Optical solitons in birefringent fibers having anti-cubic nonlinearity with exp-function

Optik - International Journal for Light and Electron Optics 186 (2019) 363–368

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Original research article

Optical solitons in birefringent fibers having anti-cubic nonlinearity with exp-function

T

Anjan Biswasa,b,c, Mehmet Ekicid, , Abdullah Sonmezoglud, Milivoj R. Belice ⁎

a

Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762–7500, USA Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa d Department of Mathematics, Faculty of Science and Arts, Yozgat Bozok University, 66100 Yozgat, Turkey e Institute of Physics Belgrade, Pregrevica 118, 11080 Zemun, Serbia b c

ARTICLE INFO

ABSTRACT

OCIS: 060.2310 060.4510 060.5530 190.3270 190.4370 Keywords: Solitons Anti-cubic nonlinearity Birefringence exp-function

This paper studies optical solitons in birefringent fibers with anti-cubic nonlinearity. The expfunction scheme retrieves singular and combo solitons. The existence criteria for these solitons are also presented.

1. Introduction Optical solitons are universally studied in the field of telecommunications with a wide variety of non-Kerr law fibers. There are several forms of nonlinearity that are addressed in this context. One such form of non-Kerr law is the anti-cubic (AC) form that has been extensively studied during the past decade [1–14,16–19,22–25], since it was first proposed by Fedele et al during 2003 [14]. While most of the results appeared with polarization preserving fibers, it is about time to start digging further along into birefringent fibers with AC nonlinearity. Elementary results, in this context, have been reported by extended trial function method [10]. This paper uncovers additional secret of AC nonlinearity in birefringent fibers by the aid of exp-function scheme which has been successfully studied previously in the context of optics and fluid dynamics [15,20,21]. This integration methodology recovers singular and bright-singular combo solitons for AC nonlinearity. The second form of solitons can only be recovered by the aid of exp-function which is an unique capacity of this algorithm. The details of the scheme and the spectrum of soliton solutions are all sculpted and enumerated in the remainder of the paper after a quick intro to the governing model with AC nonlinearity for birefringent fibers. 1.1. Governing model For polarization preserving fibers, the dynamics of soliton propagation through optical fibers with AC nonlinearity is governed by

⁎

Corresponding author. E-mail address: [email protected] (M. Ekici).

https://doi.org/10.1016/j.ijleo.2019.04.121 Received 24 February 2019; Accepted 25 April 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.

Optik - International Journal for Light and Electron Optics 186 (2019) 363–368

A. Biswas, et al.

the nonlinear Schrödinger's equation (NLSE) [1–9,11,12,14,16–19,22–25]:

iqt + aq xx +

m1 + m2 |q|2 + m3 |q|4 q = 0 |q|4

(1)

where the first term is the linear temporal evolution, while a is the group velocity dispersion (GVD) coefficient and the constant coefficients mj for j = 1, 2, 3 form nonlinear terms. In case of birefringent fibers, Eq. (1) splits into the following set of coupled equations, after neglecting 4WM effects [10]:

iut + a1 u xx + ivt + a2 vxx +

u + (p1 |u|2 + q1 |v|2 ) u + ( b1 |u|4 + c1 |u|2 |v|2 + d1 |v|4 v + (p2 |v|2 + q2 |u|2 ) v + ( b2 |v|4 + c2 |v|2 |u|2 + d2 |u|4

1

2

|u|4 + |v|4 +

|u|2 |v|2 +

1

2

|v|2 |u|2 +

|v|4 ) u = 0

1

2

|u|4 ) v = 0.

(2) (3)

From Eqs. (2) and (3), aj are the coefficients of GVD for the two components. From the nonlinear terms, bj, pj and ξj are the coefficients of self–phase modulation, while cj, dj, qj, ηj and ζj are from cross–phase modulation effects. Eqs. (2) and (3) are now going to be analyzed after a quick re-visitation of the basics. 2. Preliminaries To overcome (2) and (3), the starting hypothesis is:

u (x , t ) = P1 ( ) ei

1

(4)

v (x , t ) = P2 ( ) ei

2

(5)

where Pl(η) for l = 1, 2 denote the amplitude portion of the wave and

=x

(6)

ct

with c is the speed of the soliton. Next the phase part ϕl reads as l

=

lx

+

lt

+

(7)

l

for l = 1, 2. From (7), κl are the frequencies of the solitons, ωl are the wave numbers and finally θl are the phase constants. Now insert (4) and (5) into (2) and (3) and split into real and imaginary parts. Thus real parts are

