), 2 3 1 3∆ 27∆

Jo

v u  u ϕ(x, t) = χ1 ±t−

0

0



6∆20 l0 l4 i(−kx+ωt+θ0 ) [ ( ) ] , e 4∆21 −12∆20 l0 l4 8∆31 −36∆20 ∆1 l0 l4 ∆3 3∆0 ℘ (x − vt), , + ∆ 2 3 1 3∆ 27∆ 0

(129)

0

provided

∆3 l0 l4 < 0.

(130)

Family 19. Substituting (57) along with (51) into Eq. (54), one gets the Weierstrass elliptic function solutions of the coupled system (2) and (3) in the form: ( )    2∆2 ∆3   √ 3∆0 ℘′ (x − vt), 9∆12 , − 54∆1 3 0 0 0   ei(−kx+ωt+θ0 ) , ( ) (131) ψ(x, t) = ± − 2∆ 2 3 ∆3 2∆ ∆  6∆0 ℘ (x − vt), 9∆12 , − 54∆1 3 − ∆1  0

0

16

and

( )    2∆2 ∆3  √  3∆0 ℘′ (x − vt), 9∆12 , − 54∆1 3 0 0 0   ei(−kx+ωt+θ0 ) , ( ) ϕ(x, t) = χ1 ± − 2∆ 2 3 ∆3 2∆ ∆  6∆0 ℘ (x − vt), 9∆12 , − 54∆1 3 − ∆1  0

(132)

0

provided ∆0 ∆3 < 0.

(133)

0

and

0

ro of

Family 20. Substituting (57) along with (52) into Eq. (54), one gets the Weierstrass elliptic function solutions of the coupled system (2) and (3) in the form: ( )    2∆2 ∆3  √ 3∆0 ℘ (x − vt), 9∆12 , − 54∆1 3 − ∆1  2 5∆ 0 0  ei(−kx+ωt+θ0 ) , ( ) ψ(x, t) = ± − 18∆31∆3  (134) 2∆21 ∆31 0   ′ 3℘ (x − vt), 9∆2 , − 54∆3 ( )    2∆21 ∆31  √ 3∆ ℘ (x − vt), , − − ∆1  2 3 0 9∆0 54∆0 5∆21  ( )  ei(−kx+ωt+θ0 ) , ϕ(x, t) = χ1 ± − 2∆3 ∆0 2∆2 ∆3   9∆0 ℘′ (x − vt), 9∆21 , − 54∆1 3 0

-p

0

√

A0 = 0, A1 = 0, B0 =

−

re

provided the same constraint condition (133) is satisfied.

Result 2.

(135)

2∆0 ∆1 , B1 = 0, l0 = l0 , l2 = , l4 = l4 , ∆3 2∆0

(136)

 √

2∆0 (2m −1)

 (√ ) (√ )  ∆1 ∆1   sn 2∆ (2m 2 −1) (x−vt) dn 2 −1) (x−vt) ∆1 2∆ (2m 0 0  ei(−kx+ωt+θ0 ) , (138) ) (√ − ∆1  ∆3 (2m2 − 1)  cn 2∆ (2m 2 −1) (x−vt)

ur

and

na

lP

provided the same constraint condition (133) is satisfied. By the aid of the solutions (33) − (52), we have the following families of the Jacobian elliptic solutions for the coupled system (2) and (3) as follows: w4 m2 (m2 − 1) ∆1 Family 1. If l0 = , l4 < 0, w2 = , 0 < m < 1, then substituting (136) along l4 2∆0 (2m2 − 1) with (33) into Eq. (54), we have:  (√  √ ) (√ )  ∆1 ∆1  sn 2∆ (2m  2 −1) (x−vt) dn 2 −1) (x−vt) ∆ 2∆ (2m 1 0 0  ei(−kx+ωt+θ0 ) , (137) (√ ) ψ(x, t) = ± − ∆1  ∆3 (2m2 − 1)  cn (x−vt) 2

0

Jo

ϕ(x, t) = χ1 ±

provided ∆1 ∆3 (2m2 − 1) < 0 and ∆0 ∆1 (2m2 − 1) > 0. In particular, if m → 1, then we have the same dark soliton solution (70) and (71). While, if m → 0, then we have the same periodic wave solution (94) and (95). w4 m2 ∆1 Family 2. If l0 = , l4 > 0, w2 = − , 0 < m < 1, then substituting (136) along with l4 2∆0 (1 + m2 ) (34) into Eq. (54), we have:  (√  √ ) (√ )  ∆1 ∆1  cn − 2∆ (1+m  − 2∆ (1+m 2 ) (x−vt) dn 2 ) (x−vt) ∆ 1 0 0  ei(−kx+ωt+θ0 ) , (139) (√ ) ψ(x, t) = ±  ∆3 (1 + m2 )  sn − ∆1 2 (x−vt) 2∆0 (1+m )

