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Dynamics of optical solitons in the perturbed Gerdjikov–Ivanov equation
Journal Pre-proof Dynamics of optical solitons in the perturbed Gerdjikov–Ivanov equation K. Hosseini, M. Mirzazadeh, M. Ilie, S. Radmehr
PII:
S0030-4026(20)30184-4
DOI:
https://doi.org/10.1016/j.ijleo.2020.164350
Reference:
IJLEO 164350
To appear in:
Optik
Received Date:
27 January 2020
Accepted Date:
1 February 2020
Please cite this article as: Hosseini K, Mirzazadeh M, Ilie M, Radmehr S, Dynamics of optical solitons in the perturbed Gerdjikov–Ivanov equation, Optik (2020), doi: https://doi.org/10.1016/j.ijleo.2020.164350
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Dynamics of optical solitons in the perturbed Gerdjikov–Ivanov equation K. Hosseini1*, M. Mirzazadeh2*, M. Ilie1, S. Radmehr3 1
Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, PC 44891-63157 Rudsar-Vajargah, Iran 3 Department of Computer Engineering, University of Guilan, Rasht, Iran
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*Corresponding Authors: [email protected], [email protected]
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2
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Abstract:
The dynamics of solitons in the perturbed Gerdjikov–Ivanov (PGI) equation is explored in the current
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work. The investigation is carried out formally by considering particular transformations and exerting newly well-established methods to obtain optical solitons of the model. The kink, bright, and dark
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solitons of the PGI equation with diverse applications in optical fibers are successfully recovered. Keywords: Perturbed Gerdjikov–Ivanov equation; Optical fibers; Newly well-established methods; Kink,
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bright, and dark solitons
1. Introduction
Nonlinear partial differential equations are utilized to describe many problems in quantum mechanics, plasma physics, and nonlinear optics. Due to the availability of computer algebra systems and their effectiveness to handle symbolic computations, a wide range of capable methods has been introduced and
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developed [1-32] for dealing with nonlinear PDEs. Our aim of the present paper is to consider and study the perturbed Gerdjikov–Ivanov equation [1-6] 𝑖𝜓𝑡 + 𝛼1 𝜓𝑥𝑥 + 𝛼2 |𝜓|4 𝜓 + 𝑖𝛼3 𝜓 2 𝜓𝑥∗ − 𝑖(𝛼4 𝜓𝑥 + 𝛼5 (|𝜓|2 𝜓)𝑥 + 𝛼6 (|𝜓|2 )𝑥 𝜓) = 0,
(1)
by using the exp𝑎 -function and Kudryashov methods [25-32]. Here 𝜓 is a complex-valued function, 𝜓𝑡 is the linear temporal evolution, 𝜓𝑥𝑥 is the group velocity dispersion, |𝜓|4 𝜓 is the quintic nonlinearity, 𝛼4 is the inter-modal dispersion, (|𝜓|2 𝜓)𝑥 is the self-steepening term, and ultimately (|𝜓|2 )𝑥 𝜓 is the nonlinear dispersion. It should be noted that when 𝛼4 = 𝛼5 = 𝛼6 = 0, the PGI equation is reduced to the GI 1
equation. Furthermore, compared to the classical nonlinear Schrödinger equation that includes typically the cubic nonlinearity, the PGI equation benefits from the quintic nonlinearity. The perturbed Gerdjikov– Ivanov equation has been examined by a number of effective techniques, for example, Biswas and Alqahtani [1] utilized the semi-inverse variational principle to extract chirp-free bright optical solitons of the PGI equation. Optical solitons of the PGI equation were derived by Kaur and Wazwaz [2] using the exp(−𝜙(𝜉)) and (𝐺 ′ ⁄𝐺 2 ) expansion methods. Yaşar et al. [3] listed new optical solitons of the PGI equation by means of the sine-Gordon expansion method. Arshed et al. [4] applied the sine-cosine method to generate bright and dark solitons of the PGI equation. Bright and singular solitons of the PGI equation
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were obtained by Biswas et al. [5] through the extended trial equation method. Biswas et al. [6] also reported optical solitons of the PGI equation using the trial equation method. The organization of this
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paper is as follows: In Section 2, we summarize the key steps of the exp𝑎 -function and Kudryashov methods. In Section 2, we derive optical solitons of the perturbed Gerdjikov–Ivanov equation. Section 4
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finalizes the outcomes of the current study. 2. Methods and their ideas
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The present section formally summarizes the key steps of the exp𝑎 -function and Kudryashov methods. For this purpose, consider the following nonlinear PDE 𝜕𝜓 𝜕𝜓 𝜕𝑥
,
𝜕𝑡
, … ) = 0.
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𝑃 (𝜓,
(2)
Suppose that the above nonlinear PDE can be converted to a nonlinear ODE as (3)
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𝑂(𝜌, 𝜌′ , 𝜌′′ , … ) = 0,
through adopting the transformation 𝜓(𝑥, 𝑡) = 𝜌(𝜉)𝑒 𝑖(−𝜅𝑥+𝑡𝜔+𝜃) ,
𝜉 = 𝑥 − 𝑣𝑡.
2.1. 𝑬𝒙𝒑𝒂 -function method
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The exp𝑎 -function method benefits from adopting a test function in the following form 𝜌(𝜉) =
𝑎0 +𝑎1 𝑎𝜉 +⋯+𝑎𝑁 𝑎𝑁𝜉 𝑏0 +𝑏1 𝑎𝜉 +⋯+𝑏𝑁 𝑎𝑁𝜉
,
𝑎𝑁 ≠ 0,
𝑏𝑁 ≠ 0,
(4)
in which 𝑎𝑖 (0 ≤ 𝑖 ≤ 𝑁) and 𝑏𝑖 (0 ≤ 𝑖 ≤ 𝑁) are found later and 𝑁 ∈ ℕ. Inserting the test function (4) into Eq. (3) and exerting symbolic computations results in an algebraic system in the form 𝐶𝑖 = 0,
𝑖 = 0,1, . . . , 𝜎,
which solving it using Maple gives optical solitons of the nonlinear PDE (2). 2
2.2. Kudryashov method (New version) The Kudryshov method assumes the solution of the nonlinear ODE (3) can be presented as 𝜌(𝜉) = 𝑎0 + 𝑎1 𝑅(𝜉) + ⋯ + 𝑎𝑁 𝑅(𝜉)𝑁 ,
𝑎𝑁 ≠ 0,
(5)
where 𝑎𝑖 (0 ≤ 𝑖 ≤ 𝑁) are unknows, 𝑁 is the order of the pole of the general solution, and 𝑅(𝜉) is the solution of the equation 𝑅′ (𝜉)2 = 𝑅(𝜉)2 (1 − 𝜂𝑅(𝜉)2 ),
𝑅(𝜉) =
4𝐴 4𝐴2 𝑒 𝜉 +𝜂𝑒 −𝜉
,
𝜂 = 4𝐴𝐵.
3. Perturbed Gerdjikov–Ivanov equation and its optical solitons To solve the PGI equation, we adopt a wave transformation as 𝜉 = 𝑥 − 𝑣𝑡,
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𝜓(𝑥, 𝑡) = 𝜌(𝜉)𝑒 𝑖(−𝜅𝑥+𝜔𝑡+𝜃) ,
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system whose solution leads to optical solitons of the nonlinear PDE (2).
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Substituting the function (5) into Eq. (3) and exerting symbolic computations results in an algebraic
(6)
where v is the soliton velocity, 𝜅 is the frequency of the soliton, 𝜔 is the wave number, and 𝜃 is the phase
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constant.
Due to the PGI equation and considering the wave transformation (6), we find d2 𝜌(𝜉) d𝜉 2
− (𝜔 + 𝛼1 𝜅 2 + 𝛼4 𝜅)𝜌(𝜉) − (𝛼3 𝜅 + 𝛼5 𝜅)𝜌(𝜉)3 + 𝛼2 𝜌(𝜉)5 = 0,
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𝛼1
(𝜈 + 2𝛼1 𝜅 + 𝛼4 ) − (𝛼3 − 3𝛼5 − 2𝛼6 )𝜌(𝜉)2 = 0.
(7) (8)
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Setting the coefficients of Eq. (8) equal to zero yields 𝜈 = −(2𝛼1 𝜅 + 𝛼4 ), 𝛼3 = 3𝛼5 + 2𝛼6 .
Now, by substituting the above relations into Eq. (7) and considering the transformation 𝜌(𝜉) = √𝑉 (𝜉) , the second-order nonlinear ODE (7) is converted to d𝑉(𝜉) 2 d𝜉
) − 2𝛼1 𝑉(𝜉)
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𝛼1 (
d2 𝑉(𝜉) + 4(𝛼1 𝜅 2 d𝜉 2
+ 𝛼4 𝜅 + 𝜔)𝑉(𝜉)2 + 8(2𝛼5 𝜅 + 𝛼6 𝜅)𝑉(𝜉)3 − 4𝛼2 𝑉(𝜉)4 =
0.
