- Email: [email protected]
Optical solitons to Biswas-Arshed model in birefringent fibers using modified simple equation architecture
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical solitons to Biswas-Arshed model in birefringent fibers using modified simple equation architecture
T
Yakup Yildirim Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059 Bursa, Turkey
ARTICLE INFO
ABSTRACT
Keywords: Optical solitons Modified simple equation architecture Biswas-Arshed equation Birefringent fibers
The Biswas-Arshed equation in birefringent fibers is considered to get optical solitos of its in this paper. Optical dark, bright and singular solitons are given by modified simple equation approach. Also, singular periodic solutions are located.
1. Introduction Solitons in optics are pulses or molecules that compose of the basic fabric for the purpose of soliton transmission technology along with optical fibers, telecommunications industry, data transmission across, transcontinental and transoceanic distances which are constituted by the aid of a lot of models such as complex Ginzburg–Landau equation, Fokas-lenells equation, Kaup–Newell model, Lakshmanan–Porsezian–Daniel model, Kundu–Eckhaus equation, Radhakrishnan–Kundu–Lakshmanan equation, Schrödinger–Hirota model, Gerdjikov–Ivanov equation [17–28]. Biswas-Arshed model also is one of these governing models and has been employed along with strategic algorithms that are traveling wave hypothesis, exponetial function, extended trial equation procedure, complex amplitude ansatz procedure, trial equation technique, modified simple equation algorithm [1–6] where the solitons along the one or two component of the model are contemplated. The most striking part of this model is that self-phase modulation is neglected and also group velocity dispersion is negligibly small. Another remarkable feature of the model is that both spatio-temporal dispersions of second-order and third-order are contained in the model in the cause of the low group velocity dispersion to be compensated. 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 in birefringent fibers without four-wave mixing terms (4WM) is employed in this paper through the medium of modified simple equation procedure that discovers highly important solitons with the inclusion of the existence of circumstances for these important solitons. In addition, this method also provides additional solutions namely singular-periodic solutions by the reverse form of the constraints. The analysis of the model will be given in detail in the rest of the article. 2. Succinct pen-picture of modified simple equation architecture Quick recapitulation of modified simple equation architecture [7–16] is presented in this section before its possibility to any model for the purpose of providing very valuable optical solitons. Step-1: A nonlinear evolution equation in general form is going to be taken into account of
( , t,
x,
tt ,
xt ,
xx ,
…) = 0
and subsequently the substitution of the wave transformation https://doi.org/10.1016/j.ijleo.2019.02.013 Received 2 February 2019; Accepted 2 February 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
(1)
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
(x , t ) =
(2)
( )
with the inclusion of
=x
(3)
t
comes up the following ordinary differential equation:
( ,
,
,
(4)
, …) = 0.
In (4), the dependent function is shown as = ( ) while a polynomial containing dependent function and its derivatives is referred to as . Step-2: The most critical step of this technique is the ancillary function which will be given by N
( )=
i i=0
Q( ) Q( )
i
(5)
along with the necessary constant coefficients shown as 0, 1, …, N . It should be emphasized that these constant coefficients have to be rescued to get crucial optical solitons of Eq. (1). Step-3: The utilization of Eq. (5) is based on the existence of the N number which is obtained for the sake of the performing of the balancing condition in (4). Step-4: An overdeterminet equation system is recovered in consequence of putting the ancillary function (5) into the ordinary differential equation (4) and subsequently setting of the coefficient of the same functions namely Q i ( ), i = 1, 2, … to zero. The essential constant coefficients are emerged from the solution of the equation system, for this reason, the very valuable optical solitons of the model (1) are imparted. 2.1. FORM-1 The Biswas-Arshed model [1–6] is yield as
i
t
+ a1
xx
+ a2
xt
+ i (b1
xxx
+ b2
= i { (| |2 )x + µ (| |2 )x
xxt)
+
| |2 x}.
(6)
The existence of group velocity dispersion with spatio-temporal dispersion sequentially stem from the coefficient of a1 and a2 when the first term purports the temporal evolution of pulses in this model. The complex valued function (x , t ) implies to the wave profile. The self-steepening and nonlinear dispersions are ensured by the coefficient of , µ , sequentially. Once and for all, the existence of third order dispersion and third order spatio-temporal dispersion are ensured with the coefficient of b1, b2 sequentially. The Biswas-Arshed model without FWM in birefringent fibers is given as
i
t
+ a1
xx
+ b1
+ (µ1 (| |2 ) x +
i
t
+ a2
xx
1 (|
+ b2
+ (µ2 (| |2 ) x +
xt
xt
2 (|
+ i (c 1
xxx
+ d1
|2 ) x )
+(
1
+ i (c2
xxx
+ d2
|2 ) x ) + (
2
xxt)
= i { 1 (| |2 )x +
| |2 + xxt )
| |2 +
1
|2 )x
| |2 ) x},
= i { 2 (| |2 )x + 2
1 (|
2 (|
|2 ) x (7)
| |2 ) x }.
In the same way, the first terms are referred to as the temporal evolution of pulses when the existence of group velocity dispersion and spatio-temporal dispersion sequentially are supplied by the coefficient of aj and bj in this equation. The complex valued functions (x , t ), (x , t ) are referred to as the wave profile. The nonlinear terms that signifies self-steepening and nonlinear dispersions are ensured by the coefficient of j , j µj , j , j , j sequentially. Once and for all, the existence of third order dispersion and third order spatio-temporal dispersion are ensured with the coefficient of cj, dj sequentially. The modified simple equation algorithm is employed in the cause of discovering critical solitons of the Biswas-Arshed model (7) in this section, for this reason, the following solution condition
(x , t ) = P ( ) e i
(8)
(x , t )
with
=x
(9)
t
is going to be taken into account. In Eq. (8), the function P ( ) along with the speed of the soliton implies to the amplitude component whilst the function (x , t ) means the phase component and lastly this function will be presumed as
(x , t ) =
(10)
x+ t+ .
In Eq. (10), the parameter corresponds to soliton frequency when the parameter purports soliton wave number and also the parameter is referred to as soliton phase. The real component with the imaginary component respectively are emerged from
( 2
dj + 3 cj
dj
bj + aj ) P j + (
3c j
+
2
dj
2a
1150
j
+
bj
) Pj
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
(
j
3 j ) Pj
+
3 j P j˜
2 j Pj P ˜j
=0
(11)
and
(
2
dj + cj ) P j + ( (3
j
dj
2 j ) Pj Pj
+ 2 µj +
2c j
3 2
+2
dj +
bj
2 j P j P j˜
j Pj P j˜ P j˜
2 aj +
bj
) Pj
3 j P j˜ P ˜2j = 0
(12)
on account of putting the solution condition (8) into the model (7) along with j = 1, 2 and ˜j = 3 must be used as
j . The following circumstance (13)
P j˜ = Pj in the cause of using the balancing condition in the real part (11) as well as the imaginary part (12) that can be given by
( 2
dj + 3 cj
(
j
+
+
j
dj 3 j ) Pj
+
j
( 3cj
bj + aj ) P j
2
2a
dj +
bj + ) Pj
j
=0
(14)
and
(
2
dj + cj ) P j + (
(3
j
+ 2 µj +
dj
+2
j
2c j
3 j
+
+2
dj +
+ 3 j)
j
bj
2 aj +
bj
) Pj
P 3j = 0
3
(15)
respectively. In the cause of the amplitude component function Pj holds both the real component (14) and the imaginary component (15), the following result is imparted as 2
dj + 3 cj
dj
bj + aj
3 ( j+
=
dj + cj
j + j + j)
3 j + 2 µj + j + 2 j + j + 3 j
=
3c + 2 d 2a + j j j 3 2cj + 2 dj + bj
2 d j
bj 2 aj +
(16)
bj
as long as
aj =
2(
4 3
4
2
8
2
j dj
j
+6
2
j bj j
+6
6
2
bj
j j
+2
6
2
bj
j
j
3
8
j
2d
1 + µj
j
+4
j
2d
j
j
j d j µj
2
+8
j
2
6
j bj j
6
+6
bj j µj
6
+6
bj µj
9
j µj
9
j
+4
2
2
bj j µj
8
2
bj
j
bj µj 2 + 2
2
bj µj
2
j bj j
bj j µj + 6
2
j
9
µj
j
j j
+6
j
j
+3
j j
+3
j j
+9
j
2
+ 15
j µj
+3
j
+8
3
j
j
j
2d
j
1 + µj
16
j)
3
2 3
j dj µ j
(8
2
bj j µj
j
4
2
bj
+3
3
3
j µj
+ 15
j
2
j
2
4
bj
6 j
j
+6
3
+8
j
j
2
+ 18
+2
2
j bj j
j
2
2b j
+6
bj
j j
+6
j
6
j
bj 2
+6
j
j
2
6
2
bj
j j
bj µj 2 9
j j
9
j µj
+ 12
+ 15
bj j µj j bj j
j j
j
j j
j µj
j µj
+3
j
j
j ), j j dj
3
2b j
j j
j j
2
j d j µj
9
+ 15
j
2
9
j
dj
j
+8
bj j
3
j dj
j
bj
2
6
j j
µj
j dj
j
9
µj 2 + 3 16
2
9
j j
j
j
+2
2d
3
8
+4
j j
bj j µj + 6
+9
2d
2
j bj
6
j j j µj
j
+6
j
9
+6
+ 16
2
9
+3 j
+6
j
j j
18
j
8(
bj
9
j j
j
j bj µj
2
j
j bj
d j µj
4
j
2
j d j µj
j
dj
+2
j bj µj
3
8
j dj 2
3
+8
d j µj 2
4
j
j j dj
2
+8
d j µj
12
3
+4
j d j µj
j bj j
3
8
j
j
2
+3
cj =
j dj 2
8
2d
2
9 j
3
+8
j
dj µ j 2 + 8
9
j
3
(4
2
j j dj
4
+2
j)
+ 16
d j µj 2
3
16
j dj µ j
3
1151
dj µ j
16 j
+8
3
3
j dj
dj
j
j
2
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
+4
2
2
2
bj
j
6
2
bj
j
j
2b j
2
2
2
6
2
2b j
+2
2
+6
4
j
+2
bj
j
+6
bj
j
bj 2
+6
j
j bj j
2
6
2
j j
bj j µj
bj
2
+6
2
j j
2
2
bj
6
j
bj
bj j µj + 6
j j
6
j j
6
j j
6
6
j µj
6
j
j
9
j
2
18
6
j µj
12
+8
j µj
+ 12
j
+4
j µj
j
+6
+
j
2
j
+9
j
j
6
j
µj +6 j
2
j
+ 12
3
j
j
j µj
j
6
j j
2
+6
2
j j j
2
µj + 4
2
j
bj
12
j
9
j
2
j j j
+9
µj
j
bj j µj
+ 12 j
bj j bj
j
j
+2
2
+2
j
j
bj j µj
6
j j
+4
+4
2
j
12
j µj
j
+2
12
2
j
+4 j
j bj µj
bj µj
j µj
+4
j j
j
j
j bj
2
+6
bj µj
bj
2
2
j
j
8
bj µj 2 + 2
4
+6
j j
2
bj j µj + 4
j
j bj µj
bj
2
j bj j
3
j
2
4
bj µj 2
6
j
6
2
2
+8
bj j µj
+2
j j
bj
j bj j
+4
j
j bj j
+6
j j
j bj j
bj
6
j bj j
6
j
+6
2
j j
+ 12
2
+ 18
j
+
j
j j j j
2).
(17)
Under the condition of Eq. (17), the modified simple equation methodology will be implemented in the real component (14). Case-1: The auxiliary equation (5) is given as
P( ) =
0
+
Q( ) Q( )
1
(18)
by virtue of N = 1 that is recovered as a result of balancing P with The following equations is recovered by Q 3 coeff.: 1 (Q
Q
2
)3 (
2
j 1
Q
2 1 j
+
2 1
+
+4
j
dj
6 cj + 2
in Eq. (14).
dj + 2 bj
2 aj ) = 0,
(19)
coeff.:
3 1 Q ((2 1
2 1 j
+
P3
dj
3 cj +
dj +
bj
aj ) Q + (
j 0 1
0 1 j
0 1 j
0 1
j)Q
) = 0,
(20)
coeff.: 1 ((2
dj
+ ( 3cj
2
3 cj +
dj +
2 j 0
dj + 3
bj
aj ) Q
2 0 j
+3
+3
2 0 j
j
0
2 0
+3
j
+
2a j
bj + ) Q ) = 0,
(21)
Q0 coeff.: 3c j 0
2
+
dj
0
j 0
3
0
3
3
j
0
3
j
2a j 0
+
bj
0
0
= 0,
(22)
by virtue of inserting the auxiliary equation (18) into the ordinary differential equation (14) and later on setting of the coefficient of the same functions to zero. As long as solving the above equations, the following results are given by 0
3c j
=±
2 j
1
4
=±
dj
2a
dj +
+
j
bj +
j
+
j
6 cj + 2 j
+
+
dj + 2 bj
+
j
,
j
(23)
j
+
2 aj (24)
j
and
2( 3cj
Q =±
Q
=
2
2
dj
2( 3cj
2
2
dj
2a
dj + 3 cj +
dj + 3 cj +
bj + )
j
dj +
2a j
dj +
bj
aj
bj + ) bj
aj
Q,
(25)
Q.
