Optical solitons in birefringent fibers for Lakshmanan–Porsezian–Daniel model by modified extended direct algebraic method

Journal Pre-proof Optical solitons in birefringent fibers for Lakshmanan-Porsezian-Daniel model by modified extended direct algebraic method Gawarai Dieu-donne, Malwe Boudoue Hubert, Douvagai, Gambo Betchewe, Serge Y. Doka

PII:

S0030-4026(19)32034-0

DOI:

https://doi.org/10.1016/j.ijleo.2019.164135

Reference:

IJLEO 164135

To appear in:

Optik

Received Date:

26 May 2019

Accepted Date:

24 December 2019

Please cite this article as: Gawarai Dieu-donne, Malwe Boudoue Hubert, Douvagai, Gambo Betchewe, Serge Y. Doka, Optical solitons in birefringent fibers for Lakshmanan-Porsezian-Daniel model by modified extended direct algebraic method, (2019), doi: https://doi.org/10.1016/j.ijleo.2019.164135

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*Manuscript

Optical solitons in birefringent fibers for Lakshmanan-Porsezian-Daniel model by modified extended direct algebraic method

Department of Physics, Faculty of Science, the University of Maroua, P.O. Box 814, Cameroon Teachers’ Training College of Maroua, the University of Maroua, P.O. Box 55, Cameroon

c Department

ro

b Higher

of Physics, Faculty of Science, the University of Ngaoundere, P.O. Box 454, Cameroon

-p

a

of

Gawarai Dieu-donne a , Malwe Boudoue Hubert a , Douvagai a , Gambo Betchewe a,b , Serge Y. Doka c ,

re

Abstract

lP

In this paper, the modified extended direct algebraic has been used to investigate the Lakshmanan-Porsezian-Daniel model. Many solutions such as bright, dark and combo optical soliton solutions are obtained. These solutions may be useful in the explanation of solitons propagation in birefringent fibers.

1

ur na

Key words: Modified extended direct algebraic method; soliton solutions; birefringent fibers; Lakshmanan-Porsezian-Daniel model.

Introduction

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Since the discovery of soliton by John Scott Russel, they appear in many fields of research. In optics, to mention this example, they play an important role in the transmission of information. The propagation in optical fiber is modelized by several types of equations that are generally the shape spread of the nonlinear Schr¨odinger equation. Email addresses: [email protected]. (Gawarai Dieu-donne), [email protected]. (Malwe Boudoue Hubert), [email protected]. (Douvagai), [email protected] (Gambo Betchewe), [email protected] (Serge Y. Doka).

Preprint submitted to Journal

26 May 2019

Among the shapes generalized of the nonlinear Schr¨odinger equation, we can mention the Lakshmanan-Porsezian-Daniel equation (LPD). The dimensionless form of the LPD model with higher order dispersion, full nonlinearity and spatio-temporal dispersion (STD) is given by [1, 2]

ro

of

∗ iqt +aqxx +bqxt +cF (| q |2 )q = σqxxxx +α(qx )2 q ∗ +β | qx |2 q+γ | q |2 qxx +λq 2 qxx +δ | q |4 q, (1) where q(x, t) represents the complex-valued wave profile with x and t are the independent spatial and temporal variables respectively. a is group velocity dispersion, b is the coefficient of quintic nonlinearity, c is the coefficient of nonlinear dispersion, α is the inter-modal dispersion, λ accounts for selfsteepening with short pulses.

re

-p

The exact soliton solutions of this equation have been obtained by through several methods among which the modified trial equation [3], the modified simple equation method [4], the sine-Gordon expansion method [5], the undetermined coefficients method [6] and the modified extended direct algebraic method [7].

2

ur na

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The aim of this work is to retrieve exact soliton solutions in birefringent fibers for Lakshmanan-Porsezian-Daniel model which was recently presented by Ashed et al. [8] and Ekici [9] by using the modified extended direct algebraic scheme [7, 10, 11]. To achieve such a goal, we organized our work as follows: In section 2, we shall present the equation of Lakshmanan-Porsezian-Daniel model; section 3 is devoted to the application of the modified extended direct algebraic method to the derived equation. Finally, the conclusion and remarks are given in section 4.