17

and

 ϕ(x, t) = χ1 ±

or

√

∆1 ∆3 (1+m2 )

 (√  cn − 2∆ 

) ∆1 2 (x−vt) 0 (1+m )

(√ sn − 2∆

0

(√ − 2∆

dn

∆1 2 (x−vt) 0 (1+m )

) ∆1 (x−vt) (1+m2 )

) 

  ei(−kx+ωt+θ0 ) , 

(140)

 (√ ) (√ )  ∆1 ∆1   sd − 2∆ (1+m − 2∆ (1+m ( 2 ) √ ∆1 2 ) (x−vt) nd 2 ) (x−vt) 0 0  ei(−kx+ωt+θ0 ) , (141) (√ ) ψ(x, t) = ± m − 1 2 ∆3 (1+m )   cd − ∆1 2 (x−vt) 

2∆0 (1+m )

and (

2∆0 (1+m )

ro of

 (√ ) (√ )  ∆1 ∆1  sd − 2∆ (1+m  √ − 2∆ (1+m ) 2 ) (x−vt) nd 2 ) (x−vt) 0 0 ∆1  ei(−kx+ωt+θ0 ) , (√ ) ϕ(x, t) = χ1 ± m2 − 1 2 ∆3 (1+m )   cd − ∆1 2 (x−vt) 

(142)

and

[ √

{

(√ coth

) (√ )}] ∆1 ∆1 − (x − vt) − tanh − (x − vt) ei(−kx+ωt+θ0 ) , (144) 4∆0 4∆0

lP

ϕ(x, t) = χ1 ±

∆1 2∆3

re

-p

provided the same constraint condition (76) is satisfied. In particular, if m → 1 in (139) and (140), then we have the hyperbolic solution of the coupled system (2) and (3) in the form: [ √ { (√ ) (√ )}] ∆1 ∆1 ∆1 ψ(x, t) = ± coth − − ei(−kx+ωt+θ0 ) , (143) (x − vt) − tanh (x − vt) 2∆3 4∆0 4∆0

na

provided the same constraint condition (76) is satisfied. While, if m → 0 in (139) − (142), then we have the same periodic wave solutions (79), (80), (94) and (95), respectively. w4 (1 − m2 ) ∆1 Family 3. If l0 = , l4 < 0, w2 = , 0 < m < 1, then substituting (136) along l4 2∆0 (2 − m2 ) with (35) into Eq. (54), we have:  (√  ) (√ )  √ ∆1 ∆1  cn 2∆ (2−m  2 ) (x−vt) sn 2 ) (x−vt) ∆ 2∆ (2−m 1 0 0  ei(−kx+ωt+θ0 ) , (145) (√ ) ψ(x, t) = ±m2 − ∆1  ∆3 (2 − m2 )  dn 2 (x−vt)

ur

and

2∆0 (2−m )



Jo

ϕ(x, t) = χ1 ±m2

 (√ ) (√ )  ∆1 ∆1  cn 2∆ (2−m  2 ) (x−vt) sn 2 ) (x−vt) ∆1 2∆ (2−m 0 0  ei(−kx+ωt+θ0 ) , (146) ) (√ − ∆1  ∆3 (2 − m2 )  dn 2∆ (2−m 2 ) (x−vt)

√

0

provided the same constraint condition (69) is satisfied. In particular, if m → 1, then we have the same dark soliton solution (70) and (71). w4 (1 − m2 ) ∆1 Family 4. If l0 = , l4 > 0, w2 = , 0 < m < 1, then substituting (136) along l4 2∆0 (2 − m2 ) with (36) into Eq. (54), we have:  (√  √ ) (√ )  ∆1 ∆1  dn 2∆ (2−m  2 ) (x−vt) sn 2 ) (x−vt) ∆ 2∆ (2−m 1 0 0  ei(−kx+ωt+θ0 ) , (√ ) ψ(x, t) = ± − (147) ∆1  ∆3 (2 − m2 )  cs (x−vt) 2 2∆0 (2−m )

18

and

 √ ϕ(x, t) = χ1 ± −

 (√  dn 2∆

∆1 ∆3 (2 − m2 ) 