(26)
1152
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
One can easily reach out to
2
Q =±
dj
3 cj +
2( 3cj
2
dj
3 cj +
dj + 2a
dj +
bj
aj
k1 e
bj + )
j
±
2( 3cj 2 dj + 2aj bj + ) 2 dj 3 c j + dj + bj aj
,
(27)
and
2
Q=
2( 3cj
2
dj + 2a
dj +
bj
aj
bj + )
j
k1 e
±
2( 3cj 2 dj + 2aj bj + ) 2 dj 3 cj + dj + bj aj
+ k2,
(28)
under the condition of employing Eqs. (25) and (26). In consequence of putting Eqs. (27) and (28) with the inclusion of k1, k2 integration constants in Eq. (18), the optical solitons for the model (7) are given as 3c
(x , t ) = ±
2
1
1
± ×
2
d1
2( 3c1
2 2( 3c1
d1
2a 1
d1 + +
3 c1 + d1 + 2 d + 2a 1 1
b1
1 +
a1
k1 e
b1 + )
b1 + 1
+
4
±
2( 3c1 2 d1+ 2a1 b1+ ) (x 2 d1 3 c1+ d1+ b1 a1
±
2( 3c1 2 d1+ 2a1 b1 + ) (x 2 d1 3 c1+ d1+ b1 a1
± 3 c1 + d1 + b1 a1 ke 2 d + 2a b1 + ) 1 1 1
d1
6 c1 + 2 d1 + 2 b1 1 + 1 1 + 1+
1
t)
ei ( t)
2 a1
x+ t+ )
+ k2
(29)
and 3c 2
(x , t ) = ±
2 2
± ×
2
d2
2( 3c 2
2 2( 3c2
d2
3 c2 +
d2 + b2
2 d + 2a 2 2
3 c2 +
2a 2
d2 + +
2
+
a2
k1 e
b2 + )
d2 + b2
a2
ke b2 + ) 1
2 d + 2a 2 2
b2 + 2
±
+
4
±
2( 3c 2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c2 + d2 + b2 a2
6 c2 + 2 d2 + 2 b2 2 + 2 2 + 2 +
t)
2 a2
t)
ei (
2( 3c2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c 2+ d2 + b2 a2
±
d2
2
x+ t+ )
+ k2
(30)
If we set
k1 =
2( 3cj 2
2
dj
2a j
dj + 3 cj +
bj + )
dj +
bj
d1 +
2a 1
aj
e
2( 3c j 2 dj + 2aj bj + ) 0 2 dj 3 cj + dj + bj aj ,
±
k2 = ± 1,
we get:
(x , t ) = ±
× ei (
3c
2
1
1 +
b1 + 1
+
tanh
1
3c
2
1
2(2
d1
d1 + 2a1 b1 + (x 3 c1 + d1 + b1 a1 )
t+
0)
(31)
x+ t+ ) ,
(x , t ) = ±
3c 2
2 2
× ei (
1 +
2a 2
d2 + +
2
+
b2 + 2
+
tanh
2
3c 2
2(2
2
d2
d2 + 2a2 b2 + (x 3 c2 + d2 + b2 a2 )
t+
0)
(32)
x+ t+ ) ,
The consequences (31)-(32) imply to dark soliton solutions on condition that
( 3cj
2
dj +
(x , t ) = ±
2a 3c
1
bj + )(2
j 2 1
× ei (
1
+
3 cj + b1 +
1
+
coth
1
dj +
bj 3c
aj ) > 0. 2
1
2(2
d1
d1 + 2a1 b1 + (x 3 c1 + d1 + b1 a1 )
t+
0)
(33)
x+ t+ ) ,
(x , t ) = ±
3c 2
2 2
× ei (
2a 1
d1 + +
dj
2a 2
d2 + +
2
+
b2 + 2
+
2
coth
3c 2
2(2
2
d2
d2 + 2a2 b2 + (x 3 c2 + d2 + b2 a2 )
t+
0)
(34)
x+ t+ ) ,
1153
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
The consequences (33)-(34) stem from singular soliton solutions on condition that
( 3cj
2
3c
(x , t ) = ±
× ei (
2a
dj +
bj + )(2
j
2
1
1 +
3 cj +
b1 + +
1
dj +
bj 3c
tan
aj ) > 0. 2
1
2(2
1
d1 + 2a1 b1 + (x 3 c1 + d1 + b1 a1 )
d1
t+
0)
(35)
x+ t+ ) , 3c 2
(x , t ) = ±
2
2a 2
d2 + +
2
ei (
2a 1
d1 +
1 +
dj
+
2
b2 + +
2
3c 2
× tan
2(2
2
2
d2 + 2a2 b2 + (x 3 c2 + d2 + b2 a2 )
d2
t+
(36)
x+ t+ ) , 3c
(x , t ) = ±
× ei (
2
1
2a 1
d1 +
1 +
1 +
b1 + +
1
3c
cot
2
1
2(2
1
d1 + 2a1 b1 + (x 3 c1 + d1 + b1 a1 )
d1
t+
0)
(37)
x+ t+ ) , 3c 2
(x , t ) = ±
2
2a 2
d2 + +
2
× ei (
0)
+
2
b2 + +
2
3c 2
cot
2
2(2
2
d2 + 2a2 b2 + (x 3 c2 + d2 + b2 a2 )
d2
t+
0)
(38)
x+ t+ ) ,
The consequences (35)-(38) correspond to singular periodic solutions on condition that
( 3b1
2
2a 1
b2 +
a2 + )(2
b2
3 b1 +
b2 +
a2
a1 ) < 0.
Case-2: The real component (14) is given as
( 2
dj + 3
4 (
j
+
j
cj
+
bj + aj )( (Q ) 2 + 2QQ )
dj
+
j
j )Q
3
0
+
1
Q( ) + Q( )
Q
5
2 6 2 (Q ) (4
dj +
2
(40)
1 2
(Q )3 (( 8
2
+ 18
1 2
dj + 18
24
0 2
12 3
+
+
2 j
2 j
+
2
)2
or QQ with
6 cj + 2
j
Q3
in Eq. (39).
dj + 2 bj
2 aj ) = 0,
(41)
3
cj +
2
+ 3 cj
j
dj +
2
1
dj
1
2
dj
bj
2
bj
2 aj ) Q 1
+( 2
dj
1
j 1 2
1 2 j
(42)
+ aj 1) Q ) = 0,
coeff.:
12
Q
dj
2
1 2 j
Q
+ ) Q2
coeff.:
12 2 (Q ) 4 ((2
4
2a j
+
The auxiliary equation (5) is given as
Q( ) Q( )
2
j 2
3c j
dj +
(39)
1 Q2.
by virtue of N = 2 that is recovered as a result of balancing (Q The following equations is recovered by Q 6 coeff.:
4
2
bj
=0
through the medium of using P =
Q( ) =
4(
j 1
2
dj + 12
dj 2
0 2
2
bj
4
dj
12
2 j 1 2
12
2
2 1
cj 18
1
12
2
bj
1 2
6
3
2
bj
4
2
2
2
1 2 aj ) Q
4
2
2
2
2 3c j
+4
+( 4
2
2
12
j 1
2 j 1 2
+ 36
j 0 2
+ 12
0 2 aj
2
0 2
+3
2a ) Q j
bj + 4
1
2a
2
2 2
12
2
cj + 9 j) Q
+ (36
2 1
dj
cj
4
2
2
12
12
0 2
j 0 2
dj
1 2 2 2a
j
54
+4
j 0 2
dj
2
2
3
1 2
2
bj
12 1
2
cj
j 0 2
2
dj (43)
)=0
coeff.:
2(Q ) 2 (( 6 30
0 2
dj + 9
1 2
cj
6
2 1
1 2
cj + 10
cj 0 2
3
1 2
dj + 2
dj 2 1
3
1 2
dj + 10
bj + 3 0 2
1 2 aj ) Q
bj + 2 1154
2
1
bj
+ (20 10
0 2
0 2 aj
dj + 4 2
2 1
2 1 aj ) Q
dj
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
+( 4 12
2 Q
2
dj
0 1
12
2
2
1 2
dj
4
12
j 0 1 2
bj
0 1
4
2a j
1 2
+4
1 2
j 0 1 2
2
+2
1 2
0 1 aj ) Q
bj
4
3 j 1
dj
0 1 3 j 1
2
12
3 j 1
2
+6
3 j 1
2
j 0 1 2 0 1
cj (44)
) = 0,
coeff.:
( 8
0 2
+4
0 2 aj
4
2 1
2a
j) Q
2
bj + 4
dj +2
1
bj +
1
4
0 2
6
0 1 aj ) Q
bj
dj + 12 Q
0 2
+4
+( 8
2
12
0 1
4 3cj j 0
12
j
2
cj
2 1
2
1
4
2
2
+8
2 0 1
j
bj
aj ) Q
dj
0 2
8
2
12 4
0 2
18
2 1
0
dj 2
dj
1
cj
0 2
+4
2
2
dj
0 1
2 j 0 1
12
2
dj
0 2
dj + 12
)2 + (12
2a )(Q j
1
12
1
1
dj + 2
0 2
0 2 aj
2
bj
cj + 6
0 2
3c j 0 2
Q +( 8
2 0 1 j
12 1
+4
j 0 1 2
+8
Q
3c j
1 2
2 1
3 0 1
2
bj
0 2
cj
4
cj + 6
0 1
8
2a
12
0
1
2 j
4
2
2
0 2
2a
4
2
12
2 j
2 1
dj +
dj + 6
j 0 2
bj
1
)(Q )2 = 0,
dj bj
0 1
2
j 1 0
2
2
j
(45)
coeff.:
2
0 1 ((2
+6
dj
3 cj +
+6
j 0
+6
j 0
dj + +6
j 0
j 0
3c j
+ (4
2
4
dj + 4
2a
4
j
bj
+ 4 ) Q ) = 0,
(46)
Q0 coeff.:
4
3c 2 j 0
4
0
3
+4
2
4
j
dj 0
3
0
2
4
j
2a
4 0
3
j 0
2
+4
4
j
bj 0
2
0
2
4
j 0
3
= 0,
(47)
by virtue of inserting the auxiliary equation (40) into the ordinary differential equation (39) and later on setting of the coefficient of the same functions to zero. As long as solving the above equations, the following results are given by 0
1
= 0,
2
2(2
=
3 cj +
( j+
16( 3cj
=±
dj
2
2a
dj +
dj +
+
j
j
+
bj
aj )
j)
bj + )(2
j
( j+
j
+
j
,
(48)
dj 2 j)
+
3 cj +
dj +
bj
aj )
2
,
(49)
and
4( 3cj
Q =±
Q
=
2
2
dj
4( 3cj
2
2
dj
2a j
dj + 3 cj + 2a
dj + 3 cj +
bj + )
dj +
bj
bj + )
j
dj +
bj
Q,
aj
aj
(50)
Q.
(51)
One can easily reach out to
2
Q =±
dj
3 cj +
4( 3cj
2
dj
3 cj +
dj + 2a j
dj +
bj
aj
bj + )
k1 e
±
4( 3cj 2 dj + 2aj bj + ) 2 dj 3 c j + dj + bj aj
,
(52)
and
Q=
2 4( 3cj
2
dj +
dj + 2a j
bj
aj
bj + )
k1 e
±
4( 3c j 2 dj + 2aj bj + ) 2 dj 3 cj + dj + bj aj
+ k2
(53)
under the condition of employing Eqs. (50) and (51). In consequence of putting Eqs. (52) and (53) with the inclusion of k1, k2 integration constants in Eq. (40), the optical solitons for the model (7) are given as
(x , t ) = ±
16( 3c1
2
d1 +
2a 1
(
1
b1 + )(2 + 1+ 1+
d1 1)
2 2
1155
3 c1 +
d1 +
b1
a1 )
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
4( 3c1 2 d1+ 2a1 b1+ ) (x d1 3 c1+ d1+ b1 a1 2
± 2 d1 3 c1 + d1 + b1 a1 ke 2 d + 2a 4( 3c1 b1 + ) 1 1 1
± ×
4( 3c1 2 d1+ 2a1 b1+ ) (x 2 d1 3 c1+ d1+ b1 a1
± 2 d1 3 c1 + d1 + b1 a1 ke 2 d + 2a 4( 3c1 b1 + ) 1 1 1
2(2
d1 (
2
±
d1
4( 3c1
×
3 c1 + d1 + b1 + 1 + 1 + 1)
1
3 c1 +
d1 + b1
a1
2 d1 3 c1 + d1 + b1 a1 ke 2 d + 2a 4( 3c1 b1 + ) 1 1 1
+ k2
a1 )
ke b1 + ) 1
2 d + 2a 1 1
t)
t)
4( 3c1 2 d1+ 2a1 b1+ ) (x 2 d1 3 c1+ d1+ b1 a1
±
ei (
4( 3c1 2 d1+ 2a1 b1+ ) (x 2 d1 3 c1+ d1+ b1 a1
±
1 2 2
t)
t)
x+ t+ )
+ k2
(54)
and
16( 3c2
(x , t ) = ±
d2
4( 3c2
× 2
d2
2(2
±
d2 (
2
2
d2
2
d2
4( 3c2
b2
a2
b2 +
b2
a2
b2 + )
k1 e
ke ) 1
3 c2 +
d2 + b2
3 c2 +
d2 + b2
a2
a2
ke b2 + ) 1
2 d + 2a 2 2
b2 + )(2 + 2+ 2+
d2 2)
3 c2 +
4( 3c 2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c 2 + d2 + b2 a2
±
d2 +
b2
a2 )
2 2
t)
t)
+ k2
a2 )
ke b2 + ) 1
2 d + 2a 2 2
2
4( 3c 2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c 2 + d2 + b2 a2
±
3 c2 + d2 + b2 + 2 + 2 + 2)
4( 3c 2
×
3 c 2 + d2 + 2 d + 2a 2 2
3 c2 + d2 + 2 d + 2a 2 2
4( 3c 2
2a 2
d2 +
(
2
±
2
4( 3c 2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c2 + d2 + b2 a2
±
ei (
4( 3c 2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c2 + d2 + b2 a2
±
1 2 2
t)
t)
x+ t+ )
+ k2
(55)
If we set
k1 =
4( 3cj 2
2
dj
2a
dj + 3 cj +
bj + )
j
dj +
bj
aj
e
±
4( 3cj 2 dj + 2aj bj + ) 0 2 dj 3 cj + dj + bj aj ,
k2 = ± 1,
we get:
(x , t ) =
× ei (
2
d1 + 2a1 ( 1+ 1+ 1+
b1 + ) 1)
3c
sech2
2
1
2
d1
d1 + 2a1 3 c1 + d1 +
b1 + (x b1 a1
t+
1 2
0)
(56)
x+ t+ ) ,
(x , t ) =
× ei (
2( 3c1
2( 3c2
2
(
2
d2 + 2a2 + 2+ 2+
b2 + ) 2)
3c 2
sech2
2
2
d2
d2 + 2a2 3 c2 + d2 +
b2 + (x b2 a2
t+
1 2
0)
(57)
x+ t+ ) ,
The consequences (56)-(57) signify bright soliton solutions on condition that
( 3cj
2
(x , t ) =
× ei (
dj +
2a
bj + )(2
j
2( 3c1
2
(
1
d1 + 2a1 + 1+ 1+
dj
3 cj +
b1 + ) 1)
csch2
dj +
bj 3c
aj ) < 0. 2
1
2
d1
d1 + 2a1 3 c1 + d1 +
b1 + (x b1 a1
t+
0)
1 2
(58)
x+ t+ ) ,
(x , t ) =
2( 3c2
2
(
2
d2 + 2a2 + 2+ 2+
b2 + ) 2)
csch2
3c 2
2 1156
2
d2
d2 + 2a2 3 c2 + d2 +
b2 + (x b2 a2
t+
0)
1 2
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
× ei (
(59)
x+ t+ ) ,
The consequences (58) and (59) purport singular soliton solutions as long as
( 3cj
2
dj +
d1 + 2a1 ( 1+ 1+ 1+
3 cj + b1 + )
1)
dj +
bj 3c
sec2
aj ) < 0. 2
1
2
d1
d1 + 2a1 3 c1 + d1 +
b1 + (x b1 a1
t+
0)
1 2
(60) 2
(
d2 + 2a2 + 2+ 2+
2
b2 + ) 2)
3c 2
sec2
2
2
d2 + 2a2 3 c2 + d2 +
d2
b2 + (x b2 a2
t+
1 2
0)
(61)
x+ t+ ) ,
2( 3c1
2
d1 + 2a1 ( 1+ 1+ 1+
b1 + ) 1)
3c 1
csc2
2
2
d1
d1 + 2a1 3 c1 + d1 +
b1 + (x b1 a1
t+
0)
1 2
(62)
x+ t+ ) ,
2( 3c2
(x , t ) =
× ei (
2
2( 3c2
(x , t ) =
× ei (
dj
x+ t+ ) ,
(x , t ) =
× ei (
bj + )(2
j
2( 3c1
(x , t ) =
× ei (
2a
2
(
d2 + 2a2 + 2+ 2+
2
b2 + ) 2)
3c 2
csc2
2
2
d2
d2 + 2a2 3 c2 + d2 +
b2 + (x b2 a2
t+
0)
1 2
(63)
x+ t+ ) ,
The consequences (60)–(63) account for singular periodic solutions as long as
( 3cj
2
dj +
2a
bj + )(2
j
dj
3 cj +
dj +
bj
aj ) > 0.