Presentation of the model

The Lakshmanan-Porsezian-Daniel model in two-component form is read as [8, 9]:

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iqt + a1 qxx + b1 qxt + [c1 | q |2 +d1 | v |2 ]q = σ1 qxxxx + (α1 qx2 + β1 vx2 )q ∗ + (γ1 | qx |2 +δ1 | vx |2 )q ∗ + (λ1 | q |2 +θ1 | v |2 )qxx + (ε1 q 2 + η1 v 2 )qxx

+ (f1 | q |4 +g1 | v |2 | q |2 +h1 | v |4 )q, (2) 2

ivt + a2 vxx + b2 vxt + [c2 | v |2 +d2 | q |2 ]v = σ2 qxxxx + (α2 vx2 + β2 qx2 )v ∗ + (γ2 | vx |2 +δ2 | qx |2 )v ∗ + (λ2 | v |2 +θ2 | q |2 )vxx + (ε2 v 2 + η2 v 2 )vxx

+ (f2 | v |4 +g2 | v |2 | q |2 +h2 | q |4 )v, (3)

-p

ro

of

where the independent variables are x and t that represents spatial and temporal variables respectively. The dependent √ variables q(x, t), v(x, t) give the complex valued wave profile and i = −1. ai and bi (i = 1, 2) are group velocity dispersion and spatio-temporal dispersion coefficients respectively; ci and fi represent the coefficients of Kerr law nonlinearity; σi is the fourth order dispersion coefficient; αi are the inter-modal dispersion coefficients; εi account for self-steepening with short pulses; di , gi , hi accounted for cross-phase modulation.

Exact solutions of the model by the modified extended direct algebraic method

re

3

ur na

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In this section, we apply the modified extended direct algebraic method [7,10, 11] to Eq. (2) and Eq. (3) to retrieve exact solutions to the presented model. To start, the initial assumption of the solution structure of Eq. (2) and Eq. (3) are taken to be (4) q(x, t) = V1 (ξ)eiφ(x,t) ,

v(x, t) = V2 (ξ)eiφ(x,t) ,

(5)

where φ(x, t) = −kx + ωt + θ0 and ξ = x − νt; with V1 and V2 the amplitude component of wave; ν is the soliton velocity; k is the soliton wave number, ω is the soliton frequency and θ0 is the phase constant.

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Introducing Eq. (4) and Eq. (5) into Eq. (2) and Eq. (3) respectively, Eq. (2) and Eq. (3) lead to real real and imaginary parts. The imaginary parts can written as





′

′′

′

ν(−1 + bm k) − 2am k + bm ω − σm k 3 Vm + 4σm kVm + 2k(αm + γm + λm − ξm )Vm2 Vm ′

′

+2k(βm + δm )Vm Vm∗ Vm∗ + 2k(θm − ηm )Vm Vm2 ∗ = 0,

3

m = 1, 2;

m∗ = 3 − m. (6)

Setting the coefficients of the linearly independent functions to zero gives the following relation

v=

2am k − bm ω + 4σm k 3 , bm k − 1

(7)

(8)

αm + γm + λm − ξm = 0,

(9)

of

σm = 0,

-p

θm − ηm = 0.

ro

βm + δm = 0,

(10)

(11)

b1 = b2 = b.

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a1 = a2 = a

re

Now, we assume that the two expressions of wave velocity are identic

(12)

Substituting Eq. (12) and Eq. (8) into Eq. (7) leads to

ur na

ν=

2ak − bω . bk − 1

(13)

The real parts yields

′′

′

(a − bν)Vm − (ω + ak 2 − bkω)Vm + cm Vm3 + dm Vm Vm2 ∗ = (αm + βm )(Vm (Vm )2 − k 2 Vm3 ) ′

+ (ξm + λm )(Vm2 Vm − k 2 Vm3 )

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′′

+ (θm + ηm )(Vm2 Vm − k 2 Vm2 ∗ Vm )