)

∆1 2 (x−vt) 0 (2−m )

(√

sn

(√

∆1 (x−vt) 2∆0 (2−m2 )

) 

  ei(−kx+ωt+θ0 ) , (148) 

)

∆1 (x−vt) 2∆0 (2−m2 )

cs

2∆0 (2m −1)

and

 (√  ns 2∆

 √

∆1 ∆3 (2m2 − 1) 

(√

ds

2∆0

(√

∆1 (x−vt) 2∆0 (2m2 −1)

cs

) ∆1 (x−vt) (2m2 −1)

) 

  ei(−kx+ωt+θ0 ) , (150) 

-p

ϕ(x, t) = χ1 ± −

) ∆1 (x−vt) 2 0 (2m −1)

ro of

provided the same constraint condition (69) is satisfied. In particular, if m → 1, then we have the same singular soliton solution (100) and (101). While, if m → 0, then we have the same periodic wave solution (88) and (89). ∆1 w4 m2 (m2 − 1) , l4 > 0, w2 = , 0 < m < 1, then substituting (136) along Family 5. If l0 = l4 2∆0 (2m2 − 1) with (37) into Eq. (54), we have:  (√  √ ) (√ )  ∆1 ∆1  ns 2∆ (2m  (x−vt) 2 −1) (x−vt) cs 2 ∆ 2∆0 (2m −1) 1 0  ei(−kx+ωt+θ0 ) , (149) (√ ) ψ(x, t) = ± − ∆1  ∆3 (2m2 − 1)  ds (x−vt) 2

lP

re

provided the same constraint condition (139) is satisfied. In particular, if m → 1, then we have the same singular soliton solution (100) and (101). While, if m → 0, then we have the same periodic wave solution (79) and (80). w4 m2 (m2 − 1) ∆1 Family 6. If l0 = , l4 > 0, w2 = , 0 < m < 1, then substituting (136) along l4 2∆0 (2m2 − 1) with (38) into Eq. (54), we have:  (√  √ ) (√ )  ∆1 ∆1  dc 2∆ (2m  2 −1) (x−vt) sc 2 −1) (x−vt) ∆ 2∆ (2m 1 0 0  ei(−kx+ωt+θ0 ) , (151) (√ ) ψ(x, t) = ± − ∆1  ∆3 (2m2 − 1)  nc (x−vt) 2 2∆0 (2m −1)

 (√  dc 2∆

 √

ϕ(x, t) = χ1 ± −

na

and

∆1 ∆3 (2m2 − 1) 

)

∆1 (x−vt) 2 0 (2m −1)

(√

nc

2∆0

sc

(√ ∆1 (x−vt) 2∆0 (2m2 −1)

) ∆1 (x−vt) (2m2 −1)

) 

  ei(−kx+ωt+θ0 ) , (152) 

Jo

ur

provided the same constraint condition (139) is satisfied. In particular, if m → 1, then we have the same dark soliton solution (70) and (71). While, if m → 0, then we have the same periodic wave solution (94) and (95). ∆1 w4 (1 − m2 ) , l4 < 0, w2 = , 0 < m < 1, then substituting (136) along Family 7. If l0 = l4 2∆0 (2 − m2 ) with (39) into Eq. (54), we have:  (√  ) (√ )  √ ∆1 ∆1  cd 2∆ (2−m  2 ) (x−vt) sd 2 ) (x−vt) ∆ 2∆ (2−m 1 0 0  ei(−kx+ωt+θ0 ) , (153) ) (√ ψ(x, t) = ±m2 − ∆1  ∆3 (2 − m2 )  nd 2∆ (2−m 2 ) (x−vt) 0

and



ϕ(x, t) = χ1 ±m2

 (√ ) (√ )  ∆1 ∆1  cd 2∆ (2−m  2 ) (x−vt) sd 2 ) (x−vt) ∆1 2∆ (2−m 0 0  ei(−kx+ωt+θ0 ) , (154) (√ ) − ∆1  ∆3 (2 − m2 )  nd 2∆ (2−m (x−vt) 2)