2.2. FORM-2 The Biswas-Arshed model with full nonlinearity [1–6] is given as
i
t
+ a1
+ a2
xx
+ i (b1
xt
xxx
+ b2
xxt)
= i { (| |2n )x + µ (| |2n ) x
+
| |2n
(64)
x }.
This model is yield as the first form (7) as long as full nonlinearity parameter n = 1. The Biswas-Arshed model with full nonlinearity and without FWM in birefringent fibers is imparted as
i
t
+ a1
+ b1
xx
+ (µ1 (| |2n )x + i
t
+ a2
+ b2
xx
+ (µ 2 (| |2n ) x +
+ i (c 1
xt
1 (| xt
|2n )x )
+ i (c2
2 (|
xxx
+ d1
+( xxx
1
+ d2
|2n )x ) + (
2
xxt)
= i { 1 (| |2n ) x +
| |2n + xxt )
1
2
|2n ) x
| |2n ) x},
= i { 2 (| |2n )x +
| |2n +
1 (|
2 (|
|2n )x (65)
| |2n ) x }.
For the sake of recovering a generalized perspective, the modified simple equation technique is employed for the sake of obtaining highly important solitons of Eq. (65) in this section, for this reason, the real component with the imaginary component respectively are emerged from
(2
dj
+ P 2j n + 1
3
cj +
j
+ P j2n + 1
dj + j
bj
aj ) P j
+ Pj P j2˜ n
j
(
+ P j2˜ n + 1
bj + j
2
3c j
dj
2a
j
) Pj
=0
(66)
and
( dj
bj + 3 2cj +
cj ) P j + (2 aj
+ (2 n j +
j ) P j˜
P j2˜ n + (
j
+
j
+ 2n
j
bj
+
2 µj n) P j P j2n
2
dj
+2
2
dj ) P j + P j2˜ n P j
2n 1 nP P j P j˜ ˜j j
j
=0
on account of putting the solution condition (8) into the model (65) along with j = 1, 2 and ˜j = 3 must be used by
(67)
j . The following circumstance (68)
P j˜ = Pj
1157
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
in the cause of using the balancing condition in the real part (66) as well as the imaginary part (67) that can be given by
( 2
dj + 3 (
+
j
cj
+
j
j
dj
bj + aj ) P j
2n + 1 j ) Pj
+
(
2
bj
3c j
dj +
2a j
+
+ ) Pj
=0
(69)
and
(
dj + cj ) P j + ( 2 aj + ( j + 2n j +
j
+
+
j
3 2cj
bj + 2n
j
+
+ 2 µj n + 2
j
2
bj + j n)
dj + 2
dj ) Pj
P j2n + 1 = 0
2n + 1
(70)
respectively. In the cause of the amplitude component function Pj holds both the real component (69) and the imaginary component (70), the following result is imparted as 2
dj
3
cj +
dj + bj
dj
aj
(2n + 1) ( j +
=
cj
j+
=
j + j + j)
j + 2n µj + 2 n j + (2 n + 1) j + (2n + 1) j 2 d + 3c + 2a + bj j j j 2 d bj + 3 2cj + bj j
2 aj
2
(71)
dj
as long as
1
aj =
4 2n2 (
+ 4 3n2
j
2d
j
j
3n2
8
j
2n2
j
2n2
8
2n2
j dj
+4
2n2
j bj j
+4
2n2
+4
2n2
bj j µj
4
2n2
+2
2n
j bj j
2
2n
bj
j j
2
2n
bj
j
+8
j
dj µ j
j bj µj
2
2n2
bj
+4
2n2
bj j µj
bj
j
j
2n
+2
2n
bj j µj
4
2n
+ 8 n2 bj j µj
bj µj
12 n2
j bj µj
4 n2 bj
j
8 n2
j j
+ 2 n2
j
+ 2 n bj
+ 2 n2
j
j
6 n2
2
2 n
j j
2
3 n
j
8 n
j j
+ 6 n2
j
2
2 n2
j j
2 n2
j
n
j
3 n
2 n2 2 n2 j
j
j bj j
2 n
2
j j
+ 6 n2 µj 2
j
2
2n2
4
j
j j j
2 n2 µj
2
j
2n2
j
bj bj
2n2
4
j j j
+2
2n
bj j µj
2
2n
bj
j
2b j
+ 8 n2
2 n2 bj
2
j j
6 n2
j µj
6 n2
j µj
2
j bj j
+ 2 n bj
j
3 n
j j
+ 10 n2 j j
2
j bj
2n
3 n
j µj
2 n
j µj
2 n2
1158
j
j j
4 n2
j
2 n2 j j
j j
j
2
j
2 n2
j
+ 4 n2 j
j
2
j
j
j
3 n
j j
2
3 n µj
j
j
2
j j
4 n
j
j
6 n2 µj
j
3 n j
j
j j
j
j
2
j bj j
+ 2 n bj
j
+ 12n2
2
bj
+ 4 n2 bj
j
j j
j j
j µj
j
bj j µj
2 n bj j µj + 2 n bj 2
j
j j
4 n2 j j
j bj j
bj
bj
6 n2
j
5 n
j j
2n
2
j dj µj
2n2
2
+2
2 n2
j µj
2
2n2
4
2
j j
3 n
+ 10 n2
2n2
j bj j
2
j
+4
+ 4 n2
2 n j
2b j
j
j
4 n2 bj j µj + 4 n2 bj
j
6 n2
j
2n2
+ 4 n2 bj
j j
j
4 n
2
6 n2
2 n
j
j
j
+8
2
j
dj
j
bj µj 2 j
j
2d
j j
2n2
+2
j
bj
bj
j
j
+2
2n
j bj
+ 8 n2
2
2
5 n j
j
j
j bj j
j
2n2
j bj
4 n2
j
4
j dj
3n2
+4
j
j
2n
6 n2
j
8 3n2 dj µj
j
8 3n2
j dj µ j
+2
2 n2
j j
j j
5 n
+ 4 n2
dj
2n2
4
2
2 n bj µj
+ 10 n2
j µj
j
+ 8 3n2
j dj
2n2
4
j
2 n bj j µj + 4 n bj
j j
5 n
j µj
bj
2 n2 bj
j
2 n2
j
j
j
2n2
+8
4 n2 bj j µj + 4 n2 bj
j
2 n2
2 n bj j µj + 2 n bj
bj
+ 8 n2
j µj
+ 2 n bj
j bj j
2n
2
j
6 n2 bj µj 2 + 8 n2 bj µj j
j d j µj
j
j j dj
+ 4 3n2 dj µj 2
j
2n2
+2
2n
8 3n2
d j µj 2 + 8
j bj j
+2
2d j
j dj
2n2
8
2n
j
j
3n2
+8
+2
j bj j
6 n2
(4 3n2
2
j j dj
2n2
4
j
j)
j dj µ j
4
4 n2
2d
+ µj
n
j
j
2
j j j
2
+ 10 n2 j
2
2
j µj
j j
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
2
j
j
+ 3n
j j
+ 5n
j µj
+2
+ 3n
2
2
j
2
j
j
8 3n2 (
j
j
2d
+ 8 3n2
j
+4
2n2
j
4
2n2
bj
j j
4
2n2
bj
j
2b j
j
j bj
+2
2n
bj j µj
2n
2
j
2b j
+ 4 n2 bj
j j
+ 4 n2 bj
j
4 n2
j
2 n
j bj j
j
+ 8 n2
2 n
j 2
j j
j
2 n
j j
+ 4n
j
j j
j µj
+ 8 n2
j µj
+ 4 n2
2
j j
2
j
j
j
j µj
j
j
+
j
j
j
2
2n
bj
2
2
+ 8 n2
2
j j
j j
+ 8n
+2
+2
j
+
bj
4 n2
2n
+2
j j j
+2
n
j
j
j
j
+ 4n
2
+2
j
+
j
j
2n
j bj j
4 n2
j µj
4 n2
j
4 n2 µj
+ 2 n bj
j
2
2 n
j j j
2
8 n
2
j
2
j j
+ 8 n2
2
j
j j
+ 4n
+2
j j
j
j µj
j
2
2
j
j
+ 4n
j j
j µj
+2
+ 4n j
j
j
j
j
j j
4 n2
j µj
2 n
2
6 n
2
6 n
j bj j
j
j
j j
j
+ 4n 2
j
j µj
2 j
j
j j j
+ 8 n2
+ 4n j
j bj
2 n bj µj
j
j j
j
2
4 n2 bj j µj
+ 8 n2
+ 4n
j
2 n bj j µj
j 2
bj
j bj j
j µj
2
j
+
j
j
j
2n
j j
2 n
+ 4 n2
j
j bj j
4 n2
j j
+ 4 n2
2
bj j µj
+ 4 n2
2 n
2 n
j bj
2n2
bj
j
j
2n2
2n
2
8 n2
j
j j
2 n
j
j j
+ 4n
j
j
2n
j
j j
2 n bj j µj + 2 n bj
j
+ 8 3n2 dj
+2
j bj µj
j
j
+4
4
bj µj
+ 2 n bj
j j
j
bj j µj
4 n2
j j
j
j dj
4
j
4 n2 bj j µj + 4 n2 bj
4 n2
2
bj
8 n2
j bj j
j
16 3n2
j bj µj
2n2
2n
+2
j
j
j j j j
j
2n
2 n µj
j
2
j
j
+
j
+ 4n
+ 4n
+ 4 n µj
+2
2
+2
4 n
j
j
6 n
j
+ 3n
j dj µ j
2n2
+8
j bj j
2 n
j j
4 n
j µj
+ 4n j
2 n
j j
2 n
2
+ 4n
j
j bj j
2n
4 n2
j µj
2 n bj j µj + 2 n bj
j
2
j bj
j j
j
+ 4 n2 µj 2 j
4 n2
j µj
16 3n2 dj µj
+2
j bj j
+ 2 n bj
2 n
2
4 n2
2
j j
+ 16 3n2
j j dj
4
2
4 n2 bj µj 2 + 4 n2 bj µj
j
+2
j j
bj j µj
+ 4 n2 bj j µj
j j
+2
+ 5n
j j
+ 5n
j j
2n2
bj j µj
4 n2
j bj j
j
4 n
bj
2n
+2
j
j
4 n2
j j
2n
2
+ 8n
2n2
+4
2
j j
6 n
j bj j
j j
4 n2
j
2n2
j
2
2
2
+ 5n
j j
+ 8 3n2 dj µj 2
j
bj µj
j
+ 2 n bj
j dj
2n2
j
j
3n2
4
j
j
16 3n2
j
bj µj 2
4 n2
+ 4 n bj
2d
bj j µj + 4
+ 4 n2 bj j
j
2n2
bj
+ 4 n2
+
4
bj
2n
2
2
j j
bj
2n2
+4
j
j
+ 16
+4
+ 4n
j
j
+ 5n
j j
2 ),
(8 3n2
2
j bj j
2n2
4
j
j d j µj
2n2
4
2n
j
j)
16 3n2
j
+2
4 n2
+ µj
+ 5n
n
j
+
j
2
2n
+ 5 n µj
j
+2
j
j
j µj
1
cj =
+2
j
+ 5n
j j
+ 3n
+
j
j
j
j
j j
+ 4n
j µj
2
2).