4 , + fm Vm5 + gm Vm3 Vm2 ∗ + hm Vm Vm∗ (14)

with m = 1, 2 and m∗ = 3 − m. Considering the balancing principle, the following relation can be written Vm = Vm∗ , 4

(15)

and Eq. (14) can be reduced to ′′

′

(a − bν)Vm − (ω + ak 2 − bkω)Vm + (cm + dm + 2k 2 (ξm + θm ))Vm3 + (λm − ξm )Vm (Vm )2 ′

−(ξm + λm + 2θm )Vm2 Vm − (fm + gm + hm )Vm5 = 0.

(16)

Using the assumption λm = ξ m ,

(17)

ξm + λm + 2θm = 0,

(18)

and

of

Eq. (16) can be transformed to ′′

ro

(a − bν)Vm − (ω + ak 2 − bkω)Vm2 + (cm + dm )Vm3 − (fm + gm + hm )Vm5 = 0. (19) Assuming Eq. (19) can be written as 

′′

′



q

Um ,

-p

Vm =

(20)

re

2 3 4 (a−bν) 2Um Um − (Um )2 −4(ω+ak 2 −bkω)Um +4(cm +dm )Um −4(fm +gm +hm )Um = 0. (21) ′′

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The highest power of Um Um appears as 2N + 2 and the higher nonlinearity 4 Um appears as 4N. Thus, one can find N = 1. Consequently we reach

and

ur na

U1 (ξ) = a−1 (φ(ξ))−1 + a0 + a1 φ(ξ), U2 (ξ) = ρ−1 (φ(ξ))−1 + ρ0 + ρ1 φ(ξ),

where ′

φ (ξ) =

p

(22) (23)

r0 + r1 φ(ξ) + r2 φ2 (ξ) + r3 φ3 (ξ) + r4 φ4 (ξ) + r5 φ5 (ξ) + r6 φ6 (ξ), (24)

and ri are constants and can be discussed as in [12].

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Substituting Eq. (21), Eq. (22) and Eq. (23) into Eq. (20) and using the results in [12–16], the solutions of the system of Eq. (1) and Eq. (2) are: case 1

r0 = r1 = r3 = r5 = r6 = 0

a−1 = 0,

a0 =

3(d1 +c1 ) 8(h1 +f1 +g1 ) ,

ρ−1 = 0,

ρ0 =

3(d2 +c2 ) 8(h2 +f2 +g2 ) ,

q

3(c1 +d1 )

a1 = ± ρ1 = ±

2

8(f1 +g1q +h1 )

3(c2 +d2 )

5

r

− r4

,

ω=

64r2 k 2 bν(fm +gm +hm )−3(cm +dm )2 (4k 2 +5r2 ) 64r2 (fm +gm +hm )(bk−1)

,

a=

16bνr2 (fm +gm +hm )−3(cm +dm )2 . 16r2 (fm +gm +hm )

r

− r4

8(f2 +g2 +h2 )

2

(i1.1) : When r2 > 0 and r4 < 0 Eq. (2) and (3) have bright 1-soliton as solutions and are  1 2 √ 3(d1 + c1 ) 3(c1 + d1 ) q1,1 (x, t) = ± sech ( r2 (x − νt)) eiφ(x,t) , (25) 8(h1 + f1 + g1 ) 8(f1 + g1 + h1 )



√ 3(c2 + d2 ) 3(d2 + c2 ) ± sech ( r2 (x − νt)) v1,1 (x, t) = 8(h2 + f2 + g2 ) 8(f2 + g2 + h2 )

1

2

eiφ(x,t) . (26)

ro

of

(i1.2) : When r2 < 0 and r4 > 0, there are singular periodic solutions 1 
2 iφ(x,t) √ 3(d1 + c1 ) 3(c1 + d1 ) e , (27) q1,2 (x, t) = ± sec −r2 (x − νt) 8(h1 + f1 + g1 ) 8(f1 + g1 + h1 )