(9)
𝑬𝒙𝒑𝒂 -function method: In order to obtain optical solitons of the PGI equation using the exp𝑎 -function method, a test function is considered as below 𝑉(𝜉) =
𝑎0 +𝑎1 𝑎𝜉 𝑏0 +𝑏1 𝑎𝜉
,
which 𝑎0 , 𝑎1 , 𝑏0 , and 𝑏1 are constants to be determined later. Inserting the above test function into Eq. (9) and adopting specific operations yields 3
4𝜅 2 𝑎02 𝛼1 𝑏02 + 16𝜅𝑎03 𝛼5 𝑏0 + 8𝜅𝑎03 𝛼6 𝑏0 + 4𝜅𝑎02 𝛼4 𝑏02 + 4𝜔𝑎02 𝑏02 − 4𝑎04 𝛼2 = 0, 2
2
2(ln(𝑎)) 𝑎02 𝛼1 𝑏0 𝑏1 − 2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏02 + 8𝜅 2 𝑎02 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎0 𝑎1 𝛼1 𝑏02 + 16𝜅𝑎03 𝛼5 𝑏1 + 8𝜅𝑎03 𝛼6 𝑏1 + 48𝜅𝑎02 𝑎1 𝛼5 𝑏0 + 24𝜅𝑎02 𝑎1 𝛼6 𝑏0 + 8𝜅𝑎02 𝛼4 𝑏0 𝑏1 + 8𝜅𝑎0 𝑎1 𝛼4 𝑏02 + 8𝜔𝑎02 𝑏0 𝑏1 + 8𝜔𝑎0 𝑎1 𝑏02 − 16𝑎03 𝑎1 𝛼2 = 0, 2
2
2
−(ln(𝑎)) 𝑎02 𝛼1 𝑏12 + 2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏0 𝑏1 − (ln(𝑎)) 𝑎12 𝛼1 𝑏02 + 4𝜅 2 𝑎02 𝛼1 𝑏12 + 16𝜅 2 𝑎0 𝑎1 𝛼1 𝑏0 𝑏1 + 4𝜅 2 𝑎12 𝛼1 𝑏02 + 48𝜅𝑎02 𝑎1 𝛼5 𝑏1 + 24𝜅𝑎02 𝑎1 𝛼6 𝑏1 + 4𝜅𝑎02 𝛼4 𝑏12 + 48𝜅𝑎0 𝑎12 𝛼5 𝑏0 + 24𝜅𝑎0 𝑎12 𝛼6 𝑏0 + 16𝜅𝑎0 𝑎1 𝛼4 𝑏0 𝑏1 + 4𝜅𝑎12 𝛼4 𝑏02 + 4𝜔𝑎02 𝑏12 + 16𝜔𝑎0 𝑎1 𝑏0 𝑏1 + 4𝜔𝑎12 𝑏02 − 24𝑎02 𝑎12 𝛼2 = 0, 2
2
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−2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏12 + 2(ln(𝑎)) 𝑎12 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎0 𝑎1 𝛼1 𝑏12 + 8𝜅 2 𝑎12 𝛼1 𝑏0 𝑏1 + 48𝜅𝑎0 𝑎12 𝛼5 𝑏1 + 24𝜅𝑎0 𝑎12 𝛼6 𝑏1 + 8𝜅𝑎0 𝑎1 𝛼4 𝑏12 + 16𝜅𝑎13 𝛼5 𝑏0 + 8𝜅𝑎13 𝛼6 𝑏0 + 8𝜅𝑎12 𝛼4 𝑏0 𝑏1 + 8𝜔𝑎0 𝑎1 𝑏12 + 8𝜔𝑎12 𝑏0 𝑏1 − 16𝑎0 𝑎13 𝛼2 = 0,
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4𝜅 2 𝑎12 𝛼1 𝑏12 + 16𝜅𝑎13 𝛼5 𝑏1 + 8𝜅𝑎13 𝛼6 𝑏1 + 4𝜅𝑎12 𝛼4 𝑏12 + 4𝜔𝑎12 𝑏12 − 4𝑎14 𝛼2 = 0, which admits the following solutions
𝜔=
1 3
√− 𝛼1 𝛼2 ln(𝑎) 2𝛼5 +𝛼6 ln(𝑎)
12(2𝛼5 +𝛼6 )2
,
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𝜅=±
ln(𝑎)𝛼1
,
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𝑏1 = ∓2
1 3
√− 𝛼1 𝛼2 𝑎1
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𝑎0 = 0,
(4ln(𝑎)𝛼12 𝛼2 + 12ln(𝑎)𝛼1 𝛼52 + 12ln(𝑎)𝛼1 𝛼5 𝛼6 + 3ln(𝑎)𝛼1 𝛼62 ∓
1
1
3
3
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24√− 𝛼1 𝛼2 𝛼4 𝛼5 ∓ 12√− 𝛼1 𝛼2 𝛼4 𝛼6 ).
Now, one can derive the following solitons to the PGI equation
√
𝑎1 𝑎𝑥+(2𝛼1 𝜅+𝛼4 )𝑡
1 √− 𝛼1 𝛼2 𝑎1 3
𝑏0 ∓2
ln(𝑎)𝛼1
𝑒 𝑖(−𝜅𝑥+𝜔𝑡+𝜃) ,
𝛼1 𝛼2 < 0,
𝑎𝑥+(2𝛼1 𝜅+𝛼4 )𝑡
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𝜓1,2 (𝑥, 𝑡) =
where the soliton frequency and the wave number have been presented above. Three-dimensional and density plots of |𝜓1 (𝑥, 𝑡)| presenting a kink soliton have been given in Figure 1 for suitable values of the involved parameters. To extract other solitons of the PGI equation by using the exp𝑎 -function method, we adopt another test function as follows 𝑉(𝜉) =
𝑎0 +𝑎1 𝑎𝜉 +𝑎2 𝑎2𝜉 𝑏0 +𝑏1 𝑎𝜉 +𝑏2 𝑎2𝜉
,
(10) 4
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Figure 1: Three-dimensional and density plots of |𝜓1 (𝑥, 𝑡)| for 𝑎1 = 1, 𝑏0 = 1, 𝛼1 = −1, 𝛼2 = 1, 𝛼4 = −1, 𝛼5 = −1, 𝛼6 = −1, 𝜃 = 0, and 𝑎 = 2.7.
which 𝑎0 , 𝑎1 , 𝑎2 , 𝑏0 , 𝑏1 , and 𝑏2 are constants to be found later. Setting a test function like (10) in Eq. (9)
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and adopting specific operations results in
4𝜅 2 𝑎02 𝛼1 𝑏02 + 16𝜅𝑎03 𝛼5 𝑏0 + 8𝜅𝑎03 𝛼6 𝑏0 + 4𝜅𝑎02 𝛼4 𝑏02 + 4𝜔𝑎02 𝑏02 − 4𝑎04 𝛼2 = 0, 2
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2
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2(ln(𝑎)) 𝑎02 𝛼1 𝑏0 𝑏1 − 2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏02 + 8𝜅 2 𝑎02 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎0 𝑎1 𝛼1 𝑏02 + 16𝜅𝑎03 𝛼5 𝑏1 + 8𝜅𝑎03 𝛼6 𝑏1 + 48𝜅𝑎02 𝑎1 𝛼5 𝑏0 + 24𝜅𝑎02 𝑎1 𝛼6 𝑏0 + 8𝜅𝑎02 𝛼4 𝑏0 𝑏1 + 8𝜅𝑎0 𝑎1 𝛼4 𝑏02 + 8𝜔𝑎02 𝑏0 𝑏1 + 8𝜔𝑎0 𝑎1 𝑏02 − 16𝑎03 𝑎1 𝛼2 = 0, 4𝜅 2 𝑎02 𝛼1 𝑏12 + 4𝜅 2 𝑎12 𝛼1 𝑏02 + 16𝜅𝑎03 𝛼5 𝑏2 + 8𝜅𝑎03 𝛼6 𝑏2 + 4𝜅𝑎02 𝛼4 𝑏12 + 4𝜅𝑎12 𝛼4 𝑏02 + 8𝜔𝑎02 𝑏0 𝑏2 + 2
2
8𝜔𝑎0 𝑎2 𝑏02 − (ln(𝑎)) 𝑎02 𝛼1 𝑏12 − (ln(𝑎)) 𝑎12 𝛼1 𝑏02 + 16𝜅 2 𝑎0 𝑎1 𝛼1 𝑏0 𝑏1 + 16𝜅𝑎0 𝑎1 𝛼4 𝑏0 𝑏1 + 2
2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎02 𝛼1 𝑏0 𝑏2 + 8𝜅 2 𝑎0 𝑎2 𝛼1 𝑏02 + 48𝜅𝑎02 𝑎1 𝛼5 𝑏1 + 24𝜅𝑎02 𝑎1 𝛼6 𝑏1 + 48𝜅𝑎02 𝑎2 𝛼5 𝑏0 + 24𝜅𝑎02 𝑎2 𝛼6 𝑏0 + 8𝜅𝑎02 𝛼4 𝑏0 𝑏2 + 48𝜅𝑎0 𝑎12 𝛼5 𝑏0 + 24𝜅𝑎0 𝑎12 𝛼6 𝑏0 + 8𝜅𝑎0 𝑎2 𝛼4 𝑏02 + 2