(72)
Under the condition of Eq. (72), the modified simple equation methodology will be implemented in the real component (69) that can given as
( 2
dj + 3
4n2 (
j
+
cj j
+
dj j
+
bj + aj )((1 j )Q
3
through the medium of using P =
Q( ) =
0
+
1
Q( ) + Q( )
2
2n)(Q ) 2 + 2nQQ )
4n2 (
= 0.
1 (Q) 2n .
bj
2
dj +
3c j
+
2a j
+ ) Q2 (73)
The auxiliary equation (5) is given as
Q( ) Q( )
2
(74) 1159
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
by virtue of N = 2 that is recovered as a result of balancing (Q )2 or QQ with Q3 in Eq. (73). The following equations is recovered by Q 6 coeff.: 2 6 2 (Q ) (
4
n2
+ n dj + n bj
Q
5
3 cj
Q
+2
2 aj ) Q
+2
1
dj + 2 n
2
2
2
2
dj
+ ( 20 n dj
8
dj
4 n2
+ 2n +
1
dj + 2 n
dj
12 n
cj
2
n2
bj
aj ) = 0,
4
2
+ 2 n dj
j
dj
2
3
1
cj + 4 n
2
n
+ 3 n2
j 1 2
bj
1
2
1
2 2
2 1
2
1 2
+ 30 ncj
2
2
2
2
bj
cj
3 ncj + 2
dj (75)
dj
2
n
bj + 6
2
+ 3 n2
j 1 2
2 n 1 aj +
2
dj + 4 n
16
1 2) Q
2 j 1 2
2
dj
2
2n
1
1 2
j 0 2
2a
2
cj + n 2 aj
2
+ 4n
j 1 2
dj +
1
2
2
bj
1
4n
2a j
+4
10 n dj
1 2
2
bj
1
2
2
2
2a ) Q j
2
dj
1
1 aj ) Q
2 j 1 2
6n
2
2
dj
1
cj
2
2
bj
(76)
) = 0,
2
2 2a
+ ( 8n
dj + 4
+ 12 n2
2
bj
1 2
j 0 2
12 n 2 1 aj
j
2
2
dj + 12 n 2
2
+ 12 n2 2 1
6n
+ 10 naj
1 2
2 1
bj + 4 n
0 2
2
2
cj
)2
2a )(Q j
+ 24 cj
0 2 aj
+ 24 n
2
4
dj + 12 n
0 2
2 j 1 2
+ 4 n2
Q
10 n bj
cj + 12 n
0 2
+ 12 n2
2 3c j
3
j
2
+ 12 n2
j 0 2
dj + 4 n2
1
2
cj + 4 n
Q + ( 36 n
+ 12 n2
2
2 2
12
bj + 12 n2
dj + 4 n2
1
2
1 2
bj + 2
1
2
+ 8 aj
1 2
2
4n
+ 12 n2
dj + 2 n
bj +
dj
4 n2
dj
j 0 2
2
2
2 2
4n
1 2
dj
2 j 1 2
cj
dj
0 2
(77)
cj )(Q ) 2) = 0,
coeff.:
2Q ((6 n dj
+ 6 ncj + 2 bj
+n
1
dj
2a
3 j 1
4 n2
1 2
+ 4 n2
1 2
2 1
dj
1
+ 2 n2
2
dj
4 n2
3c j
+ 4 n2
2a
bj +
3 j 1
1
j) Q
1 2
dj
2n
0 2
bj
10 n
Q + ( 2n
bj + 12 n2
2 1
2 1
n
+ 2 naj
1 2
bj + 3
+ 4 n2
6n
1 2)Q
dj + 30 n
j 0 1 2
j 0 1 2
3 naj
1 2
6 cj
0 1 aj
+ 12 n2
j 0 1 2
2a )(Q )2) j
2 n bj
0 2
dj
+ 12 n2
1 2
1 2
2 1
n
+ 3 n bj
1 2
1 2
)2 + ( 20 n
dj
0 2
+ 3 n dj
1 2
2 n dj
1 2
1 2 )(Q
10 n 2
j
+ 2 n2
dj
2 aj
1 2
9 ncj
1 2
+4
1 2
2 1
2
2
dj +
+ 3 n2
j 1 2
8 bj
1 2
+ 12 n2
Q
+
2 j
coeff.:
+8
3
naj +
dj + 3 n
2
+ (3 n2
(Q ) 2 ((8 n
Q
n2
+
2 j
coeff.:
4 2 (Q ) 4 (( 2 n
4
n2
+
j 2
0 1
2
1 2
+ 12 n2
3 j 1
j 0 1 2
cj + 2 n
+2
cj + 10 n
+ 2 n2
1 2
+ ( 4 n dj
cj + 3 n
0 2
1
Q
dj 1
2
1 2
cj
0 2 aj
+ 2 n2
+ 4n
dj + 2 n
0 1
1 2
3 j 1
dj
0 1
bj
0 1
(78)
= 0,
coeff.:
( 8n
2n
2 1
2 1
2
2
6n
dj 1
2n2a 2 j 1
12 8 n2
4n
0 1 aj ) Q
n2
4n
bj + 4 n
dj
1
4
dj
0 2
0
2
0 2
2
2 1
0 2 aj 0 2
bj +
dj + 12 n 2a
+ 2n
1
dj + 2 n
Q + ( 8 3n2cj
2 j
12
4 n2
n2
0 2 0
2
2
1
j
cj
4n
+ ( 8n
0 2
dj
4n
bj + 2 n
4 3n2cj
12
0 2
dj
0 1
+ 4 n2 bj
2
2 2 1 )(Q )
0 2
2 1
Q
)2 + (12 n
2 1 aj )(Q
+ 8 n2 bj
j) Q
cj + 6 n
0 2
n2
1
2 1
2
0 1
2
+8
dj + 4 n
18 n
0 1
2n2
dj
12 n2
j 0
12
n2
j
2 1
2
2
dj
0 2
2n
2
2 1
cj + 6 n +4
12 n2
2 0 1 j
2
dj
dj + 12 n
1
bj + 3
0 2
1
12
0 2
cj + 4 n
dj
j 0 1
n2
1
2
0 1
cj
2
8
bj
6n
2 1
2n
2 1 aj
cj
bj
0 1
2n2a j 0 2
12 n2 2
0 2
0 2 aj
dj + 6 n
0 1 2n2
4n
0
2
2 j
j
(79)
= 0, 1160
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
Q
1
coeff.:
2n
0 1 ((2
+6 n
j 0
dj
3 cj +
+6 n
dj +
+6 n
0 j
bj
aj ) Q
+ (4
0 j
+6 n
0
4
2n2a 2 j 0
+ 4 n2 bj
j
3nc
4
j
2n
2 na
dj + 4
4 n bj
j
+ 4 n ) Q ) = 0,
(80)
Q0 coeff.:
4
3n2c 2 j 0
4 n2
0
3
2n2
+4
4 n2
j
dj 0
3
0
2
4 n2
j
0
2
0
2
4 n2
j 0
3
4 n2
0
3
j
(81)
= 0,
by virtue of inserting the auxiliary equation (74) into the ordinary differential equation (73) and later on setting of the coefficient of the same functions to zero. As long as solving the above equations, the following results are given by 0
= 0,
2
=
2
1
= 0,
n2 ( j + 3c d j j
2
dj
3
2(
=
2d
bj
1
,
1 + j )( 2dj + bj
j
+
j
2 nb
j cj
+3
aj bj
j
cj + aj ) +
2
bj dj +
n bj 2
n bj + 3 cj + naj
1)
(2 3n dj 2
3
2b c j j
2
3nc
2 n dj
j dj
+2
3
dj 2 + 3
najbj +
2n
bj dj
bj 2 + 3 ncj (82)
bj + aj ).
The trivial solutions are reached out by using of these results. It is not possible for the purpose of recovering optical soliton pulses in case of full nonlinearity. 3. Conclusions The Biswas-Arshed model in birefringent fibers without FWM was perused in the cause of catching up optical solitons of its in this paper. As long as parameter restrictions, optical dark, bright as well as singular solitons were imparted by modified simple equation approach. It is not possible for the sake of recovering these very valuable optical solitons in case of full nonlinearity. Singular periodic solutions also were granted along with the reverse form of the parameter constraints. The results obtained in this study give rise to peruse the model as more elaborate. The Biswas-Arshed equation firstly will be generalized in DWDM system that subsequently is going to be handled with strategic procedures for example extended Kudryashov's method, trial equation integration architecture and modified simple equation algorithm. And furthermore, perturbation terms for example Raman scattering, saturable amplifiers, multi–photon absorption is going to be added. These precious consequences sequentially are going to be reported. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
A. Biswas, S. Arshed, Optical solitons in presence of higher order dispersions and absence of self-phase modulation, Optik 174 (2018) 452–459. M. Ekici, A. Sonmezoglu, Optical solitons with Biswas-Arshed equation by extended trial function method, Optik 177 (2019) 13–20. S. Aouadi, A. Bouzida, A.K. Daoui, H. Triki, Q. Zhou, S. Liu, W-shaped, bright and dark solitons of Biswas-Arshed equation, Optik (2019). Y. Yildirim, Optical solitons of Biswas-Arshed equation by trial equation technique, Optik (2019). Y. Yildirim, Optical solitons of Biswas-Arshed equation in birefringent fibers by trial equation technique, Optik (2019). Y. Yildirim, Optical solitons of Biswas-Arshed equation by modified simple equation technique, Optik (2019). Y. Yildirim, Bright, dark and singular optical solitons to Kundu–Eckhaus equation having four-wave mixing in the context of birefringent fibers by using of modified simple equation methodology, Optik (2019). Y. Yildirim, Optical solitons to Schrödinger–Hirota equation in DWDM system with modified simple equation integration architecture, Optik (2019). Y. Yildirim, Optical solitons of Gerdjikov–Ivanov equation in birefringent fibers with modified simple equation scheme, Optik (2019). Y. Yildirim, Optical solitons of Gerdjikov–Ivanov equation with four-wave mixing terms in birefringent fibers by modified simple equation methodology, Optik (2019). 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. E. Yaşar, Y. Yıldırım, Q. Zhou, S.P. Moshokoa, M.Z. Ullah, H. Triki, A. Biswas, M. Belic, Perturbed dark and singular optical solitons in polarization preserving fibers by modified simple equation method, Superlattices Microstruct. 111 (2017) 487–498. A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with Gerdjikov–Ivanov equation by modified simple equation method, Optik 157 (2018) 1235–1240. A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with full nonlinearity for Kundu–Eckhaus equation by modified simple equation method, Optik 157 (2018) 1376–1380. A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with resonant nonlinear Schrödinger's equation having full nonlinearity by modified simple equation method, Optik 160 (2018) 33–43. A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for complex Ginzburg–Landau equation with modified simple equation method, Optik 158 (2018) 399–415. A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton solutions to Fokas-lenells equation using some different methods, Optik 173 (2018) 21–31. A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, M.F. Mahmood, S.P. Moshokoa, M. Belic, Optical solitons with differential group delay for coupled Fokas–Lenells equation using two integration schemes, Optik 165 (2018) 74–86. A. Biswas, Y. Yıldırım, E. Yaşar, R.T. Alqahtani, Optical solitons for Lakshmanan–Porsezian–Daniel model with dual-dispersion by trial equation method, Optik 168 (2018) 432–439.
1161
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
[20] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with complex Ginzburg–Landau equation using trial solution approach, Optik 160 (2018) 44–60. [21] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, S.P. Moshokoa, M. Belic, Sub pico-second pulses in mono-mode optical fibers with Kaup–Newell equation by a couple of integration schemes, Optik 167 (2018) 121–128. [22] A. Biswas, Y. Yildirim, E. Yasar, M.F. Mahmood, A.S. Alshomrani, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for Radhakrishnan–Kundu–Lakshmanan equation with a couple of integration schemes, Optik 163 (2018) 126–136. [23] M. Mirzazadeh, Y. Yıldırım, E. Yaşar, H. Triki, Q. Zhou, S.P. Moshokoa, M.Z. Ullah, A.R. Seadawy, A. Biswas, M. Belic, Optical solitons and conservation law of Kundu–Eckhaus equation, Optik 154 (2018) 551–557. [24] A. Biswas, Y. Yıldırım, E. Yaşar, M.M. Babatin, Conservation laws for Gerdjikov–Ivanov equation in nonlinear fiber optics and PCF, Optik 148 (2017) 209–214. [25] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with full nonlinearity for Gerdjikov–Ivanov equation by trial equation method, Optik 157 (2018) 1214–1218. [26] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Solitons for perturbed Gerdjikov–Ivanov equation in optical fibers and PCF by extended Kudryashov's method, Opt. Quantum Electron. 50 (3) (2018) 149. [27] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Dispersive optical solitons with Schrödinger–Hirota model by trial equation method, Optik 162 (2018) 35–41. [28] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Dispersive optical solitons with differential group delay by a couple of integration schemes, Optik 162 (2018) 108–120.