Pl

bl (

+ al

l

2 5 l ) Pl

cl (

l

+ al

2 3 2 l ) Pl Pl¯

dl (

+ (cl ql + dl pl ) Pl3 Pl¯4 + dl ql Pl Pl¯6 + bl l Pl9 + (bl + (c l l +

dl l ) Pl3 Pl¯6

+

dl l Pl Pl¯8

al bl Pl4 Pl

+

+

l

l

+ al

2 4 l ) Pl Pl¯

+ bl pl Pl7 + (bl ql + cl pl ) Pl5 Pl¯2

+ cl l ) Pl7 Pl¯2 + (bl l + cl

al cl Pl2 Pl¯2 Pl

+

al dl Pl¯4 Pl

l

+ dl l ) Pl5 Pl¯4

=0

(8)

and imaginary parts give

(c + 2al l )(bl Pl4 + cl Pl2 Pl¯2 + dl Pl¯4 ) Pl = 0 for l = 1, 2 and l¯ = 3

(9)

l . Next, utilizing the balancing principle cause (10)

P l¯ = Pl. Therefore, Eqs. (8) and (9) change to:

Pl

(bl + cl + dl )(

l

+ al

2 5 l ) Pl

+ (bl + cl + dl )(pl + ql ) Pl7 + (bl + cl + dl )( l +

l

+ l ) Pl9 + al (bl + cl + dl ) Pl4 Pl = 0

(11) (12)

(bl + cl + dl )(c + 2al l ) Pl4 Pl = 0. From (12)

c=

2al

(13)

l

together with the constraint relation

a1

1

a2

2

(14)

=0

as long as

bl + cl + dl

Now, balancing the terms

N=

(15)

0.

Pl4 Pl

and

Pl9

in Eq. (11) gives

1 . 2

(16)

In order to reduce Eq. (11) to the suitable form, the applied transformation is 364

Optik - International Journal for Light and Electron Optics 186 (2019) 363–368

A. Biswas, et al.

(17)

Pl = Ul1/2 and then (11) changes to

4(bl + cl + dl )(

4

2 2 l ) Ul + 4(bl 2 (Ul ) ) = 0.

+ al

l

+ al (bl + cl + dl )(2Ul Ul 3. exp(

+ cl + dl )(pl + ql ) Ul3 + 4(bl + cl + dl )( l +

l

+ l ) Ul4 (18)

( )) -expansion scheme

To extract solitons to the model by the adopted scheme [13,15,20,21], the starting assumption is: N

(l ) j

Ul ( ) =

( )]) j

(exp[

(19)

j=1

where

(l) j

are constants to be detected later, such that

( ) = exp[

(l) N

0 , and the function ψ(η) holds (20)

( )] + exp[ ( )] + .

Here, it is important to remark that the solutions of Eq. (20) are exhibited as: If ζ ≠ 0 and ϑ2 − 4ζ > 0, 2

2

4 tanh

4

2

( ) = ln

( + c) + .

2

(21) 2

For ζ ≠ 0 and ϑ − 4ζ < 0, 2

4

2

4

tan

2

( ) = ln

( + c) .

2

(22) 2

When ζ = 0, ϑ ≠ 0 and ϑ − 4ζ > 0,

( )=

ln

exp( ( + c ))

1

.

(23)

2

Whenever ζ ≠ 0, ϑ ≠ 0 and ϑ − 4ζ = 0,

( ) = ln

2( ( + c ) + 2) . 2 ( + c)

(24)

2

However, if ζ = 0, ϑ = 0 and ϑ − 4ζ = 0, (25)

( ) = ln[ + c]. Here c is the integration constant. Balancing

Ul ( ) =

(l) 0

(l ) 1

+

exp[

Ul4

with Ul Ul in (18) yields N = 1. Thus, one reaches (26)

( )].

Putting (26) into (18), designing the coefficients of exp( Set-1: (l ) 0

(l ) 1

= 0,

(l ) 1 ,

2

4

2 +

2 l

, + cl + dl )( 1(l) ) 2 4 al = , 2 (b + c + d )( (l) )2 l l l 1 4 pl = ql 2 (b + c + d )( l l l l

l

=

=

=

( )) , and accomplishing the resulting system, one has two solutions set as:

2 (b l

3+

2 (b l

+ cl + dl )(

2 (b l

l

+ cl + dl )(

(l) 3 1 )

,

+ l )( (l) 4 1 )

(l ) 4 1 )

,

(27)

365

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A. Biswas, et al.