1
2 iφ(x,t) √ 3(c2 + d2 ) 3(d2 + c2 ) ± sec −r2 (x − νt) v1,2 (x, t) = e , (28) 8(h2 + f2 + g2 ) 8(f2 + g2 + h2 )

-p



and


√ 3(c1 + d1 ) 3(d1 + c1 ) ± csc −r2 (x − νt) q1,3 (x, t) = 8(h1 + f1 + g1 ) 8(f1 + g1 + h1 )

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re



1

eiφ(x,t) , (29)

1

eiφ(x,t) . (30)


√ 3(d2 + c2 ) 3(c2 + d2 ) v1,3 (x, t) = ± csc −r2 (x − νt) 8(h2 + f2 + g2 ) 8(f2 + g2 + h2 ) case 2

ur na



r1 = r3 = r5 = r6 = 0, q

3(c1 +d1 )

• (i) a−1 =

− r2r

3(c1 +d1 )

ρ−1 =

− r2r

4

16(f1 +g1 +h1 )

• (ii) a−1 = 0, ρ−1 = 0,

4

16(f1q +g1 +h1 )

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,

ρ0 =

−

2r4 r2

8(hq 1 +f1 +g1 )

3(d2 +c2 )

−

2r4 r2

8(h2 +f2 +g2 )

q

2r4 r2

32r2 (h q1 +f1 +g1 ) 2r4 r2

,

,

a1 = 0, ρ1 = 0,

a=

q

a1 = ±

a0 =

2

8(fq 1 +g1 +h1 )

3(c2 +d2 )

ρ1 = ±

8bνr2 (fm +gm +hm )−3(cm +dm )2 . 8r2 (fm +gm +hm ) 16r2 k 2 bν(fm +gm +hm )−3(cm +dm )2 (2k 2 +r2 ) 16r2 (fm +gm +hm )(bk−1)

,ω =

r

− r4 2

8(f2 +g2 +h2 )

3(d1 +c1 ) 8(h1 +f1 +g1 ) ,

16r2 k 2 bν(fm +gm +hm )−3(cm +dm )2 (2k 2 +r2 ) 16r2 (fm +gm +hm )(bk−1)

r

− r4

3(c1 +d1 )

,

ω=

,

a=

8bνr2 (fm +gm +hm )−3(cm +dm )2 . 8r2 (fm +gm +hm )

q

3(c1 +d1 )

a1 =

2r4 r2

16(f q1 +g1 +h1 ) r − r4 2

,

3(c2 +d2 ) 3(d2 +c2 ) , ρ = 1 8(h2 +f2 +g2 ) 8(f2 +g2 +h2 ) , 64r2 k 2 bν(fm +gm +hm )−3(cm +dm )2 (4k 2 +5r2 ) 32bνr2 (fm +gm +hm )+3(cm +dm )2 , ω= . 16r2 (fm +gm +hm ) 64r2 (fm +gm +hm )(bk−1) 3r2 (d2 +c2 )

ρ−1 = a=

3(d2 +c2 ) 8(h2 +f2 +g2 ) ,

q

3r2 (d1 +c1 )

• (iii) a−1 =

ρ0 =

2

r22 4r4

3(d1 +c1 ) 8(h1 +f1 +g1 ) ,

a0 =

,

3(d1 +c1 )

a0 =

r0 =

2

32r2 (h2 +f2 +g2 ) ,

ρ0 =

6

q

3r2 (d1 +c1 )

• (vi)a−1 = −

a=

−

2r4 r2

32r2 (h q1 +f1 +g1 )

3r2 (d2 +c2 )

ρ−1 = −

2r − r4 2

a0 =

,

ρ0 =

,

32r2 (h2 +f2 +g2 ) 32bνr2 (fm +gm +hm )−3(cm +dm )2 , 16r2 (fm +gm +hm )

q

3(c1 +d1 )

3(d1 +c1 ) 8(h1 +f1 +g1 ) ,

a1 =

−

2r4 r2

16(f 1 +g1 +h1 ) q

,

r − r4 2

3(c2 +d2 ) 3(d2 +c2 ) , ρ = 1 8(h2 +f2 +g2 ) 8(f2 +g2 +h2 ) , 32r2 k 2 bν(fm +gm +hm )−3(cm +dm )2 (4k 2 +5r2 ) ω= . 32r2 (fm +gm +hm )(bk−1)