1162
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical solitons to Biswas-Arshed model in birefringent fibers using modified simple equation architecture
T
Yakup Yildirim Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059 Bursa, Turkey
ARTICLE INFO
ABSTRACT
Keywords: Optical solitons Modified simple equation architecture Biswas-Arshed equation Birefringent fibers
The Biswas-Arshed equation in birefringent fibers is considered to get optical solitos of its in this paper. Optical dark, bright and singular solitons are given by modified simple equation approach. Also, singular periodic solutions are located.
1. Introduction Solitons in optics are pulses or molecules that compose of the basic fabric for the purpose of soliton transmission technology along with optical fibers, telecommunications industry, data transmission across, transcontinental and transoceanic distances which are constituted by the aid of a lot of models such as complex Ginzburg–Landau equation, Fokas-lenells equation, Kaup–Newell model, Lakshmanan–Porsezian–Daniel model, Kundu–Eckhaus equation, Radhakrishnan–Kundu–Lakshmanan equation, Schrödinger–Hirota model, Gerdjikov–Ivanov equation [17–28]. Biswas-Arshed model also is one of these governing models and has been employed along with strategic algorithms that are traveling wave hypothesis, exponetial function, extended trial equation procedure, complex amplitude ansatz procedure, trial equation technique, modified simple equation algorithm [1–6] where the solitons along the one or two component of the model are contemplated. The most striking part of this model is that self-phase modulation is neglected and also group velocity dispersion is negligibly small. Another remarkable feature of the model is that both spatio-temporal dispersions of second-order and third-order are contained in the model in the cause of the low group velocity dispersion to be compensated. 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 in birefringent fibers without four-wave mixing terms (4WM) is employed in this paper through the medium of modified simple equation procedure that discovers highly important solitons with the inclusion of the existence of circumstances for these important solitons. In addition, this method also provides additional solutions namely singular-periodic solutions by the reverse form of the constraints. The analysis of the model will be given in detail in the rest of the article. 2. Succinct pen-picture of modified simple equation architecture Quick recapitulation of modified simple equation architecture [7–16] is presented in this section before its possibility to any model for the purpose of providing very valuable optical solitons. Step-1: A nonlinear evolution equation in general form is going to be taken into account of
( , t,
x,
tt ,
xt ,
xx ,
…) = 0
and subsequently the substitution of the wave transformation https://doi.org/10.1016/j.ijleo.2019.02.013 Received 2 February 2019; Accepted 2 February 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
(1)
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
(x , t ) =
(2)
( )
with the inclusion of
=x
(3)
t
comes up the following ordinary differential equation:
( ,
,
,
(4)
, …) = 0.
In (4), the dependent function is shown as = ( ) while a polynomial containing dependent function and its derivatives is referred to as . Step-2: The most critical step of this technique is the ancillary function which will be given by N
( )=
i i=0
Q( ) Q( )
i
(5)
along with the necessary constant coefficients shown as 0, 1, …, N . It should be emphasized that these constant coefficients have to be rescued to get crucial optical solitons of Eq. (1). Step-3: The utilization of Eq. (5) is based on the existence of the N number which is obtained for the sake of the performing of the balancing condition in (4). Step-4: An overdeterminet equation system is recovered in consequence of putting the ancillary function (5) into the ordinary differential equation (4) and subsequently setting of the coefficient of the same functions namely Q i ( ), i = 1, 2, … to zero. The essential constant coefficients are emerged from the solution of the equation system, for this reason, the very valuable optical solitons of the model (1) are imparted. 2.1. FORM-1 The Biswas-Arshed model [1–6] is yield as
i
t
+ a1
xx
+ a2
xt
+ i (b1
xxx
+ b2
= i { (| |2 )x + µ (| |2 )x
xxt)
+
| |2 x}.
(6)
The existence of group velocity dispersion with spatio-temporal dispersion sequentially stem from the coefficient of a1 and a2 when the first term purports the temporal evolution of pulses in this model. The complex valued function (x , t ) implies to the wave profile. The self-steepening and nonlinear dispersions are ensured by the coefficient of , µ , sequentially. Once and for all, the existence of third order dispersion and third order spatio-temporal dispersion are ensured with the coefficient of b1, b2 sequentially. The Biswas-Arshed model without FWM in birefringent fibers is given as
i
t
+ a1
xx
+ b1
+ (µ1 (| |2 ) x +
i
t
+ a2
xx
1 (|
+ b2
+ (µ2 (| |2 ) x +
xt
xt
2 (|
+ i (c 1
xxx
+ d1
|2 ) x )
+(
1
+ i (c2
xxx
+ d2
|2 ) x ) + (
2
xxt)
= i { 1 (| |2 )x +
| |2 + xxt )
| |2 +
1
|2 )x
| |2 ) x},
= i { 2 (| |2 )x + 2
1 (|
2 (|
|2 ) x (7)
| |2 ) x }.
In the same way, the first terms are referred to as the temporal evolution of pulses when the existence of group velocity dispersion and spatio-temporal dispersion sequentially are supplied by the coefficient of aj and bj in this equation. The complex valued functions (x , t ), (x , t ) are referred to as the wave profile. The nonlinear terms that signifies self-steepening and nonlinear dispersions are ensured by the coefficient of j , j µj , j , j , j sequentially. Once and for all, the existence of third order dispersion and third order spatio-temporal dispersion are ensured with the coefficient of cj, dj sequentially. The modified simple equation algorithm is employed in the cause of discovering critical solitons of the Biswas-Arshed model (7) in this section, for this reason, the following solution condition
(x , t ) = P ( ) e i
(8)
(x , t )
with
=x
(9)
t
is going to be taken into account. In Eq. (8), the function P ( ) along with the speed of the soliton implies to the amplitude component whilst the function (x , t ) means the phase component and lastly this function will be presumed as
(x , t ) =
(10)
x+ t+ .
In Eq. (10), the parameter corresponds to soliton frequency when the parameter purports soliton wave number and also the parameter is referred to as soliton phase. The real component with the imaginary component respectively are emerged from
( 2
dj + 3 cj
dj
bj + aj ) P j + (
3c j
+
2
dj
2a
1150
j
+
bj
) Pj
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
(
j
3 j ) Pj
+
3 j P j˜
2 j Pj P ˜j
=0
(11)
and
(
2
dj + cj ) P j + ( (3
j
dj
2 j ) Pj Pj
+ 2 µj +
2c j
3 2
+2
dj +
bj
2 j P j P j˜
j Pj P j˜ P j˜
2 aj +
bj
) Pj
3 j P j˜ P ˜2j = 0
(12)
on account of putting the solution condition (8) into the model (7) along with j = 1, 2 and ˜j = 3 must be used as
j . The following circumstance (13)
P j˜ = Pj in the cause of using the balancing condition in the real part (11) as well as the imaginary part (12) that can be given by
( 2
dj + 3 cj
(
j
+
+
j
dj 3 j ) Pj
+
j
( 3cj
bj + aj ) P j
2
2a
dj +
bj + ) Pj
j
=0
(14)
and
(
2
dj + cj ) P j + (
(3
j
+ 2 µj +
dj
+2
j
2c j
3 j
+
+2
dj +
+ 3 j)
j
bj
2 aj +
bj
) Pj
P 3j = 0
3
(15)
respectively. In the cause of the amplitude component function Pj holds both the real component (14) and the imaginary component (15), the following result is imparted as 2
dj + 3 cj
dj
bj + aj
3 ( j+
=
dj + cj
j + j + j)
3 j + 2 µj + j + 2 j + j + 3 j
=
3c + 2 d 2a + j j j 3 2cj + 2 dj + bj
2 d j
bj 2 aj +
(16)
bj
as long as
aj =
2(
4 3
4
2
8
2
j dj
j
+6
2
j bj j
+6
6
2
bj
j j
+2
6
2
bj
j
j
3
8
j
2d
1 + µj
j
+4
j
2d
j
j
j d j µj
2
+8
j
2
6
j bj j
6
+6
bj j µj
6
+6
bj µj
9
j µj
9
j
+4
2
2
bj j µj
8
2
bj
j
bj µj 2 + 2
2
bj µj
2
j bj j
bj j µj + 6
2
j
9
µj
j
j j
+6
j
j
+3
j j
+3
j j
+9
j
2
+ 15
j µj
+3
j
+8
3
j
j
j
2d
j
1 + µj
16
j)
3
2 3
j dj µ j
(8
2
bj j µj
j
4
2
bj
+3
3
3
j µj
+ 15
j
2
j
2
4
bj
6 j
j
+6
3
+8
j
j
2
+ 18
+2
2
j bj j
j
2
2b j
+6
bj
j j
+6
j
6
j
bj 2
+6
j
j
2
6
2
bj
j j
bj µj 2 9
j j
9
j µj
+ 12
+ 15
bj j µj j bj j
j j
j
j j
j µj
j µj
+3
j
j
j ), j j dj
3
2b j
j j
j j
2
j d j µj
9
+ 15
j
2
9
j
dj
j
+8
bj j
3
j dj
j
bj
2
6
j j
µj
j dj
j
9
µj 2 + 3 16
2
9
j j
j
j
+2
2d
3
8
+4
j j
bj j µj + 6
+9
2d
2
j bj
6
j j j µj
j
+6
j
9
+6
+ 16
2
9
+3 j
+6
j
j j
18
j
8(
bj
9
j j
j
j bj µj
2
j
j bj
d j µj
4
j
2
j d j µj
j
dj
+2
j bj µj
3
8
j dj 2
3
+8
d j µj 2
4
j
j j dj
2
+8
d j µj
12
3
+4
j d j µj
j bj j
3
8
j
j
2
+3
cj =
j dj 2
8
2d
2
9 j
3
+8
j
dj µ j 2 + 8
9
j
3
(4
2
j j dj
4
+2
j)
+ 16
d j µj 2
3
16
j dj µ j
3
1151
dj µ j
16 j
+8
3
3
j dj
dj
j
j
2
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
+4
2
2
2
bj
j
6
2
bj
j
j
2b j
2
2
2
6
2
2b j
+2
2
+6
4
j
+2
bj
j
+6
bj
j
bj 2
+6
j
j bj j
2
6
2
j j
bj j µj
bj
2
+6
2
j j
2
2
bj
6
j
bj
bj j µj + 6
j j
6
j j
6
j j
6
6
j µj
6
j
j
9
j
2
18
6
j µj
12
+8
j µj
+ 12
j
+4
j µj
j
+6
+
j
2
j
+9
j
j
6
j
µj +6 j
2
j
+ 12
3
j
j
j µj
j
6
j j
2
+6
2
j j j
2
µj + 4
2
j
bj
12
j
9
j
2
j j j
+9
µj
j
bj j µj
+ 12 j
bj j bj
j
j
+2
2
+2
j
j
bj j µj
6
j j
+4
+4
2
j
12
j µj
j
+2
12
2
j
+4 j
j bj µj
bj µj
j µj
+4
j j
j
j
j bj
2
+6
bj µj
bj
2
2
j
j
8
bj µj 2 + 2
4
+6
j j
2
bj j µj + 4
j
j bj µj
bj
2
j bj j
3
j
2
4
bj µj 2
6
j
6
2
2
+8
bj j µj
+2
j j
bj
j bj j
+4
j
j bj j
+6
j j
j bj j
bj
6
j bj j
6
j
+6
2
j j
+ 12
2
+ 18
j
+
j
j j j j
2).
(17)
Under the condition of Eq. (17), the modified simple equation methodology will be implemented in the real component (14). Case-1: The auxiliary equation (5) is given as
P( ) =
0
+
Q( ) Q( )
1
(18)
by virtue of N = 1 that is recovered as a result of balancing P with The following equations is recovered by Q 3 coeff.: 1 (Q
Q
2
)3 (
2
j 1
Q
2 1 j
+
2 1
+
+4
j
dj
6 cj + 2
in Eq. (14).
dj + 2 bj
2 aj ) = 0,
(19)
coeff.:
3 1 Q ((2 1
2 1 j
+
P3
dj
3 cj +
dj +
bj
aj ) Q + (
j 0 1
0 1 j
0 1 j
0 1
j)Q
) = 0,
(20)
coeff.: 1 ((2
dj
+ ( 3cj
2
3 cj +
dj +
2 j 0
dj + 3
bj
aj ) Q
2 0 j
+3
+3
2 0 j
j
0
2 0
+3
j
+
2a j
bj + ) Q ) = 0,
(21)
Q0 coeff.: 3c j 0
2
+
dj
0
j 0
3
0
3
3
j
0
3
j
2a j 0
+
bj
0
0
= 0,
(22)
by virtue of inserting the auxiliary equation (18) into the ordinary differential equation (14) and later on setting of the coefficient of the same functions to zero. As long as solving the above equations, the following results are given by 0
3c j
=±
2 j
1
4
=±
dj
2a
dj +
+
j
bj +
j
+
j
6 cj + 2 j
+
+
dj + 2 bj
+
j
,
j
(23)
j
+
2 aj (24)
j
and
2( 3cj
Q =±
Q
=
2
2
dj
2( 3cj
2
2
dj
2a
dj + 3 cj +
dj + 3 cj +
bj + )
j
dj +
2a j
dj +
bj
aj
bj + ) bj
aj
Q,
(25)
Q.