Set-2: (l ) 0 l

=

al = pl = l

=

(l ) (l ) (l ) 0 , 1 = 1 , (l ) 2 (l) (l) 6( 0 ) 6 0 1 + (2 + 2 4 l2 )( 1(l) ) 2 , (l) (l) (l ) 2 2 (bl + cl + dl )[ ( 1(l) )2 0 1 + ( 0 ) ] 4( 1(l) )2 , (l ) (l ) (l ) 2 2 (bl + cl + dl )[ ( 1(l) ) 2 0 1 + ( 0 ) ] 8 0(l) 4 1(l) ql + , (l) (l) (l ) 2 2 (bl + cl + dl )[ ( 1(l) )2 0 1 + ( 0 ) ] (l) (l) (l) 2 2 3 + (bl + cl + dl )( l + l )[ ( 1(l) ) 2 0 1 + ( 0 ) ] (l ) 2 (l ) (l ) (l ) 2 2 (bl + cl + dl )[ ( 1 ) 0 1 + ( 0 ) ]

=

.

(28)

Lastly, soliton and other solutions to (2) and (3) can be derived as: The results of (27) lead to 1 2 (1) 1

2

u (x , t ) = 2

+

2

4 tanh

exp i

4

1x

+

(c + )

2

2 + 2 (b 1

2

2 1

4

(1) 2 1 )

+ c1 + d1 )(

t+

1

(29)

1/2 (2) 1

2

v (x , t ) = 2

+

2

4 tanh

exp i

4

2x

2 (b

(c + )

2

2

2 +

+

2 2

4

+ c2 + d2 )(

2

(2) 2 1 )

t+

2

(30)

1/2 (1) 1

2

u (x , t ) = 2

4

2

4

tan

2

exp i

1x

+

(c + )

2

2 + 2 (b

1

4

2 1 (1) 2 1 )

+ c1 + d1 )(

t+

1

(31)

1/2

2

v (x , t ) = 2

4

u (x , t ) =

v (x , t ) =

(2) 1

2 (1) 1

(c + ) 2 (c + ) + 4 2 (2) 1

(c + ) 2 (c + ) + 4

tan

2

4 2

exp i

2x

+

(c + )

1/2

exp i

1x

+

exp i

2x

+

8(3 4 (b 1

1/2

2

8

2 1)

+ c1 + d1 )(

8(3 4 (b

2

2

8

2 + 2 (b

(1) 2 1 )

2 2)

+ c2 + d2 )(

(2) 2 1 )

2

2

4

2 2

+ c2 + d2 )(

t+

1

t+

2

(2) 2 1 )

t+

2

(32)

(33)

.

(34)

The results of (28) bring about 1/2

u (x , t )=

2

(1) 0 2

+ × exp i

1x

+

(1) 1 2

4 tanh 6(

(1) 2 0 )

4

2

(c + )

(1) (1) 1 + (2 0 d1 )[ ( 1(1) )2

6

(b1 + c1 +

2

+

(1) (1) 1 0

(1) 2 2 1 )( 1 ) (1) 2 2 +( 0 )]

t+

1

(2) 2 2 2 )( 1 ) (2) 2 2 +( 0 )]

t+

2

4

(35)

1/2

v (x , t )=

2

(2) 0 2

+ × exp i

2x

+

(2) 1 2

4 tanh 6(

(2) 2 0 )

(b2 + c2 +

2

4

(c + )

(2) (2) 1 + (2 0 d2 )[ ( 1(2) )2

6

+

2 (2) (2) 1 0

366

4

(36)

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A. Biswas, et al. 1/2

2

(1) 0

u (x , t )=

(1) 1

2

4 × exp i

1x

+

2

(1) 2 0 )

6(

2

4

tan

(c + )

(1) (1) 1 + (2 0 d1 )[ ( 1(1) )2

6

(b1 + c1 +

2

+

(1) 2 2 1 )( 1 ) (1) 2 2 +( 0 )]

t+

1

(2) 2 2 2 )( 1 ) (2) 2 2 +( 0 )]

t+

2

4

(1) (1) 1 0

(37)

1/2

2

(2) 0

v (x , t )=

4 × exp i

(1) 0

u (x , t )=

+

2x

(2) 0

+

(1) 0

v (x , t ) =

(2) 0

(c + )

(2) (2) 1 + (2 0 d2 )[ ( 1(2) )2

6

(b2 + c2 +

2

6(

(1) (1) 1 0

6

+(

(b1 + c1 + d1 )((

1

(2) 2 0 )