The solutions of (i), (ii), (iii) and (vi) of this case are in the form (i2.1): When r2 > 0 and r4 > 0, they are singular periodic solutions as r

2r4 cot r2

r

  12 r r r2 r2 r2 eiφ(x,t) , (x − νt) + a0 + a1 (x − νt) tan 2 2r4 2 (31)



r

2r4 cot r2

r

  21 r r r2 r2 r2 (x − νt) + ρ0 + ρ1 (x − νt) tan eiφ(x,t) . 2 2r4 2 (32)

ro

v2,1 (x, t) = ρ−1

-p

q2,1 (x, t) = a−1

of



re

(i2.2): When r2 < 0 and r4 > 0, Eq.(2) Eq. (3) have dark-singular combo optical solitons as solutions, r  r  12 r2 −r2 r2 (x − νt) + a0 + a1 tanh (x − νt) q2,2 (x, t) = a−1 eiφ(x,t) , 2 2r4 2 (33) r r    21 r r r2 −r2 r2 2r4 eiφ(x,t) . v2,2 (x, t) = ρ−1 − (x − νt) + ρ0 + ρ1 (x − νt) coth tanh r2 2 2r4 2 (34) case 3

r

2r4 coth − r2

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r

ur na



r3 = r4 = r5 = r6 = 0

0 (c1 +d1 ) a−1 = − 2r3r , 1 (f1 +g1 +h1 )

a0 = 0,

a1 = 0,

ω=

4r12 k 2 bν(fm +gm +hm )+3(cm +dm )2 (4k 2 r0 +r0 r2 ) 4r12 (fm +gm +hm )(bk−1) 2 bνr1 (fm +gm +hm )+3r0 (cm +dm )2 . r12 (fm +gm +hm )

0 (c2 +d2 ) , ρ0 = 0, ρ1 = 0, a = ρ−1 = − 2r3r 1 (f2 +g2 +h2 ) The solutions of this case are singular optical soliton in the form

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case 4

q3 (x, t) =

"

v3 (x, t) =

"

√

4r2 r0 2r2

√

4r2 r0 2r2

a−1 √ sinh[ r2 (x − νt)] − ρ−1 √ sinh[ r2 (x − νt)] −

r0 = r1 = r2 = r5 = r6 = 0

7

r1 2r2

#1

eiφ(x,t) ,

(35)

r1 2r2

#1

eiφ(x,t) .

(36)

2

2

• (i) a−1 = 0,

a0 = 0,

ρ−1 = 0,

ρ0 = 0,

• (ii) a−1 = 0,

a0 =

ρ−1 = 0,

ρ0 =

4 (c1 +d1 ) a1 = − 2r3r , 3 (f1 +g1 +h1 )

4 (c2 +d2 ) , ρ1 = − 2r3r 3 (f2 +g2 +h2 )

(c1 +d1 ) 2(f1 +g1 +h1 ) , (c2 +d2 ) ρ1 2(f2 +g2 +h2 ) ,

k 2 [bνr32 (fm +gm +hm )+3r4 (cm +dm )2 ] r32 (fm +gm +hm )(bk−1) bνr32 (fm +gm +hm )+3r4 (cm +dm )2 . r32 (fm +gm +hm )

ω=

a=

r4 (c1 +d1 ) 2r3 (f1 +g1 +h1 ) , r4 (c2 +d2 ) a 2r3 (f2 +g2 +h2 ) ,

a1 = =

12r32 k 2 bν(fm +gm +hm )+(cm +dm )2 (4k 2 r4 −3r32 ) 12r32 (fm +gm +hm )(bk−1) 3bνr32 (fm +gm +hm )+3r4 (cm +dm )2 . 3r32 (fm +gm +hm )

