Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 18, Number 2, 2022



Article Number  9  
Number of page(s)  10  
DOI  https://doi.org/10.1051/jeos/2022008  
Published online  19 September 2022 
Research Article
Cubic–quartic optical soliton perturbation and modulation instability analysis in polarizationcontrolled fibers for Fokas–Lenells equation
^{1}
University of Technology and Applied Sciences, P.O. Box 14, Ibri 516, Oman
^{2}
Department of Mathematics, Sultan Qaboos University, P.O.Box 36, AlKhod 123, Muscat, Oman
^{3}
Department of Mathematics and Physics, Grambling State University, Grambling, LA 71245, USA
^{4}
Mathematical Modeling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
^{5}
Department of Applied Mathematics, National Research Nuclear University, 31 Kashirskoe Hwy, Moscow 115409, Russian Federation
^{6}
Department of Applied Sciences, CrossBorder Faculty, Dunarea de Jos University of Galati, 111 Domneasca Street, Galati 800201, Romania
^{7}
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa 0204, Pretoria, South Africa
^{*} Corresponding author: khalil.ibr@cas.edu.om
Received:
20
January
2022
Accepted:
28
July
2022
The objective of this study is to investigate miscellaneous wave structures for perturbed Fokas–Lenells equation (FLE) with cubicquartic dispersion in polarizationpreserving fibers. Based on the improved projective Riccati equations method, various types of soliton solutions such as bright soliton, combo dark–bright soliton, singular soliton and combo singular soliton are constructed. Additionally, a set of periodic singular waves are also retrieved. The dynamical behaviors of some obtained solutions are depicted to provide a key to understanding the physics of the model. The modulation instability of the FLE is reported by employing the linear stability analysis which shows that all solutions are stable.
Key words: Perturbed Fokas–Lenells equation / Optical solitons / Cubicquartic dispersion / Improved projective Riccati equations method / Modulation instability
© The Author(s), published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Recently, nonlinear optics has become one of the important fields of science that have wide range of physical and engineering applications. The significance of this field has been enhanced since the appearance of optical fiber as a common type of optical waveguide that transmits light and signals over longe distances [1, 2]. Further to this, the continuous theoretical and experimental research works confirm that optical fiber has potential influences on developing photonic and optoelectronic devices [3–6]. One of the diagnostic tools to examine the physical properties of optical fiber is the optical pulses. The controllable interaction of dispersion and nonlinearity of the pulse propagation leads to the formation of stable and undistorted pulses known as soliton. There are several mathematical models that study the dynamic of soliton in optical fibers. One of these models that is accounted as a generalized form of the nonlinear Schrödinger equation is the Fokas–Lenells equation (FLE). In literature, FLE is dealt with by many authors to obtain exact solutions by utilizing various powerful techniques. The employed integration schemes in the previous studies are SineGordon expansion method, Riccati equation method, mapping method, trial equation method, Kudryashov’s method, semiinverse variational principle, modified simple equation method, LaplaceAdomian decomposition method, auxiliary equation method and many others. For more details, readers are referred to references [7–18].
Soliton propagation along an optical fiber can be subject to the low count of chromatic dispersion (CD) which severely affects the transmission process. To overcome this effect, a variety of novel techniques have recently been proposed. One of the most popular technologies employed in the research studies is based on adding another form of dispersion such as Bragg gratings dispersion, pure–cubic dispersion, pure–quartic dispersion, cubic–quartic dispersion and many others. For example, the combination of fourthorder dispersion (4OD) and thirdorder dispersion (3OD) terms can completely compensate for low CD and gives rise to creation of the socalled cubic–quartic (CQ) solitons, see the references [19–24]. Later, the model of FLE is developed to include 4OD and 3OD terms and that means CQ solitons can be constructed in polarization preserving fibers [25–27]. The current study mainly discusses CQFLE with perturbation terms of Hamiltonian type. The proposed model takes the form(1)where Ψ(x, t) is is a complexvalued function representing optical soliton profile. The independent variables x and t denote the distance along the fiber and the elapsed time, respectively. The first term indicates the time evolution while the terms with a and b account for the third and fourthorder dispersions. The nonlinear influence has the form of Kerr law and is given by the coefficient of c. The term with d is the coefficient of nonlinear dispersion. On the righthand side of equation (1), the perturbation terms with α, λ and μ are defined as intermodal dispersion, selfsteepening effect and higherorder dispersion, respectively. The parameter n represents the full nonlinearity effect and .