(26)
1152
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
One can easily reach out to
2
Q =±
dj
3 cj +
2( 3cj
2
dj
3 cj +
dj + 2a
dj +
bj
aj
k1 e
bj + )
j
±
2( 3cj 2 dj + 2aj bj + ) 2 dj 3 c j + dj + bj aj
,
(27)
and
2
Q=
2( 3cj
2
dj + 2a
dj +
bj
aj
bj + )
j
k1 e
±
2( 3cj 2 dj + 2aj bj + ) 2 dj 3 cj + dj + bj aj
+ k2,
(28)
under the condition of employing Eqs. (25) and (26). In consequence of putting Eqs. (27) and (28) with the inclusion of k1, k2 integration constants in Eq. (18), the optical solitons for the model (7) are given as 3c
(x , t ) = ±
2
1
1
± ×
2
d1
2( 3c1
2 2( 3c1
d1
2a 1
d1 + +
3 c1 + d1 + 2 d + 2a 1 1
b1
1 +
a1
k1 e
b1 + )
b1 + 1
+
4
±
2( 3c1 2 d1+ 2a1 b1+ ) (x 2 d1 3 c1+ d1+ b1 a1
±
2( 3c1 2 d1+ 2a1 b1 + ) (x 2 d1 3 c1+ d1+ b1 a1
± 3 c1 + d1 + b1 a1 ke 2 d + 2a b1 + ) 1 1 1
d1
6 c1 + 2 d1 + 2 b1 1 + 1 1 + 1+
1
t)
ei ( t)
2 a1
x+ t+ )
+ k2
(29)
and 3c 2
(x , t ) = ±
2 2
± ×
2
d2
2( 3c 2
2 2( 3c2
d2
3 c2 +
d2 + b2
2 d + 2a 2 2
3 c2 +
2a 2
d2 + +
2
+
a2
k1 e
b2 + )
d2 + b2
a2
ke b2 + ) 1
2 d + 2a 2 2
b2 + 2
±
+
4
±
2( 3c 2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c2 + d2 + b2 a2
6 c2 + 2 d2 + 2 b2 2 + 2 2 + 2 +
t)
2 a2
t)
ei (
2( 3c2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c 2+ d2 + b2 a2
±
d2
2
x+ t+ )
+ k2
(30)
If we set
k1 =
2( 3cj 2
2
dj
2a j
dj + 3 cj +
bj + )
dj +
bj
d1 +
2a 1
aj
e
2( 3c j 2 dj + 2aj bj + ) 0 2 dj 3 cj + dj + bj aj ,
±
k2 = ± 1,
we get:
(x , t ) = ±
× ei (
3c
2
1
1 +
b1 + 1
+
tanh
1
3c
2
1
2(2
d1
d1 + 2a1 b1 + (x 3 c1 + d1 + b1 a1 )
t+
0)
(31)
x+ t+ ) ,
(x , t ) = ±
3c 2
2 2
× ei (
1 +
2a 2
d2 + +
2
+
b2 + 2
+
tanh
2
3c 2
2(2
2
d2
d2 + 2a2 b2 + (x 3 c2 + d2 + b2 a2 )
t+
0)
(32)
x+ t+ ) ,
The consequences (31)-(32) imply to dark soliton solutions on condition that
( 3cj
2
dj +
(x , t ) = ±
2a 3c
1
bj + )(2
j 2 1
× ei (
1
+
3 cj + b1 +
1
+
coth
1
dj +
bj 3c
aj ) > 0. 2
1
2(2
d1
d1 + 2a1 b1 + (x 3 c1 + d1 + b1 a1 )
t+
0)
(33)
x+ t+ ) ,
(x , t ) = ±
3c 2
2 2
× ei (
2a 1
d1 + +
dj
2a 2
d2 + +
2
+
b2 + 2
+
2
coth
3c 2
2(2
2
d2
d2 + 2a2 b2 + (x 3 c2 + d2 + b2 a2 )
t+
0)
(34)
x+ t+ ) ,
1153
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
The consequences (33)-(34) stem from singular soliton solutions on condition that
( 3cj
2
3c
(x , t ) = ±
× ei (
2a
dj +
bj + )(2
j
2
1
1 +
3 cj +
b1 + +
1
dj +
bj 3c
tan
aj ) > 0. 2
1
2(2
1
d1 + 2a1 b1 + (x 3 c1 + d1 + b1 a1 )
d1
t+
0)
(35)
x+ t+ ) , 3c 2
(x , t ) = ±
2
2a 2
d2 + +
2
ei (
2a 1
d1 +
1 +
dj
+
2
b2 + +
2
3c 2
× tan
2(2
2
2
d2 + 2a2 b2 + (x 3 c2 + d2 + b2 a2 )
d2
t+
(36)
x+ t+ ) , 3c
(x , t ) = ±
× ei (
2
1
2a 1
d1 +
1 +
1 +
b1 + +
1
3c
cot
2
1
2(2
1
d1 + 2a1 b1 + (x 3 c1 + d1 + b1 a1 )
d1
t+
0)
(37)
x+ t+ ) , 3c 2
(x , t ) = ±
2
2a 2
d2 + +
2
× ei (
0)
+
2
b2 + +
2
3c 2
cot
2
2(2
2
d2 + 2a2 b2 + (x 3 c2 + d2 + b2 a2 )
d2
t+
0)
(38)
x+ t+ ) ,
The consequences (35)-(38) correspond to singular periodic solutions on condition that
( 3b1
2
2a 1
b2 +
a2 + )(2
b2
3 b1 +
b2 +
a2
a1 ) < 0.
Case-2: The real component (14) is given as
( 2
dj + 3
4 (
j
+
j
cj
+
bj + aj )( (Q ) 2 + 2QQ )
dj
+
j
j )Q
3
0
+
1
Q( ) + Q( )
Q
5
2 6 2 (Q ) (4
dj +
2
(40)
1 2
(Q )3 (( 8
2
+ 18
1 2
dj + 18
24
0 2
12 3
+
+
2 j
2 j
+
2
)2
or QQ with
6 cj + 2
j
Q3
in Eq. (39).
dj + 2 bj
2 aj ) = 0,
(41)
3
cj +
2
+ 3 cj
j
dj +
2
1
dj
1
2
dj
bj
2
bj
2 aj ) Q 1
+( 2
dj
1
j 1 2
1 2 j
(42)
+ aj 1) Q ) = 0,
coeff.:
12
Q
dj
2
1 2 j
Q
+ ) Q2
coeff.:
12 2 (Q ) 4 ((2
4
2a j
+
The auxiliary equation (5) is given as
Q( ) Q( )
2
j 2
3c j
dj +
(39)
1 Q2.
by virtue of N = 2 that is recovered as a result of balancing (Q The following equations is recovered by Q 6 coeff.:
4
2
bj
=0
through the medium of using P =
Q( ) =
4(
j 1
2
dj + 12
dj 2
0 2
2
bj
4
dj
12
2 j 1 2
12
2
2 1
cj 18
1
12
2
bj
1 2
6
3
2
bj
4
2
2
2
1 2 aj ) Q
4
2
2
2
2 3c j
+4
+( 4
2
2
12
j 1
2 j 1 2
+ 36
j 0 2
+ 12
0 2 aj
2
0 2
+3
2a ) Q j
bj + 4
1
2a
2
2 2
12
2
cj + 9 j) Q
+ (36
2 1
dj
cj
4
2
2
12
12
0 2
j 0 2
dj
1 2 2 2a
j
54
+4
j 0 2
dj
2
2
3
1 2
2
bj
12 1
2
cj
j 0 2
2
dj (43)
)=0
coeff.:
2(Q ) 2 (( 6 30
0 2
dj + 9
1 2
cj
6
2 1
1 2
cj + 10
cj 0 2
3
1 2
dj + 2
dj 2 1
3
1 2
dj + 10
bj + 3 0 2
1 2 aj ) Q
bj + 2 1154
2
1
bj
+ (20 10
0 2
0 2 aj
dj + 4 2
2 1
2 1 aj ) Q
dj
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
+( 4 12
2 Q
2
dj
0 1
12
2
2
1 2
dj
4
12
j 0 1 2
bj
0 1
4
2a j
1 2
+4
1 2
j 0 1 2
2
+2
1 2
0 1 aj ) Q
bj
4
3 j 1
dj
0 1 3 j 1
2
12
3 j 1
2
+6
3 j 1
2
j 0 1 2 0 1
cj (44)
) = 0,
coeff.:
( 8
0 2
+4
0 2 aj
4
2 1
2a
j) Q
2
bj + 4
dj +2
1
bj +
1
4
0 2
6
0 1 aj ) Q
bj
dj + 12 Q
0 2
+4
+( 8
2
12
0 1
4 3cj j 0
12
j
2
cj
2 1
2
1
4
2
2
+8
2 0 1
j
bj
aj ) Q
dj
0 2
8
2
12 4
0 2
18
2 1
0
dj 2
dj
1
cj
0 2
+4
2
2
dj
0 1
2 j 0 1
12
2
dj
0 2
dj + 12
)2 + (12
2a )(Q j
1
12
1
1
dj + 2
0 2
0 2 aj
2
bj
cj + 6
0 2
3c j 0 2
Q +( 8
2 0 1 j
12 1
+4
j 0 1 2
+8
Q
3c j
1 2
2 1
3 0 1
2
bj
0 2
cj
4
cj + 6
0 1
8
2a
12
0
1
2 j
4
2
2
0 2
2a
4
2
12
2 j
2 1
dj +
dj + 6
j 0 2
bj
1
)(Q )2 = 0,
dj bj
0 1
2
j 1 0
2
2
j
(45)
coeff.:
2
0 1 ((2
+6
dj
3 cj +
+6
j 0
+6
j 0
dj + +6
j 0
j 0
3c j
+ (4
2
4
dj + 4
2a
4
j
bj
+ 4 ) Q ) = 0,
(46)
Q0 coeff.:
4
3c 2 j 0
4
0
3
+4
2
4
j
dj 0
3
0
2
4
j
2a
4 0
3
j 0
2
+4
4
j
bj 0
2
0
2
4
j 0
3
= 0,
(47)
by virtue of inserting the auxiliary equation (40) into the ordinary differential equation (39) and later on setting of the coefficient of the same functions to zero. As long as solving the above equations, the following results are given by 0
1
= 0,
2
2(2
=
3 cj +
( j+
16( 3cj
=±
dj
2
2a
dj +
dj +
+
j
j
+
bj
aj )
j)
bj + )(2
j
( j+
j
+
j
,
(48)
dj 2 j)
+
3 cj +
dj +
bj
aj )
2
,
(49)
and
4( 3cj
Q =±
Q
=
2
2
dj
4( 3cj
2
2
dj
2a j
dj + 3 cj + 2a
dj + 3 cj +
bj + )
dj +
bj
bj + )
j
dj +
bj
Q,
aj
aj
(50)
Q.
(51)
One can easily reach out to
2
Q =±
dj
3 cj +
4( 3cj
2
dj
3 cj +
dj + 2a j
dj +
bj
aj
bj + )
k1 e
±
4( 3cj 2 dj + 2aj bj + ) 2 dj 3 c j + dj + bj aj
,
(52)
and
Q=
2 4( 3cj
2
dj +
dj + 2a j
bj
aj
bj + )
k1 e
±
4( 3c j 2 dj + 2aj bj + ) 2 dj 3 cj + dj + bj aj
+ k2
(53)
under the condition of employing Eqs. (50) and (51). In consequence of putting Eqs. (52) and (53) with the inclusion of k1, k2 integration constants in Eq. (40), the optical solitons for the model (7) are given as
(x , t ) = ±
16( 3c1
2
d1 +
2a 1
(
1
b1 + )(2 + 1+ 1+
d1 1)
2 2
1155
3 c1 +
d1 +
b1
a1 )
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
4( 3c1 2 d1+ 2a1 b1+ ) (x d1 3 c1+ d1+ b1 a1 2
± 2 d1 3 c1 + d1 + b1 a1 ke 2 d + 2a 4( 3c1 b1 + ) 1 1 1
± ×
4( 3c1 2 d1+ 2a1 b1+ ) (x 2 d1 3 c1+ d1+ b1 a1
± 2 d1 3 c1 + d1 + b1 a1 ke 2 d + 2a 4( 3c1 b1 + ) 1 1 1
2(2
d1 (
2
±
d1
4( 3c1
×
3 c1 + d1 + b1 + 1 + 1 + 1)
1
3 c1 +
d1 + b1
a1
2 d1 3 c1 + d1 + b1 a1 ke 2 d + 2a 4( 3c1 b1 + ) 1 1 1
+ k2
a1 )
ke b1 + ) 1
2 d + 2a 1 1
t)
t)
4( 3c1 2 d1+ 2a1 b1+ ) (x 2 d1 3 c1+ d1+ b1 a1
±
ei (
4( 3c1 2 d1+ 2a1 b1+ ) (x 2 d1 3 c1+ d1+ b1 a1
±
1 2 2
t)
t)
x+ t+ )
+ k2
(54)
and
16( 3c2
(x , t ) = ±
d2
4( 3c2
× 2
d2
2(2
±
d2 (
2
2
d2
2
d2
4( 3c2
b2
a2
b2 +
b2
a2
b2 + )
k1 e
ke ) 1
3 c2 +
d2 + b2
3 c2 +
d2 + b2
a2
a2
ke b2 + ) 1
2 d + 2a 2 2
b2 + )(2 + 2+ 2+
d2 2)
3 c2 +
4( 3c 2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c 2 + d2 + b2 a2
±
d2 +
b2
a2 )
2 2
t)
t)
+ k2
a2 )
ke b2 + ) 1
2 d + 2a 2 2
2
4( 3c 2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c 2 + d2 + b2 a2
±
3 c2 + d2 + b2 + 2 + 2 + 2)
4( 3c 2
×
3 c 2 + d2 + 2 d + 2a 2 2
3 c2 + d2 + 2 d + 2a 2 2
4( 3c 2
2a 2
d2 +
(
2
±
2
4( 3c 2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c2 + d2 + b2 a2
±
ei (
4( 3c 2 2 d2 + 2a2 b2 + ) (x 2 d2 3 c2 + d2 + b2 a2
±
1 2 2
t)
t)
x+ t+ )
+ k2
(55)
If we set
k1 =
4( 3cj 2
2
dj
2a
dj + 3 cj +
bj + )
j
dj +
bj
aj
e
±
4( 3cj 2 dj + 2aj bj + ) 0 2 dj 3 cj + dj + bj aj ,
k2 = ± 1,
we get:
(x , t ) =
× ei (
2
d1 + 2a1 ( 1+ 1+ 1+
b1 + ) 1)
3c
sech2
2
1
2
d1
d1 + 2a1 3 c1 + d1 +
b1 + (x b1 a1
t+
1 2
0)
(56)
x+ t+ ) ,
(x , t ) =
× ei (
2( 3c1
2( 3c2
2
(
2
d2 + 2a2 + 2+ 2+
b2 + ) 2)
3c 2
sech2
2
2
d2
d2 + 2a2 3 c2 + d2 +
b2 + (x b2 a2
t+
1 2
0)
(57)
x+ t+ ) ,
The consequences (56)-(57) signify bright soliton solutions on condition that
( 3cj
2
(x , t ) =
× ei (
dj +
2a
bj + )(2
j
2( 3c1
2
(
1
d1 + 2a1 + 1+ 1+
dj
3 cj +
b1 + ) 1)
csch2
dj +
bj 3c
aj ) < 0. 2
1
2
d1
d1 + 2a1 3 c1 + d1 +
b1 + (x b1 a1
t+
0)
1 2
(58)
x+ t+ ) ,
(x , t ) =
2( 3c2
2
(
2
d2 + 2a2 + 2+ 2+
b2 + ) 2)
csch2
3c 2
2 1156
2
d2
d2 + 2a2 3 c2 + d2 +
b2 + (x b2 a2
t+
0)
1 2
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
× ei (
(59)
x+ t+ ) ,
The consequences (58) and (59) purport singular soliton solutions as long as
( 3cj
2
dj +
d1 + 2a1 ( 1+ 1+ 1+
3 cj + b1 + )
1)
dj +
bj 3c
sec2
aj ) < 0. 2
1
2
d1
d1 + 2a1 3 c1 + d1 +
b1 + (x b1 a1
t+
0)
1 2
(60) 2
(
d2 + 2a2 + 2+ 2+
2
b2 + ) 2)
3c 2
sec2
2
2
d2 + 2a2 3 c2 + d2 +
d2
b2 + (x b2 a2
t+
1 2
0)
(61)
x+ t+ ) ,
2( 3c1
2
d1 + 2a1 ( 1+ 1+ 1+
b1 + ) 1)
3c 1
csc2
2
2
d1
d1 + 2a1 3 c1 + d1 +
b1 + (x b1 a1
t+
0)
1 2
(62)
x+ t+ ) ,
2( 3c2
(x , t ) =
× ei (
2
2( 3c2
(x , t ) =
× ei (
dj
x+ t+ ) ,
(x , t ) =
× ei (
bj + )(2
j
2( 3c1
(x , t ) =
× ei (
2a
2
(
d2 + 2a2 + 2+ 2+
2
b2 + ) 2)
3c 2
csc2
2
2
d2
d2 + 2a2 3 c2 + d2 +
b2 + (x b2 a2
t+
0)
1 2
(63)
x+ t+ ) ,
The consequences (60)–(63) account for singular periodic solutions as long as
( 3cj
2
dj +
2a
bj + )(2
j
dj
3 cj +
dj +
bj
aj ) > 0.