2
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16𝜔𝑎0 𝑎1 𝑏0 𝑏1 + 8(ln(𝑎)) 𝑎02 𝛼1 𝑏0 𝑏2 − 8(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏02 + 4𝜔𝑎12 𝑏02 + 4𝜔𝑎02 𝑏12 − 16𝑎03 𝑎2 𝛼2 − 24𝑎02 𝑎12 𝛼2 = 0, 16𝜅𝑎13 𝛼5 𝑏0 + 8𝜅𝑎13 𝛼6 𝑏0 + 8𝜔𝑎02 𝑏1 𝑏2 + 8𝜔𝑎0 𝑎1 𝑏12 + 8𝜔𝑎12 𝑏0 𝑏1 + 8𝜔𝑎1 𝑎2 𝑏02 − 48𝑎02 𝑎1 𝑎2 𝛼2 + 16𝜅 2 𝑎0 𝑎1 𝛼1 𝑏0 𝑏2 + 16𝜅 2 𝑎0 𝑎2 𝛼1 𝑏0 𝑏1 + 96𝜅𝑎0 𝑎1 𝑎2 𝛼5 𝑏0 + 48𝜅𝑎0 𝑎1 𝑎2 𝛼6 𝑏0 + 16𝜅𝑎0 𝑎1 𝛼4 𝑏0 𝑏2 + 2
2
16𝜅𝑎0 𝑎2 𝛼4 𝑏0 𝑏1 + 16(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏0 𝑏2 − 8(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎12 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎1 𝑎2 𝛼1 𝑏02 + 48𝜅𝑎02 𝑎1 𝛼5 𝑏2 + 24𝜅𝑎02 𝑎1 𝛼6 𝑏2 + 48𝜅𝑎02 𝑎2 𝛼5 𝑏1 + 24𝜅𝑎02 𝑎2 𝛼6 𝑏1 + 8𝜅𝑎02 𝛼4 𝑏1 𝑏2 + 48𝜅𝑎0 𝑎12 𝛼5 𝑏1 + 24𝜅𝑎0 𝑎12 𝛼6 𝑏1 + 8𝜅𝑎0 𝑎1 𝛼4 𝑏12 + 8𝜅𝑎12 𝛼4 𝑏0 𝑏1 + 8𝜅𝑎1 𝑎2 𝛼4 𝑏02 + 16𝜔𝑎0 𝑎1 𝑏0 𝑏2 + 2
2
2
16𝜔𝑎0 𝑎2 𝑏0 𝑏1 − 2(ln(𝑎)) 𝑎02 𝛼1 𝑏1 𝑏2 − 2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏12 + 2(ln(𝑎)) 𝑎12 𝛼1 𝑏0 𝑏1 − 2
6(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏02 + 8𝜅 2 𝑎02 𝛼1 𝑏1 𝑏2 + 8𝜅 2 𝑎0 𝑎1 𝛼1 𝑏12 − 16𝑎0 𝑎13 𝛼2 = 0, 5
4𝜅 2 𝑎02 𝛼1 𝑏22 + 4𝜅 2 𝑎12 𝛼1 𝑏12 + 4𝜅 2 𝑎22 𝛼1 𝑏02 + 4𝜅𝑎02 𝛼4 𝑏22 + 16𝜅𝑎13 𝛼5 𝑏1 + 8𝜅𝑎13 𝛼6 𝑏1 + 4𝜅𝑎12 𝛼4 𝑏12 + 2
2
4𝜅𝑎22 𝛼4 𝑏02 + 8𝜔𝑎0 𝑎2 𝑏12 + 8𝜔𝑎12 𝑏0 𝑏2 − 48𝑎0 𝑎12 𝑎2 𝛼2 − 4(ln(𝑎)) 𝑎02 𝛼1 𝑏22 − 4(ln(𝑎)) 𝑎22 𝛼1 𝑏02 + 16𝜅 2 𝑎0 𝑎1 𝛼1 𝑏1 𝑏2 + 16𝜅 2 𝑎0 𝑎2 𝛼1 𝑏0 𝑏2 + 16𝜅 2 𝑎1 𝑎2 𝛼1 𝑏0 𝑏1 + 96𝜅𝑎0 𝑎1 𝑎2 𝛼5 𝑏1 + 48𝜅𝑎0 𝑎1 𝑎2 𝛼6 𝑏1 + 2
16𝜅𝑎0 𝑎1 𝛼4 𝑏1 𝑏2 + 16𝜅𝑎0 𝑎2 𝛼4 𝑏0 𝑏2 + 16𝜅𝑎1 𝑎2 𝛼4 𝑏0 𝑏1 − 2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏1 𝑏2 + 2
2
8(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏0 𝑏2 − 2(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎0 𝑎2 𝛼1 𝑏12 + 8𝜅 2 𝑎12 𝛼1 𝑏0 𝑏2 + 48𝜅𝑎02 𝑎2 𝛼5 𝑏2 + 24𝜅𝑎02 𝑎2 𝛼6 𝑏2 + 48𝜅𝑎0 𝑎12 𝛼5 𝑏2 + 24𝜅𝑎0 𝑎12 𝛼6 𝑏2 + 48𝜅𝑎0 𝑎22 𝛼5 𝑏0 + 24𝜅𝑎0 𝑎22 𝛼6 𝑏0 + 8𝜅𝑎0 𝑎2 𝛼4 𝑏12 + 48𝜅𝑎12 𝑎2 𝛼5 𝑏0 + 24𝜅𝑎12 𝑎2 𝛼6 𝑏0 + 8𝜅𝑎12 𝛼4 𝑏0 𝑏2 + 16𝜔𝑎0 𝑎1 𝑏1 𝑏2 + 2
2
16𝜔𝑎0 𝑎2 𝑏0 𝑏2 + 16𝜔𝑎1 𝑎2 𝑏0 𝑏1 − 6(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏12 + 10(ln(𝑎)) 𝑎12 𝛼1 𝑏0 𝑏2 + 4𝜔𝑎12 𝑏12 + 4𝜔𝑎22 𝑏02 − 24𝑎02 𝑎22 𝛼2 + 4𝜔𝑎02 𝑏22 − 4𝑎14 𝛼2 = 0,
2
of
16𝜅𝑎13 𝛼5 𝑏2 + 8𝜅𝑎13 𝛼6 𝑏2 + 8𝜔𝑎0 𝑎1 𝑏22 + 8𝜔𝑎12 𝑏1 𝑏2 + 8𝜔𝑎1 𝑎2 𝑏12 + 8𝜔𝑎22 𝑏0 𝑏1 − 48𝑎0 𝑎1 𝑎22 𝛼2 + 16𝜅 2 𝑎0 𝑎2 𝛼1 𝑏1 𝑏2 + 16𝜅 2 𝑎1 𝑎2 𝛼1 𝑏0 𝑏2 + 96𝜅𝑎0 𝑎1 𝑎2 𝛼5 𝑏2 + 48𝜅𝑎0 𝑎1 𝑎2 𝛼6 𝑏2 + 16𝜅𝑎0 𝑎2 𝛼4 𝑏1 𝑏2 + 2
ro
16𝜅𝑎1 𝑎2 𝛼4 𝑏0 𝑏2 − 8(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏1 𝑏2 + 16(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏0 𝑏2 + 8𝜅 2 𝑎0 𝑎1 𝛼1 𝑏22 + 8𝜅 2 𝑎12 𝛼1 𝑏1 𝑏2 + 8𝜅 2 𝑎1 𝑎2 𝛼1 𝑏12 + 8𝜅 2 𝑎22 𝛼1 𝑏0 𝑏1 + 8𝜅𝑎0 𝑎1 𝛼4 𝑏22 + 48𝜅𝑎0 𝑎22 𝛼5 𝑏1 + 24𝜅𝑎0 𝑎22 𝛼6 𝑏1 + 48𝜅𝑎12 𝑎2 𝛼5 𝑏1 + 24𝜅𝑎12 𝑎2 𝛼6 𝑏1 + 8𝜅𝑎12 𝛼4 𝑏1 𝑏2 + 48𝜅𝑎1 𝑎22 𝛼5 𝑏0 + 24𝜅𝑎1 𝑎22 𝛼6 𝑏0 + 8𝜅𝑎1 𝑎2 𝛼4 𝑏12 + 2
2
2
-p
8𝜅𝑎22 𝛼4 𝑏0 𝑏1 + 16𝜔𝑎0 𝑎2 𝑏1 𝑏2 + 16𝜔𝑎1 𝑎2 𝑏0 𝑏2 − 6(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏22 + 2(ln(𝑎)) 𝑎12 𝛼1 𝑏1 𝑏2 − 2
2(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏12 − 2(ln(𝑎)) 𝑎22 𝛼1 𝑏0 𝑏1 − 16𝑎13 𝑎2 𝛼2 = 0,
2
re
48𝜅𝑎12 𝑎2 𝛼5 𝑏2 + 24𝜅𝑎12 𝑎2 𝛼6 𝑏2 + 48𝜅𝑎1 𝑎22 𝛼5 𝑏1 + 24𝜅𝑎1 𝑎22 𝛼6 𝑏1 + 8𝜅𝑎22 𝛼4 𝑏0 𝑏2 + 16𝜔𝑎1 𝑎2 𝑏1 𝑏2 − 2 2 8(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏22 + 8(ln(𝑎)) 𝑎22 𝛼1 𝑏0 𝑏2 + 8𝜅 2 𝑎0 𝑎2 𝛼1 𝑏22 + 8𝜅 2 𝑎22 𝛼1 𝑏0 𝑏2 + 48𝜅𝑎0 𝑎22 𝛼5 𝑏2 + 2
lP
24𝜅𝑎0 𝑎22 𝛼6 𝑏2 + 8𝜅𝑎0 𝑎2 𝛼4 𝑏22 − (ln(𝑎)) 𝑎12 𝛼1 𝑏22 − (ln(𝑎)) 𝑎22 𝛼1 𝑏12 + 4𝜅 2 𝑎12 𝛼1 𝑏22 + 4𝜅 2 𝑎22 𝛼1 𝑏12 + 4𝜅𝑎12 𝛼4 𝑏22 + 16𝜅𝑎23 𝛼5 𝑏0 + 8𝜅𝑎23 𝛼6 𝑏0 + 4𝜅𝑎22 𝛼4 𝑏12 + 8𝜔𝑎0 𝑎2 𝑏22 + 8𝜔𝑎22 𝑏0 𝑏2 − 24𝑎12 𝑎22 𝛼2 − 2
16𝑎0 𝑎23 𝛼2 + 4𝜔𝑎22 𝑏12 + 4𝜔𝑎12 𝑏22 + 16𝜅 2 𝑎1 𝑎2 𝛼1 𝑏1 𝑏2 + 16𝜅𝑎1 𝑎2 𝛼4 𝑏1 𝑏2 + 2(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏1 𝑏2 = 0, 2
ur na
2
−2(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏22 + 2(ln(𝑎)) 𝑎22 𝛼1 𝑏1 𝑏2 + 8𝜅 2 𝑎1 𝑎2 𝛼1 𝑏22 + 8𝜅 2 𝑎22 𝛼1 𝑏1 𝑏2 + 48𝜅𝑎1 𝑎22 𝛼5 𝑏2 + 24𝜅𝑎1 𝑎22 𝛼6 𝑏2 + 8𝜅𝑎1 𝑎2 𝛼4 𝑏22 + 16𝜅𝑎23 𝛼5 𝑏1 + 8𝜅𝑎23 𝛼6 𝑏1 + 8𝜅𝑎22 𝛼4 𝑏1 𝑏2 + 8𝜔𝑎1 𝑎2 𝑏22 + 8𝜔𝑎22 𝑏1 𝑏2 − 16𝑎1 𝑎23 𝛼2 = 0, 4𝜅 2 𝑎22 𝛼1 𝑏22 + 16𝜅𝑎23 𝛼5 𝑏2 + 8𝜅𝑎23 𝛼6 𝑏2 + 4𝜅𝑎22 𝛼4 𝑏22 + 4𝜔𝑎22 𝑏22 − 4𝑎24 𝛼2 = 0.