The model (1) is investigated with the help of the improved projective Riccati equations method [28, 29] to derive distinct exact solutions. The rest of this paper is organized as follows. In Section 2, we describe the suggested scheme. Section 3 demonstrates how the FLE is reduced to a simple form using the traveling wave transformation. In Section 4, various solution expressions illustrating different wave structures are extracted. In Section 5, the modulation instability by means of standard linear stability analysis is examined. Section 6 displays the remarks and discussion of the obtained results. Finally, our conclusion is given in Section 7.
2 Elucidation of scheme
Herein, we present the process of applying the improved projective Riccati equations method as follows. Consider a nonlinear evolution equation (NLEE) in the form(2)where u = u(x, t) is an unknown function and P is a polynomial in u and its various partial derivatives.
Based on the traveling wave transformation given by(3)the NLEE (2) reduces to a nonlinear ordinary differential equation (NLODE) of the form(4)where prime denotes the derivative with respect to ξ.
We assume that equation (4) has a solution in the form of a finite series as(5)where a _{ j }, b _{ j }, (j = 0, 1, 2, …, m) are constants to be determined. The parameter m is a positive integer which can be identified by balancing the highest order derivative term with the nonlinear term in equation (4).
The variables f(ξ) and g(ξ) satisfy the the following improved projective Riccati equations(6)where A, B and R are arbitrary constants and δ = ±1. The third equation in the system (6), which gives the relation between the functions f(ξ) and g(ξ), represents the first integral of the couple ODEs in this system.
The set of equations (6) is found to possess solutions in the form(7)which implies δ = 1, and(8)provided that δ = −1.
Substituting (5) along with (6) into equation (4) gives a polynomial in f ^{ j } (ξ) and f ^{ j } (ξ) g(ξ). Then, we equate each coefficient of f ^{ j } (ξ) and f ^{ j } (ξ)g(ξ) in this polynomial to zero to get a set of algebraic equations for a _{ j }, b _{ j }. Finally, solving this system of equations, we obtain various exact solutions of equation (2) according to (7) and (8).
3 Traveling wave reduction of the model
Now, we aim to reduce the complex form of the model (1) to an NLODE with a view to deriving the optical soliton solutions. Therefore, we assume the traveling wave transformation of the form(9)where ψ(ξ) accounts for the amplitude of the soliton while ϕ(x, y, t) denotes the phase component. The wave variable ξ is given by(10)and the function ϕ(x, y, t) is introduced as(11)where the parameters ν, κ, ω and θ represent the soliton velocity, frequency, wave number and phase constant, respectively.
Substituting (8) into equation (1) leads to a couple of equations for real and imaginary parts given, respectively, as(12) (13)where the prime denotes the derivative with respect to ξ. The system of equations (12) and (13) is reduced to(14)with the expression for the velocity of the soliton presented as(15)under the constraints(16) (17) (18)
4 Solutions of the model
Now, we embark on deriving the solutions of the perturped CQFLE through implementing the improved projective Riccati equations method stated in Section 2. The proposed technique is basically used to handle equation (14) and then its obtained solutions are plugged into the transformation (9) so as to extract the optical solitons of the governing model.
According to the series formula given in (5) and the balance between the terms ψ ^{ iv } and ψ ^{3} in equation (14), this leads to m = 2. Hence, the general solution form of equation (14) reads(19)
Substituting (19) together with equations (6) into equation (14) gives rise to an equation having different powers of f ^{ l } g ^{ s }. Collecting all the terms with the same power of f ^{ l } g ^{ s } together and equating each coefficient to zero, yields a set of algebraic equations. Solving these equations simultaneously leads to the following results.
Set I. If δ = 1, then the following cases of solutions in the hyperbolic secant and tangent functions are retrieved.
Case 1.
Inserting (20) in accompany with (7) into (19) yields(21)where b(A ^{2} − B ^{2})(c + dκ − λκ) > 0, ξ = x + (α + 8bκ ^{3})t and .
Case 2.
Plugging (22) along with (7) into (19) brings about(23)where b(c + dκ − λκ) < 0, ξ = x + (α + 8bκ ^{3})t and .
Substituting (24) together with (7) into (19) generates(25)where b(c + dκ − λκ) < 0, ξ = x + (α + 8bκ ^{ 3 })t and .
Putting (26) in addition to (7) into (19) gives us the soliton solution (23).