2

6(

(c + ) 2 (c + ) + 4 2 (2) 1

(2) (2) 1 0

6

(c + ) 2 (c + ) + 4

+(

1/2

exp i

1x

+

exp i

2x

+

1/2

u (x , t ) = {

(1) 0

+

(1) 1

(c + )}1/2exp i

1x

+

v (x , t ) = {

(2) 0

+

(2) 1

(c + )}1/2exp i

2x

+

6(

(1) 2 0 )

8(12(

1

t+

2

12

(1) (1) 0 1

(b1 + c1 + d1 )( (2) 2 0 )

8(12(

12

(2) (2) 0 1

(b2 + c2 + d2 )( (1) 2 0 )

(b1 + c1 + 6(

(2) 2 2 2 )( 1 ) (2) (2) 2 1 ) 0

4

(2) 2 0 )

(b2 + c2 + d2 )((

2 (1) 1

t+

(39)

1/2

cosh[ (c + )] + sinh[ (c + )] +

(1) 2 2 1 )( 1 ) (1) (1) 2 1 ) 0

4

(1) 2 0 )

(2) 1

2x

(38)

1/2

(1) 2 0 )

+

4

(2) (2) 1 0

1

1x

2

+

cosh[ (c + )] + sinh[ (c + )]

× exp i

u (x , t ) =

2

(2) 2 0 )

6(

2

4

tan

(1) 1

× exp i

v (x , t )=

+

2

(2) 1

(2) 2 0 )

(b2 + c2 +

(1) 2 2 1 ( 1 ) (1) 4 d1 )( 0 )

t+

1

(2) 2 2 2 ( 1 ) (2) 4 d2 )( 0 )

t+

2

4

4

(40)

+ (3 (1) 1

+ (3 (2) 1

2

2 2

2

(1) 2 2 1 )( 1 ) ) (1) 4 0 )

t+

1

8 22 )( 1(2) ) 2) (2) 4 0 )

t+

2

8

(41)

(42)

(43)

.

(44)

From the spectrum of solutions, Eqs. (29)–(30) and (35)–(36) represent singular solitons while the relations (39)–(40) give bright–singular combo solitons. Next, periodic-singular solutions are represented in (31)–(32) and (37)–(38). Finally, the plane waves are given by (33)–(34), (41)–(42) and (43)–(44). 4. Conclusions This paper is a second set of results after studying optical solitons with AC nonlinearity in birefringent fibers. The first set of results have been reported earlier during 2019 [10]. This paper applied exp-function scheme to retrieve singular and bright-singular combo solitons. The latter kind of solitons with birefringent, fibers having AC nonlinearity, are being reported for the first time in this paper. Thus, this integration scheme has its pros and cons. The unique advantage is its ability to secure such combo solitons which other algorithms fail to report, while a disadvantage is that the current algorithm fails to retrieve bright optical solitons for the model. This does not stop here. Later on, additional integration schemes will be implemented to study NLSE with AC nonlinearity in birefringent fibers. Some of them are Lie symmetry analysis, Kudryashov's method, Riccati–Bernoulli sub-ODE method and many others. It is just a hope that such additional integration norms will enable additional mysteries to be unraveled! Conflict of interest The authors also declare that there is no conflict of interest. 367