2.2. FORM-2 The Biswas-Arshed model with full nonlinearity [1–6] is given as
i
t
+ a1
+ a2
xx
+ i (b1
xt
xxx
+ b2
xxt)
= i { (| |2n )x + µ (| |2n ) x
+
| |2n
(64)
x }.
This model is yield as the first form (7) as long as full nonlinearity parameter n = 1. The Biswas-Arshed model with full nonlinearity and without FWM in birefringent fibers is imparted as
i
t
+ a1
+ b1
xx
+ (µ1 (| |2n )x + i
t
+ a2
+ b2
xx
+ (µ 2 (| |2n ) x +
+ i (c 1
xt
1 (| xt
|2n )x )
+ i (c2
2 (|
xxx
+ d1
+( xxx
1
+ d2
|2n )x ) + (
2
xxt)
= i { 1 (| |2n ) x +
| |2n + xxt )
1
2
|2n ) x
| |2n ) x},
= i { 2 (| |2n )x +
| |2n +
1 (|
2 (|
|2n )x (65)
| |2n ) x }.
For the sake of recovering a generalized perspective, the modified simple equation technique is employed for the sake of obtaining highly important solitons of Eq. (65) in this section, for this reason, the real component with the imaginary component respectively are emerged from
(2
dj
+ P 2j n + 1
3
cj +
j
+ P j2n + 1
dj + j
bj
aj ) P j
+ Pj P j2˜ n
j
(
+ P j2˜ n + 1
bj + j
2
3c j
dj
2a
j
) Pj
=0
(66)
and
( dj
bj + 3 2cj +
cj ) P j + (2 aj
+ (2 n j +
j ) P j˜
P j2˜ n + (
j
+
j
+ 2n
j
bj
+
2 µj n) P j P j2n
2
dj
+2
2
dj ) P j + P j2˜ n P j
2n 1 nP P j P j˜ ˜j j
j
=0
on account of putting the solution condition (8) into the model (65) along with j = 1, 2 and ˜j = 3 must be used by
(67)
j . The following circumstance (68)
P j˜ = Pj
1157
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
in the cause of using the balancing condition in the real part (66) as well as the imaginary part (67) that can be given by
( 2
dj + 3 (
+
j
cj
+
j
j
dj
bj + aj ) P j
2n + 1 j ) Pj
+
(
2
bj
3c j
dj +
2a j
+
+ ) Pj
=0
(69)
and
(
dj + cj ) P j + ( 2 aj + ( j + 2n j +
j
+
+
j
3 2cj
bj + 2n
j
+
+ 2 µj n + 2
j
2
bj + j n)
dj + 2
dj ) Pj
P j2n + 1 = 0
2n + 1
(70)
respectively. In the cause of the amplitude component function Pj holds both the real component (69) and the imaginary component (70), the following result is imparted as 2
dj
3
cj +
dj + bj
dj
aj
(2n + 1) ( j +
=
cj
j+
=
j + j + j)
j + 2n µj + 2 n j + (2 n + 1) j + (2n + 1) j 2 d + 3c + 2a + bj j j j 2 d bj + 3 2cj + bj j
2 aj
2
(71)
dj
as long as
1
aj =
4 2n2 (
+ 4 3n2
j
2d
j
j
3n2
8
j
2n2
j
2n2
8
2n2
j dj
+4
2n2
j bj j
+4
2n2
+4
2n2
bj j µj
4
2n2
+2
2n
j bj j
2
2n
bj
j j
2
2n
bj
j
+8
j
dj µ j
j bj µj
2
2n2
bj
+4
2n2
bj j µj
bj
j
j
2n
+2
2n
bj j µj
4
2n
+ 8 n2 bj j µj
bj µj
12 n2
j bj µj
4 n2 bj
j
8 n2
j j
+ 2 n2
j
+ 2 n bj
+ 2 n2
j
j
6 n2
2
2 n
j j
2
3 n
j
8 n
j j
+ 6 n2
j
2
2 n2
j j
2 n2
j
n
j
3 n
2 n2 2 n2 j
j
j bj j
2 n
2
j j
+ 6 n2 µj 2
j
2
2n2
4
j
j j j
2 n2 µj
2
j
2n2
j
bj bj
2n2
4
j j j
+2
2n
bj j µj
2
2n
bj
j
2b j
+ 8 n2
2 n2 bj
2
j j
6 n2
j µj
6 n2
j µj
2
j bj j
+ 2 n bj
j
3 n
j j
+ 10 n2 j j
2
j bj
2n
3 n
j µj
2 n
j µj
2 n2
1158
j
j j
4 n2
j
2 n2 j j
j j
j
2
j
2 n2
j
+ 4 n2 j
j
2
j
j
j
3 n
j j
2
3 n µj
j
j
2
j j
4 n
j
j
6 n2 µj
j
3 n j
j
j j
j
j
2
j bj j
+ 2 n bj
j
+ 12n2
2
bj
+ 4 n2 bj
j
j j
j j
j µj
j
bj j µj
2 n bj j µj + 2 n bj 2
j
j j
4 n2 j j
j bj j
bj
bj
6 n2
j
5 n
j j
2n
2
j dj µj
2n2
2
+2
2 n2
j µj
2
2n2
4
2
j j
3 n
+ 10 n2
2n2
j bj j
2
j
+4
+ 4 n2
2 n j
2b j
j
j
4 n2 bj j µj + 4 n2 bj
j
6 n2
j
2n2
+ 4 n2 bj
j j
j
4 n
2
6 n2
2 n
j
j
j
+8
2
j
dj
j
bj µj 2 j
j
2d
j j
2n2
+2
j
bj
bj
j
j
+2
2n
j bj
+ 8 n2
2
2
5 n j
j
j
j bj j
j
2n2
j bj
4 n2
j
4
j dj
3n2
+4
j
j
2n
6 n2
j
8 3n2 dj µj
j
8 3n2
j dj µ j
+2
2 n2
j j
j j
5 n
+ 4 n2
dj
2n2
4
2
2 n bj µj
+ 10 n2
j µj
j
+ 8 3n2
j dj
2n2
4
j
2 n bj j µj + 4 n bj
j j
5 n
j µj
bj
2 n2 bj
j
2 n2
j
j
j
2n2
+8
4 n2 bj j µj + 4 n2 bj
j
2 n2
2 n bj j µj + 2 n bj
bj
+ 8 n2
j µj
+ 2 n bj
j bj j
2n
2
j
6 n2 bj µj 2 + 8 n2 bj µj j
j d j µj
j
j j dj
+ 4 3n2 dj µj 2
j
2n2
+2
2n
8 3n2
d j µj 2 + 8
j bj j
+2
2d j
j dj
2n2
8
2n
j
j
3n2
+8
+2
j bj j
6 n2
(4 3n2
2
j j dj
2n2
4
j
j)
j dj µ j
4
4 n2
2d
+ µj
n
j
j
2
j j j
2
+ 10 n2 j
2
2
j µj
j j
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
2
j
j
+ 3n
j j
+ 5n
j µj
+2
+ 3n
2
2
j
2
j
j
8 3n2 (
j
j
2d
+ 8 3n2
j
+4
2n2
j
4
2n2
bj
j j
4
2n2
bj
j
2b j
j
j bj
+2
2n
bj j µj
2n
2
j
2b j
+ 4 n2 bj
j j
+ 4 n2 bj
j
4 n2
j
2 n
j bj j
j
+ 8 n2
2 n
j 2
j j
j
2 n
j j
+ 4n
j
j j
j µj
+ 8 n2
j µj
+ 4 n2
2
j j
2
j
j
j
j µj
j
j
+
j
j
j
2
2n
bj
2
2
+ 8 n2
2
j j
j j
+ 8n
+2
+2
j
+
bj
4 n2
2n
+2
j j j
+2
n
j
j
j
j
+ 4n
2
+2
j
+
j
j
2n
j bj j
4 n2
j µj
4 n2
j
4 n2 µj
+ 2 n bj
j
2
2 n
j j j
2
8 n
2
j
2
j j
+ 8 n2
2
j
j j
+ 4n
+2
j j
j
j µj
j
2
2
j
j
+ 4n
j j
j µj
+2
+ 4n j
j
j
j
j
j j
4 n2
j µj
2 n
2
6 n
2
6 n
j bj j
j
j
j j
j
+ 4n 2
j
j µj
2 j
j
j j j
+ 8 n2
+ 4n j
j bj
2 n bj µj
j
j j
j
2
4 n2 bj j µj
+ 8 n2
+ 4n
j
2 n bj j µj
j 2
bj
j bj j
j µj
2
j
+
j
j
j
2n
j j
2 n
+ 4 n2
j
j bj j
4 n2
j j
+ 4 n2
2
bj j µj
+ 4 n2
2 n
2 n
j bj
2n2
bj
j
j
2n2
2n
2
8 n2
j
j j
2 n
j
j j
+ 4n
j
j
2n
j
j j
2 n bj j µj + 2 n bj
j
+ 8 3n2 dj
+2
j bj µj
j
j
+4
4
bj µj
+ 2 n bj
j j
j
bj j µj
4 n2
j j
j
j dj
4
j
4 n2 bj j µj + 4 n2 bj
4 n2
2
bj
8 n2
j bj j
j
16 3n2
j bj µj
2n2
2n
+2
j
j
j j j j
j
2n
2 n µj
j
2
j
j
+
j
+ 4n
+ 4n
+ 4 n µj
+2
2
+2
4 n
j
j
6 n
j
+ 3n
j dj µ j
2n2
+8
j bj j
2 n
j j
4 n
j µj
+ 4n j
2 n
j j
2 n
2
+ 4n
j
j bj j
2n
4 n2
j µj
2 n bj j µj + 2 n bj
j
2
j bj
j j
j
+ 4 n2 µj 2 j
4 n2
j µj
16 3n2 dj µj
+2
j bj j
+ 2 n bj
2 n
2
4 n2
2
j j
+ 16 3n2
j j dj
4
2
4 n2 bj µj 2 + 4 n2 bj µj
j
+2
j j
bj j µj
+ 4 n2 bj j µj
j j
+2
+ 5n
j j
+ 5n
j j
2n2
bj j µj
4 n2
j bj j
j
4 n
bj
2n
+2
j
j
4 n2
j j
2n
2
+ 8n
2n2
+4
2
j j
6 n
j bj j
j j
4 n2
j
2n2
j
2
2
2
+ 5n
j j
+ 8 3n2 dj µj 2
j
bj µj
j
+ 2 n bj
j dj
2n2
j
j
3n2
4
j
j
16 3n2
j
bj µj 2
4 n2
+ 4 n bj
2d
bj j µj + 4
+ 4 n2 bj j
j
2n2
bj
+ 4 n2
+
4
bj
2n
2
2
j j
bj
2n2
+4
j
j
+ 16
+4
+ 4n
j
j
+ 5n
j j
2 ),
(8 3n2
2
j bj j
2n2
4
j
j d j µj
2n2
4
2n
j
j)
16 3n2
j
+2
4 n2
+ µj
+ 5n
n
j
+
j
2
2n
+ 5 n µj
j
+2
j
j
j µj
1
cj =
+2
j
+ 5n
j j
+ 3n
+
j
j
j
j
j j
+ 4n
j µj
2
2).