Plugging (27) and (7) into (19) leads to(28)where b(c + dκ − λκ) < 0, A ^{2} < B ^{2} , ξ = x + (α + 8bκ ^{3} )t and .
Substituting (29) in accompany with (7) into (19) produces the soliton solution (28).
Inserting (30) along with (7) into (19), we secure(31)where b(c + dκ − λκ) < 0, ξ = x + (α + 8bκ ^{ 3 })t and .
Substituting (32) as well as (7) into (19) provides the soliton solution(33)where and .
Case 9.
Substituting (34) together with (7) into (19) creates(35)where b(c + dκ − λκ) < 0, A^{2} < B ^{2}, ξ = x + (α + 8bκ ^{3})t and .
Set II. If δ = −1, then the following cases of solutions in the hyperbolic cosecant and cotangent functions are retrieved.
Inserting (36) along with (7) into (19) yields(37)where b(A ^{2} − B ^{2})(c + dκ − λκ) < 0, ξ = x + (α + 8bκ ^{3})t and .
Case 2.
Plugging (38) in addition to (7) into (19) brings about(39)where and .
Substituting (40) and (7) into (19) generates(41)where b(c + dκ − λκ) < 0, ξ = x + (α + 8bκ ^{3})t and .
Case 4.
Putting (42) as well as (7) into (19) gives us the soliton solution (23).
Case 5.
Plugging (43) along with (7) into (19) results in(44)where and .
Case 6.
Substituting (45) in accompany with (7) into (19) produces the soliton solution (44).
Inserting (46) and (7) into (19), we come by(47)where and .
Case 8.
Substituting (48) in accompany with (7) into (19) provides the soliton solution(49)where and .
Case 9.
Substituting (50) together with (7) into (19) gives rise to(51)where and .
Interestingly it can be noticed that the complex values of the constant R in some solutions obtained above generate periodic type solutions and then the amplitude function of these solutions may be complex. However, the complexvalued amplitude for some of these solutions can be converted into real value. For example, the periodic solution (23) has the form(52)
Since B is an arbitrary constant, it can be assumed as B = iΓ, where Γ is a real constant. Thus, solution (52) becomes(53)
Similarly, the periodic solution (28) given by(54)changes, after taking , into(55)where is a real constant. Consequently, the same technique can be used to the rest of periodic solutions to handle a real value for the amplitude of periodic waves.
5 Modulation instability analysis
In this section, the modulation instability of the perturbed Fokas–Lenells equation (1) is studied by means of the standard linear stability analysis.
Consider that equation (1) has the perturbed steadystate solution in the form(56)where P is the normalized optical power while U(x, t) is a small perturbation and U ≪ P. The perturbation U(x, t) is examined by utilizing linear stability analysis. Inserting equation (56) into equation (1) and linearizing, one can reach(57)where denotes the conjugate of the complex function U(x, t). Assuming that the solution of equation (57) in the form(58)where K and Ω are the normalized wave number and frequency of perturbation, respectively. Substituting ansatz (58) into equation (57), we find a couple of equations in β and γ by splitting the coefficients of and presented as(59)
The system of equations (59) can be written in the matrix form for the coefficients of β and γ. The determinant of this matrix leads to the dispersion relation in the form(60)where the constants χ_{1}, χ_{2} and χ_{4} are given as(61) (62) (63)
The dispersion relation has the solution given as(64)
This expression determines the steadystate stability that depends on the the fourthorder dispersion, nonlinear influence, selfsteepening effect, higherorder dispersion and wave number. It is clearly seen that the value of frequency is real for all values of K and hence the steady state is stable against small perturbations. Figure 1 shows the graph of dispersion relation.
6 Results and discussion
The implemented mathematical tools in terms of the improved projective Riccati equations have led to abundant exact solutions for the perturbed FLE model. All derived solutions are entirely new and different than the ones found in the literatures. Comparing the results obtained here with the corresponding results extracted in the previous studies, it is found that all solutions retrieved in [27] by using the sine–Gordon equation integration scheme can be deduced in this work when B = 0. The created traveling wave solutions include various wave structures such as bright soliton, combo dark–bright soliton, singular soliton, combo singular soliton and periodic waves.