ω= =

In this case, the plane wave solutions are obtained and are given by 1

eiφ(x,t) ,



1

eiφ(x,t) ,

r0 = r1 = r5 = r6 = 0

a0 =

ρ−1 = 0,

ρ0 =

(c1 +d1 ) 2(f1 +g1 +h1 ) , (c2 +d2 ) 2(f2 +g2 +h2 ) ,

4 (c1 +d1 ) a1 = − 2r3r , 3 (f1 +g1 +h1 )

4 (c2 +d2 ) ρ1 = − 2r3r , 3 (f2 +g2 +h2 )

ω=

a=

4r32 k 2 bν(fm +gm +hm )+3(cm +dm )2 (4k 2 r4 −r2 r4 ) 4r32 (fm +gm +hm )(bk−1) 2 bνr3 (fm +gm +hm )+3r4 (cm +dm )2 . r32 (fm +gm +hm )

lP

a−1 = 0,

re

case 5

(38)

-p

with r4 > 0.

2

(37)

ro

4ρ1 r3 v4 (x, t) = ρ0 + 2 r3 (x − νt)2 − 4r4

2

of



4a1 r3 q4 (x, t) = a0 + 2 r3 (x − νt)2 − 4r4

ur na

Using the coefficients given in (i) and (ii), the solutions of Eq. (2) and Eq. (3) are [13]: i6.1: When r32 = 4r2 r4 , dark-singular combo solitons are obtained and given by

















r2 q6,11 (x, t) = a0 − a1 r3

r2 v6,11 (x, t) = ρ0 − ρ1 r3

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1 + ǫ tanh

1 + ǫ tanh

r

 12 r2 (x − νt) × eiφ(x,t) , 2

(39)

r

 12 r2 (x − νt) × eiφ(x,t) , 2

(40)

r

 12 r2 × eiφ(x,t) , (x − νt) 2

(41)

r

 12 r2 × eiφ(x,t) . (x − νt) 2

(42)

and

r2 q6,12 (x, t) = a0 − a1 r3

r2 v6,12 (x, t) = ρ0 − ρ1 r3

1 + ǫ coth

1 + ǫ coth

i6.2: When r32 > 4r2 r4 and r2 > 0, singular solitons as solutions are

8

#1 √ 2 2r2 sech( r2 (x − νt)) q6,2 (x, t) = a0 + a1 √ × eiφ(x,t) , √ ǫ ∆ − r3 sech( r2 (x − νt)) #1 " √ 2 2r2 sech( r2 (x − νt)) × eiφ(x,t) . v6,2 (x, t) = ρ0 + ρ1 √ √ ǫ ∆ − r3 sech( r2 (x − νt)) "

(43)

(44)

i6.3: When r32 > 4r2 r4 and r2 < 0, combination of singular soliton and periodic singular wave as solutions are 1 √ 2 2r2 sec( −r2 (x − νt)) q6,31 (x, t) = a0 + a1 √ × eiφ(x,t) , √ ǫ ∆ − r3 sec( −r2 (x − νt))

(45)

of



1 √ 2 2r2 sec( −r2 (x − νt)) v6,31 (x, t) = ρ0 + ρ1 √ × eiφ(x,t) , √ ǫ ∆ − r3 sec( −r2 (x − νt))

(46)

1 √ 2 2r2 csc( −r2 (x − νt)) × eiφ(x,t) , q6,32 (x, t) = a0 + a1 √ √ ǫ −∆ − r3 csc( −r2 (x − νt))

(47)

ro



and periodic singular waves as

-p



1 √ 2 2r2 csc( −r2 (x − νt)) v6,32 (x, t) = ρ0 + ρ1 √ × eiφ(x,t) . √ ǫ −∆ − r3 csc( −r2 (x − νt))

re



(48)

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i6.4: When r32 < 4r2 r4 and r2 > 0, the solutions are singular solitons as #1 √ 2 2r2 csch( r2 (x − νt)) q6,4 (x, t) = a0 + a1 √ × eiφ(x,t) , √ ǫ −∆ − r3 csch( r2 (x − νt))

(49)