(72)
Under the condition of Eq. (72), the modified simple equation methodology will be implemented in the real component (69) that can given as
( 2
dj + 3
4n2 (
j
+
cj j
+
dj j
+
bj + aj )((1 j )Q
3
through the medium of using P =
Q( ) =
0
+
1
Q( ) + Q( )
2
2n)(Q ) 2 + 2nQQ )
4n2 (
= 0.
1 (Q) 2n .
bj
2
dj +
3c j
+
2a j
+ ) Q2 (73)
The auxiliary equation (5) is given as
Q( ) Q( )
2
(74) 1159
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
by virtue of N = 2 that is recovered as a result of balancing (Q )2 or QQ with Q3 in Eq. (73). The following equations is recovered by Q 6 coeff.: 2 6 2 (Q ) (
4
n2
+ n dj + n bj
Q
5
3 cj
Q
+2
2 aj ) Q
+2
1
dj + 2 n
2
2
2
2
dj
+ ( 20 n dj
8
dj
4 n2
+ 2n +
1
dj + 2 n
dj
12 n
cj
2
n2
bj
aj ) = 0,
4
2
+ 2 n dj
j
dj
2
3
1
cj + 4 n
2
n
+ 3 n2
j 1 2
bj
1
2
1
2 2
2 1
2
1 2
+ 30 ncj
2
2
2
2
bj
cj
3 ncj + 2
dj (75)
dj
2
n
bj + 6
2
+ 3 n2
j 1 2
2 n 1 aj +
2
dj + 4 n
16
1 2) Q
2 j 1 2
2
dj
2
2n
1
1 2
j 0 2
2a
2
cj + n 2 aj
2
+ 4n
j 1 2
dj +
1
2
2
bj
1
4n
2a j
+4
10 n dj
1 2
2
bj
1
2
2
2
2a ) Q j
2
dj
1
1 aj ) Q
2 j 1 2
6n
2
2
dj
1
cj
2
2
bj
(76)
) = 0,
2
2 2a
+ ( 8n
dj + 4
+ 12 n2
2
bj
1 2
j 0 2
12 n 2 1 aj
j
2
2
dj + 12 n 2
2
+ 12 n2 2 1
6n
+ 10 naj
1 2
2 1
bj + 4 n
0 2
2
2
cj
)2
2a )(Q j
+ 24 cj
0 2 aj
+ 24 n
2
4
dj + 12 n
0 2
2 j 1 2
+ 4 n2
Q
10 n bj
cj + 12 n
0 2
+ 12 n2
2 3c j
3
j
2
+ 12 n2
j 0 2
dj + 4 n2
1
2
cj + 4 n
Q + ( 36 n
+ 12 n2
2
2 2
12
bj + 12 n2
dj + 4 n2
1
2
1 2
bj + 2
1
2
+ 8 aj
1 2
2
4n
+ 12 n2
dj + 2 n
bj +
dj
4 n2
dj
j 0 2
2
2
2 2
4n
1 2
dj
2 j 1 2
cj
dj
0 2
(77)
cj )(Q ) 2) = 0,
coeff.:
2Q ((6 n dj
+ 6 ncj + 2 bj
+n
1
dj
2a
3 j 1
4 n2
1 2
+ 4 n2
1 2
2 1
dj
1
+ 2 n2
2
dj
4 n2
3c j
+ 4 n2
2a
bj +
3 j 1
1
j) Q
1 2
dj
2n
0 2
bj
10 n
Q + ( 2n
bj + 12 n2
2 1
2 1
n
+ 2 naj
1 2
bj + 3
+ 4 n2
6n
1 2)Q
dj + 30 n
j 0 1 2
j 0 1 2
3 naj
1 2
6 cj
0 1 aj
+ 12 n2
j 0 1 2
2a )(Q )2) j
2 n bj
0 2
dj
+ 12 n2
1 2
1 2
2 1
n
+ 3 n bj
1 2
1 2
)2 + ( 20 n
dj
0 2
+ 3 n dj
1 2
2 n dj
1 2
1 2 )(Q
10 n 2
j
+ 2 n2
dj
2 aj
1 2
9 ncj
1 2
+4
1 2
2 1
2
2
dj +
+ 3 n2
j 1 2
8 bj
1 2
+ 12 n2
Q
+
2 j
coeff.:
+8
3
naj +
dj + 3 n
2
+ (3 n2
(Q ) 2 ((8 n
Q
n2
+
2 j
coeff.:
4 2 (Q ) 4 (( 2 n
4
n2
+
j 2
0 1
2
1 2
+ 12 n2
3 j 1
j 0 1 2
cj + 2 n
+2
cj + 10 n
+ 2 n2
1 2
+ ( 4 n dj
cj + 3 n
0 2
1
Q
dj 1
2
1 2
cj
0 2 aj
+ 2 n2
+ 4n
dj + 2 n
0 1
1 2
3 j 1
dj
0 1
bj
0 1
(78)
= 0,
coeff.:
( 8n
2n
2 1
2 1
2
2
6n
dj 1
2n2a 2 j 1
12 8 n2
4n
0 1 aj ) Q
n2
4n
bj + 4 n
dj
1
4
dj
0 2
0
2
0 2
2
2 1
0 2 aj 0 2
bj +
dj + 12 n 2a
+ 2n
1
dj + 2 n
Q + ( 8 3n2cj
2 j
12
4 n2
n2
0 2 0
2
2
1
j
cj
4n
+ ( 8n
0 2
dj
4n
bj + 2 n
4 3n2cj
12
0 2
dj
0 1
+ 4 n2 bj
2
2 2 1 )(Q )
0 2
2 1
Q
)2 + (12 n
2 1 aj )(Q
+ 8 n2 bj
j) Q
cj + 6 n
0 2
n2
1
2 1
2
0 1
2
+8
dj + 4 n
18 n
0 1
2n2
dj
12 n2
j 0
12
n2
j
2 1
2
2
dj
0 2
2n
2
2 1
cj + 6 n +4
12 n2
2 0 1 j
2
dj
dj + 12 n
1
bj + 3
0 2
1
12
0 2
cj + 4 n
dj
j 0 1
n2
1
2
0 1
cj
2
8
bj
6n
2 1
2n
2 1 aj
cj
bj
0 1
2n2a j 0 2
12 n2 2
0 2
0 2 aj
dj + 6 n
0 1 2n2
4n
0
2
2 j
j
(79)
= 0, 1160
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
Q
1
coeff.:
2n
0 1 ((2
+6 n
j 0
dj
3 cj +
+6 n
dj +
+6 n
0 j
bj
aj ) Q
+ (4
0 j
+6 n
0
4
2n2a 2 j 0
+ 4 n2 bj
j
3nc
4
j
2n
2 na
dj + 4
4 n bj
j
+ 4 n ) Q ) = 0,
(80)
Q0 coeff.:
4
3n2c 2 j 0
4 n2
0
3
2n2
+4
4 n2
j
dj 0
3
0
2
4 n2
j
0
2
0
2
4 n2
j 0
3
4 n2
0
3
j
(81)
= 0,
by virtue of inserting the auxiliary equation (74) into the ordinary differential equation (73) and later on setting of the coefficient of the same functions to zero. As long as solving the above equations, the following results are given by 0
= 0,
2
=
2
1
= 0,
n2 ( j + 3c d j j
2
dj
3
2(
=
2d
bj
1
,
1 + j )( 2dj + bj
j
+
j
2 nb
j cj
+3
aj bj
j
cj + aj ) +
2
bj dj +
n bj 2
n bj + 3 cj + naj
1)
(2 3n dj 2
3
2b c j j
2
3nc
2 n dj
j dj
+2
3
dj 2 + 3
najbj +
2n
bj dj
bj 2 + 3 ncj (82)
bj + aj ).
The trivial solutions are reached out by using of these results. It is not possible for the purpose of recovering optical soliton pulses in case of full nonlinearity. 3. Conclusions The Biswas-Arshed model in birefringent fibers without FWM was perused in the cause of catching up optical solitons of its in this paper. As long as parameter restrictions, optical dark, bright as well as singular solitons were imparted by modified simple equation approach. It is not possible for the sake of recovering these very valuable optical solitons in case of full nonlinearity. Singular periodic solutions also were granted along with the reverse form of the parameter constraints. The results obtained in this study give rise to peruse the model as more elaborate. The Biswas-Arshed equation firstly will be generalized in DWDM system that subsequently is going to be handled with strategic procedures for example extended Kudryashov's method, trial equation integration architecture and modified simple equation algorithm. And furthermore, perturbation terms for example Raman scattering, saturable amplifiers, multi–photon absorption is going to be added. These precious consequences sequentially are going to be reported. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
A. Biswas, S. Arshed, Optical solitons in presence of higher order dispersions and absence of self-phase modulation, Optik 174 (2018) 452–459. M. Ekici, A. Sonmezoglu, Optical solitons with Biswas-Arshed equation by extended trial function method, Optik 177 (2019) 13–20. S. Aouadi, A. Bouzida, A.K. Daoui, H. Triki, Q. Zhou, S. Liu, W-shaped, bright and dark solitons of Biswas-Arshed equation, Optik (2019). Y. Yildirim, Optical solitons of Biswas-Arshed equation by trial equation technique, Optik (2019). Y. Yildirim, Optical solitons of Biswas-Arshed equation in birefringent fibers by trial equation technique, Optik (2019). Y. Yildirim, Optical solitons of Biswas-Arshed equation by modified simple equation technique, Optik (2019). Y. Yildirim, Bright, dark and singular optical solitons to Kundu–Eckhaus equation having four-wave mixing in the context of birefringent fibers by using of modified simple equation methodology, Optik (2019). Y. Yildirim, Optical solitons to Schrödinger–Hirota equation in DWDM system with modified simple equation integration architecture, Optik (2019). Y. Yildirim, Optical solitons of Gerdjikov–Ivanov equation in birefringent fibers with modified simple equation scheme, Optik (2019). Y. Yildirim, Optical solitons of Gerdjikov–Ivanov equation with four-wave mixing terms in birefringent fibers by modified simple equation methodology, Optik (2019). 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. E. Yaşar, Y. Yıldırım, Q. Zhou, S.P. Moshokoa, M.Z. Ullah, H. Triki, A. Biswas, M. Belic, Perturbed dark and singular optical solitons in polarization preserving fibers by modified simple equation method, Superlattices Microstruct. 111 (2017) 487–498. A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with Gerdjikov–Ivanov equation by modified simple equation method, Optik 157 (2018) 1235–1240. A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with full nonlinearity for Kundu–Eckhaus equation by modified simple equation method, Optik 157 (2018) 1376–1380. A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with resonant nonlinear Schrödinger's equation having full nonlinearity by modified simple equation method, Optik 160 (2018) 33–43. A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for complex Ginzburg–Landau equation with modified simple equation method, Optik 158 (2018) 399–415. A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton solutions to Fokas-lenells equation using some different methods, Optik 173 (2018) 21–31. A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, M.F. Mahmood, S.P. Moshokoa, M. Belic, Optical solitons with differential group delay for coupled Fokas–Lenells equation using two integration schemes, Optik 165 (2018) 74–86. A. Biswas, Y. Yıldırım, E. Yaşar, R.T. Alqahtani, Optical solitons for Lakshmanan–Porsezian–Daniel model with dual-dispersion by trial equation method, Optik 168 (2018) 432–439.
1161
Optik - International Journal for Light and Electron Optics 182 (2019) 1149–1162
Y. Yildirim
[20] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with complex Ginzburg–Landau equation using trial solution approach, Optik 160 (2018) 44–60. [21] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, S.P. Moshokoa, M. Belic, Sub pico-second pulses in mono-mode optical fibers with Kaup–Newell equation by a couple of integration schemes, Optik 167 (2018) 121–128. [22] A. Biswas, Y. Yildirim, E. Yasar, M.F. Mahmood, A.S. Alshomrani, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for Radhakrishnan–Kundu–Lakshmanan equation with a couple of integration schemes, Optik 163 (2018) 126–136. [23] M. Mirzazadeh, Y. Yıldırım, E. Yaşar, H. Triki, Q. Zhou, S.P. Moshokoa, M.Z. Ullah, A.R. Seadawy, A. Biswas, M. Belic, Optical solitons and conservation law of Kundu–Eckhaus equation, Optik 154 (2018) 551–557. [24] A. Biswas, Y. Yıldırım, E. Yaşar, M.M. Babatin, Conservation laws for Gerdjikov–Ivanov equation in nonlinear fiber optics and PCF, Optik 148 (2017) 209–214. [25] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with full nonlinearity for Gerdjikov–Ivanov equation by trial equation method, Optik 157 (2018) 1214–1218. [26] A. Biswas, Y. Yıldırım, E. Yaşar, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Solitons for perturbed Gerdjikov–Ivanov equation in optical fibers and PCF by extended Kudryashov's method, Opt. Quantum Electron. 50 (3) (2018) 149. [27] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Dispersive optical solitons with Schrödinger–Hirota model by trial equation method, Optik 162 (2018) 35–41. [28] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Dispersive optical solitons with differential group delay by a couple of integration schemes, Optik 162 (2018) 108–120.
1162