Jo
By solving the resulting system, we will derive the following result 𝑎1 = − 𝑎2 =
2
(3𝑏02 (ln(𝑎)) 𝛼1 +4𝑎02 𝛼2 )𝑎0 𝑏1 2
(3𝑏02 (ln(𝑎)) 𝛼1 −4𝑎02 𝛼2 )𝑏0 4
2
𝑏12 𝑎0 (9(ln(𝑎)) 𝛼12 𝑏04 +24(ln(𝑎)) 𝑎02 𝛼1 𝛼2 𝑏02 +16𝑎04 𝛼22 )
,
2
2
4𝑏02 (3𝑏02 (ln(𝑎)) 𝛼1 −4𝑎02 𝛼2 ) 4
𝑏2 =
,
2
𝑏12 (9(ln(𝑎)) 𝛼12 𝑏04 +24(ln(𝑎)) 𝑎02 𝛼1 𝛼2 𝑏02 +16𝑎04 𝛼22 ) 2
2
4𝑏0 (3𝑏02 (ln(𝑎)) 𝛼1 −4𝑎02 𝛼2 )
,
6
2
𝜅=
𝑏02 (ln(𝑎)) 𝛼1 +4𝑎02 𝛼2 , 4𝑎0 𝑏0 (2𝛼5 +𝛼6 )
4 2 2 −1 ((ln(𝑎)) 𝛼13 𝑏04 + 8(ln(𝑎)) 𝑎02 𝛼12 𝛼2 𝑏02 + 32(ln(𝑎)) 𝑎02 𝛼1 𝛼52 𝑏02 + 16𝑎02 (2𝛼5 +𝛼6 )2 𝑏02 2 2 2 32(ln(𝑎)) 𝑎02 𝛼1 𝛼5 𝛼6 𝑏02 + 8(ln(𝑎)) 𝑎02 𝛼1 𝛼62 𝑏02 + 8(ln(𝑎)) 𝑎0 𝛼1 𝛼4 𝛼5 𝑏03 + 2 4(ln(𝑎)) 𝑎0 𝛼1 𝛼4 𝛼6 𝑏03 + 16𝑎04 𝛼1 𝛼22 + 64𝑎04 𝛼2 𝛼52 + 64𝑎04 𝛼2 𝛼5 𝛼6 + 16𝑎04 𝛼2 𝛼62 + 32𝑎03 𝛼2 𝛼4 𝛼5 𝑏0 16𝑎03 𝛼2 𝛼4 𝛼6 𝑏0 ).
𝜔=
+
Now, one can get the following soliton to the PGI equation
𝑏0 +𝑏1 𝑎𝑥+(2𝛼1 𝜅+𝛼4 )𝑡 +𝑏2 𝑎2(𝑥+(2𝛼1 𝜅+𝛼4 )𝑡)
𝑒 𝑖(−𝜅𝑥+𝜔𝑡+𝜃) ,
of
𝑎0 +𝑎1 𝑎𝑥+(2𝛼1 𝜅+𝛼4 )𝑡 +𝑎2 𝑎2(𝑥+(2𝛼1 𝜅+𝛼4 )𝑡)
𝜓3 (𝑥, 𝑡) = √
ro
where 𝑎1 , 𝑎2 , 𝑏2 , 𝜅, and 𝜔 have been given above.
Three-dimensional and density plots of |𝜓3 (𝑥, 𝑡)| showing a dark soliton have been presented in Figure 2
ur na
lP
re
-p
for suitable values of the involved parameters.
Jo
Figure 2: Three-dimensional and density plots of |𝜓3 (𝑥, 𝑡)| for 𝑎0 = 1, 𝑏0 = 2, 𝑏1 = 3, 𝛼1 = 1, 𝛼2 = −1, 𝛼4 = 1, 𝛼5 = −1, 𝛼6 = −1, 𝜃 = 0, and 𝑎 = 2.7. Kudryashov method (New version): To retrieve optical solitons of the PGI equation by means of the Kudryashov method, we first balance the terms (
d𝑉(𝜉) 2 d𝜉
) and 𝑉(𝜉)4 for deriving 𝑁 = 1. This recommends
that the solution of the PGI equation can be presented as 𝑉(𝜉) = 𝑎0 + 𝑎1 𝑅(𝜉),
𝑎1 ≠ 0.
(11)
Substituting Eq. (11) into Eq. (9) and adopting specific operations results in 4𝜅 2 𝑎02 𝛼1 + 16𝜅𝑎03 𝛼5 + 8𝜅𝑎03 𝛼6 − 4𝑎04 𝛼2 + 4𝜅𝑎02 𝛼4 + 4𝜔𝑎02 = 0, 7
8𝜅 2 𝑎0 𝑎1 𝛼1 + 48𝜅𝑎02 𝑎1 𝛼5 + 24𝜅𝑎02 𝑎1 𝛼6 − 16𝑎03 𝑎1 𝛼2 + 8𝜅𝑎0 𝑎1 𝛼4 + 8𝜔𝑎0 𝑎1 − 2𝑎0 𝑎1 𝛼1 = 0, 4𝜅 2 𝑎12 𝛼1 + 48𝜅𝑎0 𝑎12 𝛼5 + 24𝜅𝑎0 𝑎12 𝛼6 − 24𝑎02 𝑎12 𝛼2 + 4𝜅𝑎12 𝛼4 + 4𝜔𝑎12 − 𝑎12 𝛼1 = 0, 16𝜅𝑎13 𝛼5 + 8𝜅𝑎13 𝛼6 − 16𝑎0 𝑎13 𝛼2 + 4𝜂𝑎0 𝑎1 𝛼1 = 0, −4𝑎14 𝛼2 + 3𝜂𝑎12 𝛼1 = 0, which admits the following solutions √3√𝛼1 𝛼2 , 2𝛼2
𝑎1 = ±
2√3√𝛼1 𝛼2 , 3(2𝛼5 +𝛼6 )
16√3√𝛼1 𝛼2 𝛼4 𝛼5 +8√3√𝛼1 𝛼2 𝛼4 𝛼6 +16𝛼12 𝛼2 +60𝛼1 𝛼52 +60𝛼1 𝛼5 𝛼6 +15𝛼1 𝛼62 . 12(4𝛼52 +4𝛼5 𝛼6 +𝛼62 )
-p
𝜔=−
ro
𝜅=
√3 𝜂𝛼1 √𝛼 , 2 2
of
𝑎0 =
4𝐴𝐵,
√3√𝛼1 𝛼2 2𝛼2
𝛼1 𝛼2 > 0,
±
4𝐴 √3 𝜂𝛼1 √ 𝛼 4𝐴2 𝑒 𝑥+(2𝛼1 𝜅+𝛼4)𝑡 +𝜂𝑒 −(𝑥+(2𝛼1𝜅+𝛼4)𝑡) 𝑒 𝑖(−𝜅𝑥+𝜔𝑡+𝜃) , 2 2
𝜂 > 0,
𝜂=
lP
𝜓1,2 (𝑥, 𝑡) = √
re
Now, one can gain the following solitons to the PGI equation
where the soliton frequency and the wave number have been given above.
ur na
Three-dimensional and density plots of |𝜓1 (𝑥, 𝑡)| presenting a bright soliton have been illustrated in
Jo
Figure 3 for suitable values of the involved parameters.
Figure 3: Three-dimensional and density plots of |𝜓1 (𝑥, 𝑡)| for 𝐴 = 1, 𝐵 = 1, 𝛼1 = 1, 𝛼2 = 1, 𝛼4 = −1, 𝛼5 = 1, 𝛼6 = 1, and 𝜃 = 0. 8
4. Conclusion In this paper, we studied the perturbed Gerdjikov–Ivanov equation in optical fibers and obtained a number of its optical solitons. We carried out this goal by considering specific transformations to convert the PGI equation to a nonlinear ODE of second-order such that the resultant ODE could be solved using the exp𝑎 function and Kudryashov methods. In summary, the kink, bright, and dark solitons of the perturbed Gerdjikov–Ivanov equation with diverse applications in optical fibers were effectively recovered.
of
Conflict of interest
No conflict of interest exits in the submission of this manuscript, and manuscript is approved
Jo
ur na
lP
re
-p
ro
by all authors for publication.