#1 √ 2 2r2 csch( r2 (x − νt)) √ × eiφ(x,t) . v6,4 (x, t) = ρ0 + ρ1 √ ǫ −∆ − r3 csch( r2 (x − νt))

(50)

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case 6 r2 = r4 = r5 = r6 = 0 0 (c1 +d1 ) a−1 = − r13r a0 = 0, (f1 +g1 +h1 ) , 0 (c2 +d2 ) ρ−1 = − r13r (f2 +g2 +h2 ) ,

ρ0 = 0,

k 2 [r12 bν(fm +gm +hm )+3r0 (cm +dm )2 (4k 2 r4 −r2 r4 )] r12 (fm +gm +hm )(bk−1) bνr12 (fm +gm +hm )+3r0 (cm +dm )2 3r4 (c2 +d2 ) − 2r3 (f2 +g2 +h2 ) , a = . r12 (fm +gm +hm )

a1 = 0, ρ1 =

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ω=

The solutions of this case are the Weierstrass elliptic function solutions  1 2 a−1     √ q6 (x, t) = × eiφ(x,t) , r3 k(x−νt) ℘ ; g2 , g3 2 

1

(51)

2

ρ−1

  × eiφ(x,t) . v6 (x, t) =   √ r3 k(x−νt) ; g2 , g3 ℘ 2

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(52)

4r0 1 where ℘ is the Weierstrass elliptic function and g2 = − 4r r3 , g3 = − r3 , r3 > 0.

case 7 r4 = r5 = r6 = 0 0 (c1 +d1 ) a−1 = − r13r a0 = 0, (f1 +g1 +h1 ) ,

0 (c2 +d2 ) ρ−1 = − r13r (f2 +g2 +h2 ) ,

ρ0 = 0,

4k 2 r12 bν(fm +gm +hm )+3r0 (cm +dm )2 (4k 2 −r2 ) 4r12 (fm +gm +hm )(bk−1) bνr12 (fm +gm +hm )+3r0 (cm +dm )2 3r4 (c2 +d2 ) . − 2r3 (f2 +g2 +h2 ) , a = r12 (fm +gm +hm )

a1 = 0, ρ1 =

ω=

The solutions of this case are the Weierstrass elliptic function solutions 

1



1

2

2

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ρ−1   × eiφ(x,t) . v7 (x, t) =   √ r3 k(x−νt) ℘ ; g2 , g3 2

(53)

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a−1   × eiφ(x,t) , q7 (x, t) =   √ r3 k(x−νt) ℘ ; g2 , g3 2

(54)

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4r0 1 where ℘ is the Weierstrass elliptic function and g2 = − 4r r3 , g3 = − r3 , r3 > 0.

Summary

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In this work, many type of solutions have been obtained. The solutions given by the Weierstrass elliptic function have no particular application in optics. However, bright, dark, singular optical soliton solutions and some forms of combo optical soliton solutions could be applicable to optical fibers.

This paper retrieves bright, dark and combo optical soliton solutions to the LakshmananPorsezian-Daniel model for polarization mode dispersion fibers by means the modified extended direct algebraic scheme. These solutions are very useful in optics precisely for the governing model. In the future work, the effects of some parameters of this model on modulational instability will be presented.

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References

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[3] S. T. R. Rizvi, K. Ali, U. Akram, M. Younis, Analytical study of solitons for Lakshmanan-Porsezian-Daniel model with parabolic law nonlinearity, Optik 168 (2018) 27-33. [4] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S. P. Moshokoa, M. Belic, Optical solitons for Lakshmanan-Porsezian-Daniel model by modified simple equation method, Optik 160 (2018) 24-32. [5] H. Rezazadeh, M. Mirzazadeh, S. M. Mirhosseini-Alizamini, A. Neiramed, M. Eslami, Q. Zhou, Optical solitons of Lakshmanan-Porsezian-Daniel model with a couple of nonlinearities, Optik 164 (2018) 414-423.

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[9] M. Ekici, Optical solitons in birefringent fibers for Lakshmanan-PorsezianDaniel model by extended Jacobi’s elliptic expansion scheme, Optik 172 (2018) 651-656.

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