√

0

19

provided the same constraint condition (69) is satisfied. In particular, if m → 1, then we have the same dark soliton solution (70) and (71). w4 (1 − m2 ) ∆1 Family 8. If l0 = , l4 > 0, w2 = , 0 < m < 1, then substituting (136) along l4 2∆0 (2 − m2 ) with (40) into Eq. (54), we have:  v   v   u u ∆1 ∆1 u    ncu t dct    (x−vt) (x−vt)  √   2) 2) 2∆ (2 − m 2∆ (2 − m 0 0   i(−kx+ωt+θ0 ) ∆1  v ψ(x, t) = ± − ∆3 (2−m2 ) , (155) e u   ∆ u   1   t     (x−vt) sc   2∆0 (2 − m2 ) and  √  ∆1 ϕ(x, t) = χ1 ± − ∆3 (2−m 2)     

 v   u ∆1 ∆ u  1  (x−vt)dct (x−vt)   2∆0 (2 − m2 ) 2∆0 (2 − m2 )  i(−kx+ωt+θ0 )  v , e u  ∆ u  1   (x−vt) sct  2∆0 (2 − m2 )

ro of

 v u u     nct



(156)

(

ψ(x, t) = ± 1 − m

)√ 2

∆1 ∆3 (m2 +1)



or



√

∆1 ∆3 (m2 +1)

Jo

ψ(x, t) = ±

and

)√

∆1 ∆3 (m2 +1)

ur

( ϕ(x, t) = χ1 ± 1 − m2



ϕ(x, t) = χ1 ±

√



na

and

 (√  nc −

 (√  cs − 

∆1 ∆3 (m2 +1)

) (√ )  ∆1 ∆ − 2∆ (m12 +1) (x−vt)  2∆0 (m2 +1) (x−vt) sc 0  ei(−kx+ωt+θ0 ) , (√ ) ∆1  dc − 2∆ (m 2 +1) (x−vt)

lP



re

-p

provided the same constraint condition (69) is satisfied. In particular, if m → 1, then we have the same singular soliton solution (100) and (101). While, if m → 0, then we have the same periodic wave solution (84) and (85). w 4 m2 ∆1 Family 9. If l0 = , l4 > 0, w2 = − , 0 < m < 1, then substituting (136) along with l4 2∆0 (m2 + 1) (41) into Eq. (54), we have:

 (√  nc − 

0

(157)

) (√ )  ∆1 ∆1  (x−vt) sc − (x−vt) 2 2 2∆0 (m +1) 2∆0 (m +1)  ei(−kx+ωt+θ0 ) , ) (√ ∆1  dc − 2∆ (m 2 +1) (x−vt) 0

(158)

) (√ )  ∆1 ∆1  (x−vt) ds − (x−vt) 2 2 2∆0 (m +1) 2∆0 (m +1)  ei(−kx+ωt+θ0 ) , (√ ) ∆1  ns − 2∆ (m2 +1) (x−vt)

(159)

0

 (√  cs − 

) (√ )  ∆1 ∆1  (x−vt) ds − (x−vt) 2 2 2∆0 (m +1) 2∆0 (m +1)  ei(−kx+ωt+θ0 ) , ) (√ ∆1  (x−vt) ns − 2∆ (m 2 +1)

(160)

0

provided the same constraint condition (76) is satisfied. In particular, if m → 1 in (159) and (160), then we have the same singular soliton (77) and (78). While, if 20

m → 0 in (157) − (160), then we have the same periodic wave solution (94), (95), (79) and (80), respectively. w4 m2 (m2 − 1) ∆1 Family 10. If l0 = , l4 < 0, w2 = , 0 < m < 1, then substituting (136) l4 2∆0 (2m2 − 1) along with (42) into Eq. (54), we have:  v  v   u u ∆ ∆ u u   1 1     ndt 2∆ (2m2 − 1) (x−vt)cdt 2∆ (2m2 − 1) (x−vt)   √  0 0   i(−kx+ωt+θ0 ) ∆1 v  ψ(x, t) = ± − ∆3 (2m , e 2 −1) u   ∆ u   1   t     sd (x−vt)   2∆0 (2m2 − 1) 

(161) and  √  ∆1 ϕ(x, t) = χ1 ± − ∆3 (2m 2 −1)     

 v   u ∆1 ∆1 u   t   (x−vt) cd (x−vt)   2∆0 (2m2 − 1) 2∆0 (2m2 − 1)  i(−kx+ωt+θ0 ) v  . e u  ∆ u  1  t    sd (x−vt)  2∆0 (2m2 − 1)

ro of

 v u u     ndt



-p

(162)