9
Jo
ur na
lP
re
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11
PII:
S0030-4026(20)30184-4
DOI:
https://doi.org/10.1016/j.ijleo.2020.164350
Reference:
IJLEO 164350
To appear in:
Optik
Received Date:
27 January 2020
Accepted Date:
1 February 2020
Please cite this article as: Hosseini K, Mirzazadeh M, Ilie M, Radmehr S, Dynamics of optical solitons in the perturbed Gerdjikov–Ivanov equation, Optik (2020), doi: https://doi.org/10.1016/j.ijleo.2020.164350
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Dynamics of optical solitons in the perturbed Gerdjikov–Ivanov equation K. Hosseini1*, M. Mirzazadeh2*, M. Ilie1, S. Radmehr3 1
Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, PC 44891-63157 Rudsar-Vajargah, Iran 3 Department of Computer Engineering, University of Guilan, Rasht, Iran
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*Corresponding Authors: [email protected], [email protected]
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2
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Abstract:
The dynamics of solitons in the perturbed Gerdjikov–Ivanov (PGI) equation is explored in the current
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work. The investigation is carried out formally by considering particular transformations and exerting newly well-established methods to obtain optical solitons of the model. The kink, bright, and dark
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solitons of the PGI equation with diverse applications in optical fibers are successfully recovered. Keywords: Perturbed Gerdjikov–Ivanov equation; Optical fibers; Newly well-established methods; Kink,
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bright, and dark solitons
1. Introduction
Nonlinear partial differential equations are utilized to describe many problems in quantum mechanics, plasma physics, and nonlinear optics. Due to the availability of computer algebra systems and their effectiveness to handle symbolic computations, a wide range of capable methods has been introduced and
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developed [1-32] for dealing with nonlinear PDEs. Our aim of the present paper is to consider and study the perturbed Gerdjikov–Ivanov equation [1-6] 𝑖𝜓𝑡 + 𝛼1 𝜓𝑥𝑥 + 𝛼2 |𝜓|4 𝜓 + 𝑖𝛼3 𝜓 2 𝜓𝑥∗ − 𝑖(𝛼4 𝜓𝑥 + 𝛼5 (|𝜓|2 𝜓)𝑥 + 𝛼6 (|𝜓|2 )𝑥 𝜓) = 0,
(1)
by using the exp𝑎 -function and Kudryashov methods [25-32]. Here 𝜓 is a complex-valued function, 𝜓𝑡 is the linear temporal evolution, 𝜓𝑥𝑥 is the group velocity dispersion, |𝜓|4 𝜓 is the quintic nonlinearity, 𝛼4 is the inter-modal dispersion, (|𝜓|2 𝜓)𝑥 is the self-steepening term, and ultimately (|𝜓|2 )𝑥 𝜓 is the nonlinear dispersion. It should be noted that when 𝛼4 = 𝛼5 = 𝛼6 = 0, the PGI equation is reduced to the GI 1
equation. Furthermore, compared to the classical nonlinear Schrödinger equation that includes typically the cubic nonlinearity, the PGI equation benefits from the quintic nonlinearity. The perturbed Gerdjikov– Ivanov equation has been examined by a number of effective techniques, for example, Biswas and Alqahtani [1] utilized the semi-inverse variational principle to extract chirp-free bright optical solitons of the PGI equation. Optical solitons of the PGI equation were derived by Kaur and Wazwaz [2] using the exp(−𝜙(𝜉)) and (𝐺 ′ ⁄𝐺 2 ) expansion methods. Yaşar et al. [3] listed new optical solitons of the PGI equation by means of the sine-Gordon expansion method. Arshed et al. [4] applied the sine-cosine method to generate bright and dark solitons of the PGI equation. Bright and singular solitons of the PGI equation
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were obtained by Biswas et al. [5] through the extended trial equation method. Biswas et al. [6] also reported optical solitons of the PGI equation using the trial equation method. The organization of this
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paper is as follows: In Section 2, we summarize the key steps of the exp𝑎 -function and Kudryashov methods. In Section 2, we derive optical solitons of the perturbed Gerdjikov–Ivanov equation. Section 4
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finalizes the outcomes of the current study. 2. Methods and their ideas
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The present section formally summarizes the key steps of the exp𝑎 -function and Kudryashov methods. For this purpose, consider the following nonlinear PDE 𝜕𝜓 𝜕𝜓 𝜕𝑥
,
𝜕𝑡
, … ) = 0.
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𝑃 (𝜓,
(2)
Suppose that the above nonlinear PDE can be converted to a nonlinear ODE as (3)
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𝑂(𝜌, 𝜌′ , 𝜌′′ , … ) = 0,
through adopting the transformation 𝜓(𝑥, 𝑡) = 𝜌(𝜉)𝑒 𝑖(−𝜅𝑥+𝑡𝜔+𝜃) ,
𝜉 = 𝑥 − 𝑣𝑡.
2.1. 𝑬𝒙𝒑𝒂 -function method
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The exp𝑎 -function method benefits from adopting a test function in the following form 𝜌(𝜉) =
𝑎0 +𝑎1 𝑎𝜉 +⋯+𝑎𝑁 𝑎𝑁𝜉 𝑏0 +𝑏1 𝑎𝜉 +⋯+𝑏𝑁 𝑎𝑁𝜉
,
𝑎𝑁 ≠ 0,
𝑏𝑁 ≠ 0,
(4)
in which 𝑎𝑖 (0 ≤ 𝑖 ≤ 𝑁) and 𝑏𝑖 (0 ≤ 𝑖 ≤ 𝑁) are found later and 𝑁 ∈ ℕ. Inserting the test function (4) into Eq. (3) and exerting symbolic computations results in an algebraic system in the form 𝐶𝑖 = 0,
𝑖 = 0,1, . . . , 𝜎,
which solving it using Maple gives optical solitons of the nonlinear PDE (2). 2
2.2. Kudryashov method (New version) The Kudryshov method assumes the solution of the nonlinear ODE (3) can be presented as 𝜌(𝜉) = 𝑎0 + 𝑎1 𝑅(𝜉) + ⋯ + 𝑎𝑁 𝑅(𝜉)𝑁 ,
𝑎𝑁 ≠ 0,
(5)
where 𝑎𝑖 (0 ≤ 𝑖 ≤ 𝑁) are unknows, 𝑁 is the order of the pole of the general solution, and 𝑅(𝜉) is the solution of the equation 𝑅′ (𝜉)2 = 𝑅(𝜉)2 (1 − 𝜂𝑅(𝜉)2 ),
𝑅(𝜉) =
4𝐴 4𝐴2 𝑒 𝜉 +𝜂𝑒 −𝜉
,
𝜂 = 4𝐴𝐵.
3. Perturbed Gerdjikov–Ivanov equation and its optical solitons To solve the PGI equation, we adopt a wave transformation as 𝜉 = 𝑥 − 𝑣𝑡,
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𝜓(𝑥, 𝑡) = 𝜌(𝜉)𝑒 𝑖(−𝜅𝑥+𝜔𝑡+𝜃) ,
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system whose solution leads to optical solitons of the nonlinear PDE (2).
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Substituting the function (5) into Eq. (3) and exerting symbolic computations results in an algebraic
(6)
where v is the soliton velocity, 𝜅 is the frequency of the soliton, 𝜔 is the wave number, and 𝜃 is the phase
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constant.
Due to the PGI equation and considering the wave transformation (6), we find d2 𝜌(𝜉) d𝜉 2
− (𝜔 + 𝛼1 𝜅 2 + 𝛼4 𝜅)𝜌(𝜉) − (𝛼3 𝜅 + 𝛼5 𝜅)𝜌(𝜉)3 + 𝛼2 𝜌(𝜉)5 = 0,
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𝛼1
(𝜈 + 2𝛼1 𝜅 + 𝛼4 ) − (𝛼3 − 3𝛼5 − 2𝛼6 )𝜌(𝜉)2 = 0.
(7) (8)
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Setting the coefficients of Eq. (8) equal to zero yields 𝜈 = −(2𝛼1 𝜅 + 𝛼4 ), 𝛼3 = 3𝛼5 + 2𝛼6 .
Now, by substituting the above relations into Eq. (7) and considering the transformation 𝜌(𝜉) = √𝑉 (𝜉) , the second-order nonlinear ODE (7) is converted to d𝑉(𝜉) 2 d𝜉
) − 2𝛼1 𝑉(𝜉)
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𝛼1 (
d2 𝑉(𝜉) + 4(𝛼1 𝜅 2 d𝜉 2
+ 𝛼4 𝜅 + 𝜔)𝑉(𝜉)2 + 8(2𝛼5 𝜅 + 𝛼6 𝜅)𝑉(𝜉)3 − 4𝛼2 𝑉(𝜉)4 =
0.