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Acknowledgements The research work of the fourth author (MRB) was supported by the grant NPRP 8-028-1-001 from QNRF and he is thankful for it. References [1] K.S. Al-Ghafri, E.V. Krishnan, A. Biswas, M. Ekici, Optical solitons having anti-cubic nonlinearity with a couple of exotic integration schemes, Optik 172 (2018) 794–800. [2] A.H. Arnous, A. Biswas, M. Asma, M. Belic, Dark and singular solitons in optical metamaterials with anti-cubic nonlinearity by modified simple equation approach, Optoelectron. Adv. Mater. Rapid Commun. 12 (5-6) (2018) 332–336. [3] A. Biswas, Q. Zhou, M.Z. Ullah, M. Asma, S.P. Moshokoa, M. Belic, Perturbation theory and optical soliton cooling with anti-cubic nonlinearity, Optik 142 (2017) 73–76. [4] A. Biswas, Q. Zhou, M.Z. Ullah, H. Triki, S.P. Moshokoa, M. Belic, Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle, Optik 143 (2017) 131–134. [5] A. Biswas, Q. Zhou, S.P. Moshokoa, H. Triki, M. Belic, R.T. Alqahtani, Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations, Optik 145 (2017) 14–17. [6] A. Biswas, A.J.M. Jawad, Q. Zhou, Resonant optical solitons with anti-cubic nonlinearity, Optik 157 (2018) 525–531. [7] A. Biswas, M. Ekici, A. Sonmezoglu, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Solitons in optical metamaterials with anti-cubic nonlinearity, Eur. Phys. J. Plus 133 (5) (2018) Article-204. [8] A. Biswas, Conservation laws for optical solitons with anti-cubic and generalized anti-cubic nonlinearities, Optik 176 (2019) 198–201. [9] A. Biswas, M. Ekici, A. Sonmezoglu, M. Belic, Chirped and chirp-free optical solitons with generalized anti-cubic nonlinearity by extended trial function scheme, Optik 178 (2019) 636–644. [10] A. Biswas, M. Ekici, A. Sonmezoglu, M.R. Belic, Optical solitons in birefringent fibers having anti-cubic nonlinearity with extended trial function, Optik 185 (2019) 456–463. [11] M. Ekici, M. Mirzazadeh, A. Sonmezoglu, M.Z. Ullah, Q. Zhou, H. Triki, S.P. Moshokoa, A. Biswas, Optical solitons with anti-cubic nonlinearity by extended trial equation method, Optik 136 (2017) 368–373. [12] M. Ekici, A. Sonmezoglu, Q. Zhou, S.P. Moshokoa, M.Z. Ullah, A.H. Arnous, A. Biswas, M. Belic, Analysis of optical solitons in nonlinear negative-indexed materials with anti-cubic nonlinearity, Optical Quantum Electron. 50 (2) (2018) Article-75. [13] M. Ekici, M. Mirzazadeh, A. Sonmezoglu, Q. Zhou, H. Triki, M.Z. Ullah, S.P. Moshokoa, A. Biswas, Optical solitons in birefringent fibers with Kerr nonlinearity by exp-function method, Optik 131 (2017) 964–976. [14] R. Fedele, H. Schamel, V.I. Karpman, P.K. Shukla, Envelope solitons of nonlinear Schrödinger equation with an anti-cubic nonlinearity, J. Phys. A 36 (2003) 1169–1173. [15] F. Ferdous, M.G. Hafez, A. Biswas, M. Ekici, Q. Zhou, M. Alfiras, S.P. Moshokoa, M. Belic, Oblique resonant optical solitons with Kerr and parabolic law nonlinearities and fractional temporal evolution by generalized exp(−ϕ(ξ))-expansion, Optik 178 (2019) 439–448. [16] M. Foroutan, J. Manafian, I. Zamanpour, Soliton wave solutions in optical metamaterials with anti-cubic law of nonlinearity by ITEM, Optik 164 (2018) 371–379. [17] M. Foroutan, J. Manafian, A. Ranjbaran, Solitons in optical metamaterials with anti-cubic law of nonlinearity by generalized G′/G-expansion method, Optik 162 (2018) 86–94. [18] M. Foroutan, J. Manafian, A. Ranjbaran, Solitons in optical metamaterials with anti-cubic law of nonlinearity by ETEM and IGEM, J. Eur. Optical Soc. Rapid Publ. 14 (2018) Article–16. [19] A.J.M. Jawad, M. Mirzazadeh, Q. Zhou, A. Biswas, Optical solitons with anti-cubic nonlinearity using three integration schemes, Superlattices Microstruct. 105 (2017) 1–10. [20] N. Kadkhoda, H. Jafari, Analytical solutions of the Gerdjikov–Ivanov equation by using exp(−ϕ(ξ))-expansion method, Optik 139 (2017) 72–76. ( )) -expansion method to find the exact solutions of modified Benjamin–Bona–Mahony equation, World Appl. Sci. J. [21] K. Khan, M.A. Akbar, Application of exp( 24 (10) (2013) 1373–1377. [22] S. Khan, A. Biswas, Q. Zhou, S. Adesanya, M. Alfiras, M. Belic, Stochastic perturbation of optical solitons having anti-cubic nonlinearity with bandpass filters and multi-photon absorption, Optik 178 (2019) 1120–1124. [23] E.V. Krishnan, A. Biswas, Q. Zhou, M.M. Babatin, Optical solitons with anti-cubic nonlinearity by mapping methods, Optik 170 (2018) 520–526. [24] M. Mirzazadeh, A. Biswas, A.S. Alshomrani, M.Z. Ullah, M. Asma, S.P. Moshokoa, Q. Zhou, M. Belic, Optical solitons having fractional temporal evolution having anti-cubic nonlinearity, Optoelectron. Adv. Mater. Rapid Commun. 12 (1-2) (2018) 68–70. [25] E.M.E. Zayed, M.E.M. Alngar, A.-G. Al-Nowehy, On solving the nonlinear Schrödinger equation with an anti-cubic nonlinearity in presence of Hamiltonian perturbation terms, Optik 178 (2019) 488–508.

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