and

na

lP

re

provided ∆1 ∆3 (2m2 − 1) < 0 and ∆0 ∆1 (2m2 − 1) > 0. In particular, if m → 1, then we have the same singular soliton (100) and (101). While, if m → 0, then we have the same periodic wave solution (79) and (80). w4 (1 − m2 )2 ∆1 Family 11. If l0 = , l4 > 0, w2 = , 0 < m < 1, then substituting (136) along 16l4 ∆0 (m2 + 1) with (44) into Eq. (54), we have: )   (√  ∆1    (x − vt)    dn  √ ∆0 (m2 + 1) i(−kx+ωt+θ0 ) 2∆1  (√ )  ψ(x, t) = ± − ∆3 (m2 +1) , (163) e   ∆1     cn (x − vt) ∆0 (m2 + 1) )   (√ ∆1    (x − vt)    dn 2 + 1)  √ ∆ (m 0 2∆ 1  ei(−kx+ωt+θ0 ) , (√ ) − ϕ(x, t) = χ1  ± 2 ∆3 (m +1)     ∆ 1    cn  (x − vt) ∆0 (m2 + 1)

ur



(164)

Jo

provided the same constraint condition (69) is satisfied. In particular, if m → 0, then we have the same periodic wave solution (88) and (89). ( )2 w 4 1 − m2 ∆1 , l4 < 0, w2 = , 0 < m < 1, then substituting (136) along Family 12. If l0 = 16l4 ∆0 (1 + m2 ) with (46) into Eq. (54), we have: )   (√  ∆1    (x − vt)   cn  √  ∆0 (m2 + 1) i(−kx+ωt+θ0 ) 2∆1  )  (√ ψ(x, t) = ±m − ∆3 (m2 +1) , (165) e   ∆1    dn  (x − vt) ∆0 (m2 + 1)

21

and

)   (√ ∆1    (x − vt)   cn  √ 2 + 1)  ∆ (m 0 2∆ 1  ei(−kx+ωt+θ0 ) , (√ ) − ϕ(x, t) = χ1  ±m ∆3 (m2 +1)     ∆ 1    dn  (x − vt) ∆0 (m2 + 1) 

(166)

ro of

provided the same constraint condition (69) is satisfied. Family 13. Substituting (136) along with (47) into Eq. (54), one gets the Weierstrass elliptic function solutions of the coupled system (2) and (3) in the form: √ ψ(x, t) = ± − ∆32∆3 ei(−kx+ωt+θ0 ) 0 [ ] (167) 12∆0 ℘(ξ,g2 ,g3 )[576∆20 ℘2 (ξ,g2 ,g3 )+48∆0 ∆1 ℘(ξ,g2 ,g3 )−(∆21 +48∆20 l0 l4 )]−∆1 [∆21 +48∆20 l0 l4 ]−1152∆30 ℘′2 (ξ,g2 ,g3 ) × , 96℘′ (ξ,g2 ,g3 )[12∆0 ℘((x−vt),g2 ,g3 )+∆0 ] and

√ ϕ(x, t) = ±χ1 − ∆32∆3 ei(−kx+ωt+θ0 ) 0 [ ] 12∆0 ℘(ξ,g2 ,g3 )[576∆20 ℘2 (ξ,g2 ,g3 )+48∆0 ∆1 ℘(ξ,g2 ,g3 )−(∆21 +48∆20 l0 l4 )]−∆1 [∆21 +48∆20 l0 l4 ]−1152∆30 ℘′2 (ξ,g2 ,g3 ) × , 96℘′ (ξ,g2 ,g3 )[12∆0 ℘((x−vt),g2 ,g3 )+∆0 ]

ξ = x − vt, g2 =

-p

provided

∆21 + 48∆20 l0 l4 ∆31 − 144∆20 ∆1 l0 l4 , g = − and ∆0 ∆3 < 0. 3 48∆20 1728∆30

(168)

(169)

lP

re

Family 14. Substituting (136) along with (48) into Eq. (54), one gets the Weierstrass elliptic function solutions of the coupled system (2) and (3) in the form: }] [ √ { 6℘′ (ξ,g2 ,g3 ){96∆0 ℘(ξ,g2 ,g3 )[6∆0 ℘(ξ,g2 ,g3 )+∆1 ]−5∆21 +144∆20 l0 l4 } 2∆30 ei(−kx+ωt+θ0 ) , ψ(x, t) = ± − ∆3 ℘(ξ,g ,g ) 6912∆3 ℘2 ((x−vt),g ,g )−36∆ ∆2 −1728∆3 l l +∆ ∆2 +144∆2 l l 1( 1 2 3 0 1 2 3 [ 0 0 4) 0 0 4] 0

and

6℘′ (ξ,g2 ,g3 ){96∆0 ℘(ξ,g2 ,g3 )[6∆0 ℘(ξ,g2 ,g3 )+∆1 ]−5∆21 +144∆20 l0 l4 }

na

[ √ { 2∆3 ϕ(x, t) = χ1 ± − ∆30 ℘(ξ,g

(170)