(9)
𝑬𝒙𝒑𝒂 -function method: In order to obtain optical solitons of the PGI equation using the exp𝑎 -function method, a test function is considered as below 𝑉(𝜉) =
𝑎0 +𝑎1 𝑎𝜉 𝑏0 +𝑏1 𝑎𝜉
,
which 𝑎0 , 𝑎1 , 𝑏0 , and 𝑏1 are constants to be determined later. Inserting the above test function into Eq. (9) and adopting specific operations yields 3
4𝜅 2 𝑎02 𝛼1 𝑏02 + 16𝜅𝑎03 𝛼5 𝑏0 + 8𝜅𝑎03 𝛼6 𝑏0 + 4𝜅𝑎02 𝛼4 𝑏02 + 4𝜔𝑎02 𝑏02 − 4𝑎04 𝛼2 = 0, 2
2
2(ln(𝑎)) 𝑎02 𝛼1 𝑏0 𝑏1 − 2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏02 + 8𝜅 2 𝑎02 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎0 𝑎1 𝛼1 𝑏02 + 16𝜅𝑎03 𝛼5 𝑏1 + 8𝜅𝑎03 𝛼6 𝑏1 + 48𝜅𝑎02 𝑎1 𝛼5 𝑏0 + 24𝜅𝑎02 𝑎1 𝛼6 𝑏0 + 8𝜅𝑎02 𝛼4 𝑏0 𝑏1 + 8𝜅𝑎0 𝑎1 𝛼4 𝑏02 + 8𝜔𝑎02 𝑏0 𝑏1 + 8𝜔𝑎0 𝑎1 𝑏02 − 16𝑎03 𝑎1 𝛼2 = 0, 2
2
2
−(ln(𝑎)) 𝑎02 𝛼1 𝑏12 + 2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏0 𝑏1 − (ln(𝑎)) 𝑎12 𝛼1 𝑏02 + 4𝜅 2 𝑎02 𝛼1 𝑏12 + 16𝜅 2 𝑎0 𝑎1 𝛼1 𝑏0 𝑏1 + 4𝜅 2 𝑎12 𝛼1 𝑏02 + 48𝜅𝑎02 𝑎1 𝛼5 𝑏1 + 24𝜅𝑎02 𝑎1 𝛼6 𝑏1 + 4𝜅𝑎02 𝛼4 𝑏12 + 48𝜅𝑎0 𝑎12 𝛼5 𝑏0 + 24𝜅𝑎0 𝑎12 𝛼6 𝑏0 + 16𝜅𝑎0 𝑎1 𝛼4 𝑏0 𝑏1 + 4𝜅𝑎12 𝛼4 𝑏02 + 4𝜔𝑎02 𝑏12 + 16𝜔𝑎0 𝑎1 𝑏0 𝑏1 + 4𝜔𝑎12 𝑏02 − 24𝑎02 𝑎12 𝛼2 = 0, 2
2
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−2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏12 + 2(ln(𝑎)) 𝑎12 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎0 𝑎1 𝛼1 𝑏12 + 8𝜅 2 𝑎12 𝛼1 𝑏0 𝑏1 + 48𝜅𝑎0 𝑎12 𝛼5 𝑏1 + 24𝜅𝑎0 𝑎12 𝛼6 𝑏1 + 8𝜅𝑎0 𝑎1 𝛼4 𝑏12 + 16𝜅𝑎13 𝛼5 𝑏0 + 8𝜅𝑎13 𝛼6 𝑏0 + 8𝜅𝑎12 𝛼4 𝑏0 𝑏1 + 8𝜔𝑎0 𝑎1 𝑏12 + 8𝜔𝑎12 𝑏0 𝑏1 − 16𝑎0 𝑎13 𝛼2 = 0,
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4𝜅 2 𝑎12 𝛼1 𝑏12 + 16𝜅𝑎13 𝛼5 𝑏1 + 8𝜅𝑎13 𝛼6 𝑏1 + 4𝜅𝑎12 𝛼4 𝑏12 + 4𝜔𝑎12 𝑏12 − 4𝑎14 𝛼2 = 0, which admits the following solutions
𝜔=
1 3
√− 𝛼1 𝛼2 ln(𝑎) 2𝛼5 +𝛼6 ln(𝑎)
12(2𝛼5 +𝛼6 )2
,
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𝜅=±
ln(𝑎)𝛼1
,
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𝑏1 = ∓2
1 3
√− 𝛼1 𝛼2 𝑎1
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𝑎0 = 0,
(4ln(𝑎)𝛼12 𝛼2 + 12ln(𝑎)𝛼1 𝛼52 + 12ln(𝑎)𝛼1 𝛼5 𝛼6 + 3ln(𝑎)𝛼1 𝛼62 ∓
1
1
3
3
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24√− 𝛼1 𝛼2 𝛼4 𝛼5 ∓ 12√− 𝛼1 𝛼2 𝛼4 𝛼6 ).
Now, one can derive the following solitons to the PGI equation
√
𝑎1 𝑎𝑥+(2𝛼1 𝜅+𝛼4 )𝑡
1 √− 𝛼1 𝛼2 𝑎1 3
𝑏0 ∓2
ln(𝑎)𝛼1
𝑒 𝑖(−𝜅𝑥+𝜔𝑡+𝜃) ,
𝛼1 𝛼2 < 0,
𝑎𝑥+(2𝛼1 𝜅+𝛼4 )𝑡
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𝜓1,2 (𝑥, 𝑡) =
where the soliton frequency and the wave number have been presented above. Three-dimensional and density plots of |𝜓1 (𝑥, 𝑡)| presenting a kink soliton have been given in Figure 1 for suitable values of the involved parameters. To extract other solitons of the PGI equation by using the exp𝑎 -function method, we adopt another test function as follows 𝑉(𝜉) =
𝑎0 +𝑎1 𝑎𝜉 +𝑎2 𝑎2𝜉 𝑏0 +𝑏1 𝑎𝜉 +𝑏2 𝑎2𝜉
,
(10) 4
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Figure 1: Three-dimensional and density plots of |𝜓1 (𝑥, 𝑡)| for 𝑎1 = 1, 𝑏0 = 1, 𝛼1 = −1, 𝛼2 = 1, 𝛼4 = −1, 𝛼5 = −1, 𝛼6 = −1, 𝜃 = 0, and 𝑎 = 2.7.
which 𝑎0 , 𝑎1 , 𝑎2 , 𝑏0 , 𝑏1 , and 𝑏2 are constants to be found later. Setting a test function like (10) in Eq. (9)
re
and adopting specific operations results in
4𝜅 2 𝑎02 𝛼1 𝑏02 + 16𝜅𝑎03 𝛼5 𝑏0 + 8𝜅𝑎03 𝛼6 𝑏0 + 4𝜅𝑎02 𝛼4 𝑏02 + 4𝜔𝑎02 𝑏02 − 4𝑎04 𝛼2 = 0, 2
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2
ur na
2(ln(𝑎)) 𝑎02 𝛼1 𝑏0 𝑏1 − 2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏02 + 8𝜅 2 𝑎02 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎0 𝑎1 𝛼1 𝑏02 + 16𝜅𝑎03 𝛼5 𝑏1 + 8𝜅𝑎03 𝛼6 𝑏1 + 48𝜅𝑎02 𝑎1 𝛼5 𝑏0 + 24𝜅𝑎02 𝑎1 𝛼6 𝑏0 + 8𝜅𝑎02 𝛼4 𝑏0 𝑏1 + 8𝜅𝑎0 𝑎1 𝛼4 𝑏02 + 8𝜔𝑎02 𝑏0 𝑏1 + 8𝜔𝑎0 𝑎1 𝑏02 − 16𝑎03 𝑎1 𝛼2 = 0, 4𝜅 2 𝑎02 𝛼1 𝑏12 + 4𝜅 2 𝑎12 𝛼1 𝑏02 + 16𝜅𝑎03 𝛼5 𝑏2 + 8𝜅𝑎03 𝛼6 𝑏2 + 4𝜅𝑎02 𝛼4 𝑏12 + 4𝜅𝑎12 𝛼4 𝑏02 + 8𝜔𝑎02 𝑏0 𝑏2 + 2
2
8𝜔𝑎0 𝑎2 𝑏02 − (ln(𝑎)) 𝑎02 𝛼1 𝑏12 − (ln(𝑎)) 𝑎12 𝛼1 𝑏02 + 16𝜅 2 𝑎0 𝑎1 𝛼1 𝑏0 𝑏1 + 16𝜅𝑎0 𝑎1 𝛼4 𝑏0 𝑏1 + 2
2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎02 𝛼1 𝑏0 𝑏2 + 8𝜅 2 𝑎0 𝑎2 𝛼1 𝑏02 + 48𝜅𝑎02 𝑎1 𝛼5 𝑏1 + 24𝜅𝑎02 𝑎1 𝛼6 𝑏1 + 48𝜅𝑎02 𝑎2 𝛼5 𝑏0 + 24𝜅𝑎02 𝑎2 𝛼6 𝑏0 + 8𝜅𝑎02 𝛼4 𝑏0 𝑏2 + 48𝜅𝑎0 𝑎12 𝛼5 𝑏0 + 24𝜅𝑎0 𝑎12 𝛼6 𝑏0 + 8𝜅𝑎0 𝑎2 𝛼4 𝑏02 + 2
2
Jo
16𝜔𝑎0 𝑎1 𝑏0 𝑏1 + 8(ln(𝑎)) 𝑎02 𝛼1 𝑏0 𝑏2 − 8(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏02 + 4𝜔𝑎12 𝑏02 + 4𝜔𝑎02 𝑏12 − 16𝑎03 𝑎2 𝛼2 − 24𝑎02 𝑎12 𝛼2 = 0, 16𝜅𝑎13 𝛼5 𝑏0 + 8𝜅𝑎13 𝛼6 𝑏0 + 8𝜔𝑎02 𝑏1 𝑏2 + 8𝜔𝑎0 𝑎1 𝑏12 + 8𝜔𝑎12 𝑏0 𝑏1 + 8𝜔𝑎1 𝑎2 𝑏02 − 48𝑎02 𝑎1 𝑎2 𝛼2 + 16𝜅 2 𝑎0 𝑎1 𝛼1 𝑏0 𝑏2 + 16𝜅 2 𝑎0 𝑎2 𝛼1 𝑏0 𝑏1 + 96𝜅𝑎0 𝑎1 𝑎2 𝛼5 𝑏0 + 48𝜅𝑎0 𝑎1 𝑎2 𝛼6 𝑏0 + 16𝜅𝑎0 𝑎1 𝛼4 𝑏0 𝑏2 + 2
2
16𝜅𝑎0 𝑎2 𝛼4 𝑏0 𝑏1 + 16(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏0 𝑏2 − 8(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎12 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎1 𝑎2 𝛼1 𝑏02 + 48𝜅𝑎02 𝑎1 𝛼5 𝑏2 + 24𝜅𝑎02 𝑎1 𝛼6 𝑏2 + 48𝜅𝑎02 𝑎2 𝛼5 𝑏1 + 24𝜅𝑎02 𝑎2 𝛼6 𝑏1 + 8𝜅𝑎02 𝛼4 𝑏1 𝑏2 + 48𝜅𝑎0 𝑎12 𝛼5 𝑏1 + 24𝜅𝑎0 𝑎12 𝛼6 𝑏1 + 8𝜅𝑎0 𝑎1 𝛼4 𝑏12 + 8𝜅𝑎12 𝛼4 𝑏0 𝑏1 + 8𝜅𝑎1 𝑎2 𝛼4 𝑏02 + 16𝜔𝑎0 𝑎1 𝑏0 𝑏2 + 2
2
2
16𝜔𝑎0 𝑎2 𝑏0 𝑏1 − 2(ln(𝑎)) 𝑎02 𝛼1 𝑏1 𝑏2 − 2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏12 + 2(ln(𝑎)) 𝑎12 𝛼1 𝑏0 𝑏1 − 2
6(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏02 + 8𝜅 2 𝑎02 𝛼1 𝑏1 𝑏2 + 8𝜅 2 𝑎0 𝑎1 𝛼1 𝑏12 − 16𝑎0 𝑎13 𝛼2 = 0, 5
4𝜅 2 𝑎02 𝛼1 𝑏22 + 4𝜅 2 𝑎12 𝛼1 𝑏12 + 4𝜅 2 𝑎22 𝛼1 𝑏02 + 4𝜅𝑎02 𝛼4 𝑏22 + 16𝜅𝑎13 𝛼5 𝑏1 + 8𝜅𝑎13 𝛼6 𝑏1 + 4𝜅𝑎12 𝛼4 𝑏12 + 2
2
4𝜅𝑎22 𝛼4 𝑏02 + 8𝜔𝑎0 𝑎2 𝑏12 + 8𝜔𝑎12 𝑏0 𝑏2 − 48𝑎0 𝑎12 𝑎2 𝛼2 − 4(ln(𝑎)) 𝑎02 𝛼1 𝑏22 − 4(ln(𝑎)) 𝑎22 𝛼1 𝑏02 + 16𝜅 2 𝑎0 𝑎1 𝛼1 𝑏1 𝑏2 + 16𝜅 2 𝑎0 𝑎2 𝛼1 𝑏0 𝑏2 + 16𝜅 2 𝑎1 𝑎2 𝛼1 𝑏0 𝑏1 + 96𝜅𝑎0 𝑎1 𝑎2 𝛼5 𝑏1 + 48𝜅𝑎0 𝑎1 𝑎2 𝛼6 𝑏1 + 2
16𝜅𝑎0 𝑎1 𝛼4 𝑏1 𝑏2 + 16𝜅𝑎0 𝑎2 𝛼4 𝑏0 𝑏2 + 16𝜅𝑎1 𝑎2 𝛼4 𝑏0 𝑏1 − 2(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏1 𝑏2 + 2
2
8(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏0 𝑏2 − 2(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏0 𝑏1 + 8𝜅 2 𝑎0 𝑎2 𝛼1 𝑏12 + 8𝜅 2 𝑎12 𝛼1 𝑏0 𝑏2 + 48𝜅𝑎02 𝑎2 𝛼5 𝑏2 + 24𝜅𝑎02 𝑎2 𝛼6 𝑏2 + 48𝜅𝑎0 𝑎12 𝛼5 𝑏2 + 24𝜅𝑎0 𝑎12 𝛼6 𝑏2 + 48𝜅𝑎0 𝑎22 𝛼5 𝑏0 + 24𝜅𝑎0 𝑎22 𝛼6 𝑏0 + 8𝜅𝑎0 𝑎2 𝛼4 𝑏12 + 48𝜅𝑎12 𝑎2 𝛼5 𝑏0 + 24𝜅𝑎12 𝑎2 𝛼6 𝑏0 + 8𝜅𝑎12 𝛼4 𝑏0 𝑏2 + 16𝜔𝑎0 𝑎1 𝑏1 𝑏2 + 2
2
16𝜔𝑎0 𝑎2 𝑏0 𝑏2 + 16𝜔𝑎1 𝑎2 𝑏0 𝑏1 − 6(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏12 + 10(ln(𝑎)) 𝑎12 𝛼1 𝑏0 𝑏2 + 4𝜔𝑎12 𝑏12 + 4𝜔𝑎22 𝑏02 − 24𝑎02 𝑎22 𝛼2 + 4𝜔𝑎02 𝑏22 − 4𝑎14 𝛼2 = 0,
2
of
16𝜅𝑎13 𝛼5 𝑏2 + 8𝜅𝑎13 𝛼6 𝑏2 + 8𝜔𝑎0 𝑎1 𝑏22 + 8𝜔𝑎12 𝑏1 𝑏2 + 8𝜔𝑎1 𝑎2 𝑏12 + 8𝜔𝑎22 𝑏0 𝑏1 − 48𝑎0 𝑎1 𝑎22 𝛼2 + 16𝜅 2 𝑎0 𝑎2 𝛼1 𝑏1 𝑏2 + 16𝜅 2 𝑎1 𝑎2 𝛼1 𝑏0 𝑏2 + 96𝜅𝑎0 𝑎1 𝑎2 𝛼5 𝑏2 + 48𝜅𝑎0 𝑎1 𝑎2 𝛼6 𝑏2 + 16𝜅𝑎0 𝑎2 𝛼4 𝑏1 𝑏2 + 2
ro
16𝜅𝑎1 𝑎2 𝛼4 𝑏0 𝑏2 − 8(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏1 𝑏2 + 16(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏0 𝑏2 + 8𝜅 2 𝑎0 𝑎1 𝛼1 𝑏22 + 8𝜅 2 𝑎12 𝛼1 𝑏1 𝑏2 + 8𝜅 2 𝑎1 𝑎2 𝛼1 𝑏12 + 8𝜅 2 𝑎22 𝛼1 𝑏0 𝑏1 + 8𝜅𝑎0 𝑎1 𝛼4 𝑏22 + 48𝜅𝑎0 𝑎22 𝛼5 𝑏1 + 24𝜅𝑎0 𝑎22 𝛼6 𝑏1 + 48𝜅𝑎12 𝑎2 𝛼5 𝑏1 + 24𝜅𝑎12 𝑎2 𝛼6 𝑏1 + 8𝜅𝑎12 𝛼4 𝑏1 𝑏2 + 48𝜅𝑎1 𝑎22 𝛼5 𝑏0 + 24𝜅𝑎1 𝑎22 𝛼6 𝑏0 + 8𝜅𝑎1 𝑎2 𝛼4 𝑏12 + 2
2
2
-p
8𝜅𝑎22 𝛼4 𝑏0 𝑏1 + 16𝜔𝑎0 𝑎2 𝑏1 𝑏2 + 16𝜔𝑎1 𝑎2 𝑏0 𝑏2 − 6(ln(𝑎)) 𝑎0 𝑎1 𝛼1 𝑏22 + 2(ln(𝑎)) 𝑎12 𝛼1 𝑏1 𝑏2 − 2
2(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏12 − 2(ln(𝑎)) 𝑎22 𝛼1 𝑏0 𝑏1 − 16𝑎13 𝑎2 𝛼2 = 0,
2
re
48𝜅𝑎12 𝑎2 𝛼5 𝑏2 + 24𝜅𝑎12 𝑎2 𝛼6 𝑏2 + 48𝜅𝑎1 𝑎22 𝛼5 𝑏1 + 24𝜅𝑎1 𝑎22 𝛼6 𝑏1 + 8𝜅𝑎22 𝛼4 𝑏0 𝑏2 + 16𝜔𝑎1 𝑎2 𝑏1 𝑏2 − 2 2 8(ln(𝑎)) 𝑎0 𝑎2 𝛼1 𝑏22 + 8(ln(𝑎)) 𝑎22 𝛼1 𝑏0 𝑏2 + 8𝜅 2 𝑎0 𝑎2 𝛼1 𝑏22 + 8𝜅 2 𝑎22 𝛼1 𝑏0 𝑏2 + 48𝜅𝑎0 𝑎22 𝛼5 𝑏2 + 2
lP
24𝜅𝑎0 𝑎22 𝛼6 𝑏2 + 8𝜅𝑎0 𝑎2 𝛼4 𝑏22 − (ln(𝑎)) 𝑎12 𝛼1 𝑏22 − (ln(𝑎)) 𝑎22 𝛼1 𝑏12 + 4𝜅 2 𝑎12 𝛼1 𝑏22 + 4𝜅 2 𝑎22 𝛼1 𝑏12 + 4𝜅𝑎12 𝛼4 𝑏22 + 16𝜅𝑎23 𝛼5 𝑏0 + 8𝜅𝑎23 𝛼6 𝑏0 + 4𝜅𝑎22 𝛼4 𝑏12 + 8𝜔𝑎0 𝑎2 𝑏22 + 8𝜔𝑎22 𝑏0 𝑏2 − 24𝑎12 𝑎22 𝛼2 − 2
16𝑎0 𝑎23 𝛼2 + 4𝜔𝑎22 𝑏12 + 4𝜔𝑎12 𝑏22 + 16𝜅 2 𝑎1 𝑎2 𝛼1 𝑏1 𝑏2 + 16𝜅𝑎1 𝑎2 𝛼4 𝑏1 𝑏2 + 2(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏1 𝑏2 = 0, 2
ur na
2
−2(ln(𝑎)) 𝑎1 𝑎2 𝛼1 𝑏22 + 2(ln(𝑎)) 𝑎22 𝛼1 𝑏1 𝑏2 + 8𝜅 2 𝑎1 𝑎2 𝛼1 𝑏22 + 8𝜅 2 𝑎22 𝛼1 𝑏1 𝑏2 + 48𝜅𝑎1 𝑎22 𝛼5 𝑏2 + 24𝜅𝑎1 𝑎22 𝛼6 𝑏2 + 8𝜅𝑎1 𝑎2 𝛼4 𝑏22 + 16𝜅𝑎23 𝛼5 𝑏1 + 8𝜅𝑎23 𝛼6 𝑏1 + 8𝜅𝑎22 𝛼4 𝑏1 𝑏2 + 8𝜔𝑎1 𝑎2 𝑏22 + 8𝜔𝑎22 𝑏1 𝑏2 − 16𝑎1 𝑎23 𝛼2 = 0, 4𝜅 2 𝑎22 𝛼1 𝑏22 + 16𝜅𝑎23 𝛼5 𝑏2 + 8𝜅𝑎23 𝛼6 𝑏2 + 4𝜅𝑎22 𝛼4 𝑏22 + 4𝜔𝑎22 𝑏22 − 4𝑎24 𝛼2 = 0.