[6912∆30 ℘2 ((x−vt),g2 ,g3 )−36∆0 ∆21 −1728∆30 l0 l4 ]+∆1 (∆21 +144∆20 l0 l4 )

ξ = x − vt, g2 =

ei(−kx+ωt+θ0 ) , (171)

ur

provided

2 ,g3 )

}]

∆21 + 48∆20 l0 l4 ∆31 − 144∆20 ∆1 l0 l4 , g = − and ∆0 ∆3 < 0. 3 48∆20 1728∆30

(172)

Jo

Family 15. Substituting (136) along with (49) into Eq. (54), one gets the Weierstrass elliptic function solutions of the coupled system (2) and (3) in the form: ( )    ∆2 −12∆2 l l ∆3 −18∆2 ∆ l l √   6℘′ (x − vt), 1 3∆2 0 0 4 , − 1 27∆03 1 0 4 3 ∆ 0 0  ei(−kx+ωt+θ0 ) , ) ( (173) ψ(x, t) = ± − 2∆03  6∆ ℘ (x − vt), ∆21 −12∆20 l0 l4 , − ∆31 −18∆20 ∆1 l0 l4 − ∆  2 3 1 0 3∆ 27∆ 0

and

0

( )   ∆21 −12∆20 l0 l4 ∆31 −18∆20 ∆1 l0 l4 ′ √   (x − vt), 6℘ , − 2 3 3 3∆0 27∆0 ∆  ei(−kx+ωt+θ0 ) , (174) ( ) ϕ(x, t) = χ1 ± − 2∆03 2 2 3 2  6∆ ℘ (x − vt), ∆1 −12∆0 l0 l4 , − ∆1 −18∆0 ∆1 l0 l4 − ∆  0 1 3∆2 27∆3 

0

22

0

provided the same constraint condition (121) is satisfied. Family 16. Substituting (136) along with (51) into Eq. (54), one gets the Weierstrass elliptic function solutions of the coupled system (2) and (3) in the form: }] [ √ { 12∆0 ℘(ξ,g2 ,g3 )[216∆20 ℘2 (ξ,g2 ,g3 )+18∆0 ∆1 ℘(ξ,g2 ,g3 )−∆21 ]−∆31 −432∆30 ℘′ (ξ,g2 ,g3 ) 0 ψ(x, t) = ± − ∆ ei(−kx+ωt+θ0 ) , ∆3 36∆2 ℘′ (ξ,g2 ,g3 )[12∆0 ℘(ξ,g2 ,g3 )+∆1 ] 0

(175) and }] [ √ { 12∆0 ℘(ξ,g2 ,g3 )[216∆20 ℘2 (ξ,g2 ,g3 )+18∆0 ∆1 ℘(ξ,g2 ,g3 )−∆21 ]−∆31 −432∆30 ℘′ (ξ,g2 ,g3 ) ∆0 ϕ(x, t) = χ1 ± − ∆3 ei(−kx+ωt+θ0 ) , 36∆2 ℘′ (ξ,g2 ,g3 )[12∆0 ℘(ξ,g2 ,g3 )+∆1 ] 0

ro of

(176)

provided ξ = x − vt, g2 =

CONCLUSIONS

(177)

-p

4

∆31 ∆21 , g3 = and ∆0 ∆3 < 0. 2 18∆0 432∆30

References

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In this article, we found the optical solitons and other solutions to Biswas-Arshed model in birefringent fibers. The generalized Jacobi elliptic function expansion approach has been discussed. Bright solitons, singular solitons, dark solitons, periodic wave solutions, Jacobi elliptic functions solutions and other solutions to this model have been presented. Therefore, integrability for the model was possible with phase-matching condition. The conclusions of this work are encouraging with a view to consider more comprehensively the model. Firstly, the Biswas-Arshed equation will be extended in not only birefringent fibers with FWM but also DWDM system which then is going to be studied through strategic algorithms such as modified simple equation algorithm, trial equation integration architecture and extended Kudryashov’s method. Moreover, perturbation terms such as Raman scattering, saturable amplifiers, multi-photon absorption subsequently will be added. These very valuable outcomes sequentially are going to be presented.

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