Jo
By solving the resulting system, we will derive the following result 𝑎1 = − 𝑎2 =
2
(3𝑏02 (ln(𝑎)) 𝛼1 +4𝑎02 𝛼2 )𝑎0 𝑏1 2
(3𝑏02 (ln(𝑎)) 𝛼1 −4𝑎02 𝛼2 )𝑏0 4
2
𝑏12 𝑎0 (9(ln(𝑎)) 𝛼12 𝑏04 +24(ln(𝑎)) 𝑎02 𝛼1 𝛼2 𝑏02 +16𝑎04 𝛼22 )
,
2
2
4𝑏02 (3𝑏02 (ln(𝑎)) 𝛼1 −4𝑎02 𝛼2 ) 4
𝑏2 =
,
2
𝑏12 (9(ln(𝑎)) 𝛼12 𝑏04 +24(ln(𝑎)) 𝑎02 𝛼1 𝛼2 𝑏02 +16𝑎04 𝛼22 ) 2
2
4𝑏0 (3𝑏02 (ln(𝑎)) 𝛼1 −4𝑎02 𝛼2 )
,
6
2
𝜅=
𝑏02 (ln(𝑎)) 𝛼1 +4𝑎02 𝛼2 , 4𝑎0 𝑏0 (2𝛼5 +𝛼6 )
4 2 2 −1 ((ln(𝑎)) 𝛼13 𝑏04 + 8(ln(𝑎)) 𝑎02 𝛼12 𝛼2 𝑏02 + 32(ln(𝑎)) 𝑎02 𝛼1 𝛼52 𝑏02 + 16𝑎02 (2𝛼5 +𝛼6 )2 𝑏02 2 2 2 32(ln(𝑎)) 𝑎02 𝛼1 𝛼5 𝛼6 𝑏02 + 8(ln(𝑎)) 𝑎02 𝛼1 𝛼62 𝑏02 + 8(ln(𝑎)) 𝑎0 𝛼1 𝛼4 𝛼5 𝑏03 + 2 4(ln(𝑎)) 𝑎0 𝛼1 𝛼4 𝛼6 𝑏03 + 16𝑎04 𝛼1 𝛼22 + 64𝑎04 𝛼2 𝛼52 + 64𝑎04 𝛼2 𝛼5 𝛼6 + 16𝑎04 𝛼2 𝛼62 + 32𝑎03 𝛼2 𝛼4 𝛼5 𝑏0 16𝑎03 𝛼2 𝛼4 𝛼6 𝑏0 ).
𝜔=
+
Now, one can get the following soliton to the PGI equation
𝑏0 +𝑏1 𝑎𝑥+(2𝛼1 𝜅+𝛼4 )𝑡 +𝑏2 𝑎2(𝑥+(2𝛼1 𝜅+𝛼4 )𝑡)
𝑒 𝑖(−𝜅𝑥+𝜔𝑡+𝜃) ,
of
𝑎0 +𝑎1 𝑎𝑥+(2𝛼1 𝜅+𝛼4 )𝑡 +𝑎2 𝑎2(𝑥+(2𝛼1 𝜅+𝛼4 )𝑡)
𝜓3 (𝑥, 𝑡) = √
ro
where 𝑎1 , 𝑎2 , 𝑏2 , 𝜅, and 𝜔 have been given above.
Three-dimensional and density plots of |𝜓3 (𝑥, 𝑡)| showing a dark soliton have been presented in Figure 2
ur na
lP
re
-p
for suitable values of the involved parameters.
Jo
Figure 2: Three-dimensional and density plots of |𝜓3 (𝑥, 𝑡)| for 𝑎0 = 1, 𝑏0 = 2, 𝑏1 = 3, 𝛼1 = 1, 𝛼2 = −1, 𝛼4 = 1, 𝛼5 = −1, 𝛼6 = −1, 𝜃 = 0, and 𝑎 = 2.7. Kudryashov method (New version): To retrieve optical solitons of the PGI equation by means of the Kudryashov method, we first balance the terms (
d𝑉(𝜉) 2 d𝜉
) and 𝑉(𝜉)4 for deriving 𝑁 = 1. This recommends
that the solution of the PGI equation can be presented as 𝑉(𝜉) = 𝑎0 + 𝑎1 𝑅(𝜉),
𝑎1 ≠ 0.
(11)
Substituting Eq. (11) into Eq. (9) and adopting specific operations results in 4𝜅 2 𝑎02 𝛼1 + 16𝜅𝑎03 𝛼5 + 8𝜅𝑎03 𝛼6 − 4𝑎04 𝛼2 + 4𝜅𝑎02 𝛼4 + 4𝜔𝑎02 = 0, 7
8𝜅 2 𝑎0 𝑎1 𝛼1 + 48𝜅𝑎02 𝑎1 𝛼5 + 24𝜅𝑎02 𝑎1 𝛼6 − 16𝑎03 𝑎1 𝛼2 + 8𝜅𝑎0 𝑎1 𝛼4 + 8𝜔𝑎0 𝑎1 − 2𝑎0 𝑎1 𝛼1 = 0, 4𝜅 2 𝑎12 𝛼1 + 48𝜅𝑎0 𝑎12 𝛼5 + 24𝜅𝑎0 𝑎12 𝛼6 − 24𝑎02 𝑎12 𝛼2 + 4𝜅𝑎12 𝛼4 + 4𝜔𝑎12 − 𝑎12 𝛼1 = 0, 16𝜅𝑎13 𝛼5 + 8𝜅𝑎13 𝛼6 − 16𝑎0 𝑎13 𝛼2 + 4𝜂𝑎0 𝑎1 𝛼1 = 0, −4𝑎14 𝛼2 + 3𝜂𝑎12 𝛼1 = 0, which admits the following solutions √3√𝛼1 𝛼2 , 2𝛼2
𝑎1 = ±
2√3√𝛼1 𝛼2 , 3(2𝛼5 +𝛼6 )
16√3√𝛼1 𝛼2 𝛼4 𝛼5 +8√3√𝛼1 𝛼2 𝛼4 𝛼6 +16𝛼12 𝛼2 +60𝛼1 𝛼52 +60𝛼1 𝛼5 𝛼6 +15𝛼1 𝛼62 . 12(4𝛼52 +4𝛼5 𝛼6 +𝛼62 )
-p
𝜔=−
ro
𝜅=
√3 𝜂𝛼1 √𝛼 , 2 2
of
𝑎0 =
4𝐴𝐵,
√3√𝛼1 𝛼2 2𝛼2
𝛼1 𝛼2 > 0,
±
4𝐴 √3 𝜂𝛼1 √ 𝛼 4𝐴2 𝑒 𝑥+(2𝛼1 𝜅+𝛼4)𝑡 +𝜂𝑒 −(𝑥+(2𝛼1𝜅+𝛼4)𝑡) 𝑒 𝑖(−𝜅𝑥+𝜔𝑡+𝜃) , 2 2
𝜂 > 0,
𝜂=
lP
𝜓1,2 (𝑥, 𝑡) = √
re
Now, one can gain the following solitons to the PGI equation
where the soliton frequency and the wave number have been given above.
ur na
Three-dimensional and density plots of |𝜓1 (𝑥, 𝑡)| presenting a bright soliton have been illustrated in
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Figure 3 for suitable values of the involved parameters.
Figure 3: Three-dimensional and density plots of |𝜓1 (𝑥, 𝑡)| for 𝐴 = 1, 𝐵 = 1, 𝛼1 = 1, 𝛼2 = 1, 𝛼4 = −1, 𝛼5 = 1, 𝛼6 = 1, and 𝜃 = 0. 8
4. Conclusion In this paper, we studied the perturbed Gerdjikov–Ivanov equation in optical fibers and obtained a number of its optical solitons. We carried out this goal by considering specific transformations to convert the PGI equation to a nonlinear ODE of second-order such that the resultant ODE could be solved using the exp𝑎 function and Kudryashov methods. In summary, the kink, bright, and dark solitons of the perturbed Gerdjikov–Ivanov equation with diverse applications in optical fibers were effectively recovered.
of
Conflict of interest
No conflict of interest exits in the submission of this manuscript, and manuscript is approved
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by all authors for publication.
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