To throw light on the dynamical behaviors of cubic–quartic optical solitons and other waves in polarizationpreserving fibers, the graphical representations for some of the constructed exact solutions are presented. Wave structures are displayed in 2D and 3D plots by selecting suitable values of the model parameters. Figure 2 illustrates the evolution of soliton solution (21), where the wave profile shows an Mshaped (twohump) soliton. The graph in Figure 3 demonstrates periodic singular wave of solution (23). Moreover, Figure 4 presents the plot of solution (25) that describes the bright soliton wave. In Figure 5, the graph of solution (28) depicts the structure of periodic bright soliton train. Additionally, it is clear from Figure 6 that the evolution of solution (31) characterizes the profile of Wshaped wave (darkdark soliton).
Figure 2 The dynamical behavior of solution (21) with the unity value for all parameters except A = 2. 
Figure 3 The dynamical behavior of solution (23) with the unity value for all parameters except b = −1, A = 2, B = i. 
Figure 4 The dynamical behavior of solution (25) with the unity value for all parameters except b = −1, B = 2. 
Figure 5 The dynamical behavior of solution (28) with the unity value for all parameters except A = i, B = 2i. 
Figure 6 The dynamical behavior of solution (31) with the unity value for all parameters except b = −1, B = 0. 
7 Conclusion
The present work focused on investigating distinct forms of exact solutions for cubic–quartic Fokas–Lenells equation with Hamiltonian perturbation terms in polarizationpreserving fibers. The study is carried out with the aid of the improved projective Riccati equations. The implemented approach enables us to find different wave structures including bright soliton, combo dark–bright soliton, singular soliton and combo singular soliton. Besides, the periodic singular waves are also recovered as a byproduct of executing solution method. The behaviors of some derived solutions are illustrated graphically to pave the way for understanding the physics of the model. Further to this, the stability of the retrieved solutions have been diagnosed by utilizing the linear stability analysis. The modulation instability of the perturbed FLE is discussed and confirms that all extracted solutions are stable. Overall, the proposed algorithm is rich in various solutions which are entirely new and can be exploited in the physical and engineering applications of fiber optics.
Conflict of interest
The authors declare no conflict of interest.
References
 Agrawal G.P. (2000) Nonlinear fiber optics, Nonlinear Science at the Dawn of the 21st Century, Springer, Berlin, Heidelberg, pp. 195–211. [NASA ADS] [CrossRef] [Google Scholar]
 De Angelis C. (2021) Nonlinear optics, Front. Photonics 1, 628215. [CrossRef] [Google Scholar]
 Liu W., Pang L., Han H., Liu M., Lei M., Fang S., Teng H., Wei Z. (2017) Tungsten disulfide saturable absorbers for 67 fs modelocked erbiumdoped fiber lasers, Opt. Express 25, 2950–2959. [NASA ADS] [CrossRef] [Google Scholar]
 Liu W., Pang L., Han H., Bi K., Lei M., Wei Z. (2017) Tungsten disulphide for ultrashort pulse generation in allfiber lasers, Nanoscale 9, 5806–5811. [CrossRef] [Google Scholar]
 Liu W., Zhu Y.N., Liu M., Wen B., Fang S., Teng H., Lei M., Liu L.M., Wei Z. (2018) Optical properties and applications for MoS 2Sb 2 Te 3MoS 2 heterostructure materials, Photonics Res. 6, 220–227. [CrossRef] [Google Scholar]
 Meng X., Li J., Guo Y., Liu Y., Li S., Guo H., Bi W., Lu H., Cheng T. (2020) Experimental study on a highsensitivity optical fiber sensor in widerange refractive index detection, JOSA B 37, 3063–3067. [CrossRef] [Google Scholar]
 Triki H., Wazwaz A.M. (2017) Combined optical solitary waves of the Fokas–Lenells equation, Waves Random Complex Media 27, 587–593. [NASA ADS] [CrossRef] [Google Scholar]
 Triki H., Wazwaz A.M. (2017) New types of chirped soliton solutions for the Fokas–Lenells equation, Int. J. Numer. Methods Heat Fluid Flow 27, 1596–1601. [CrossRef] [Google Scholar]
 Biswas A., Ekici M., Sonmezoglu A., Alqahtani R.T. (2018) Optical soliton perturbation with full nonlinearity for Fokas–Lenells equation, Optik 165, 29–34. [NASA ADS] [CrossRef] [Google Scholar]
 Jawad A.J.M., Biswas A., Zhou Q., Moshokoa S.P., Belic M. (2018) Optical soliton perturbation of Fokas–Lenells equation with two integration schemes, Optik 165, 111–116. [NASA ADS] [CrossRef] [Google Scholar]
 Biswas A., Rezazadeh H., Mirzazadeh M., Eslami M., Ekici M., Zhou Q., Moshokoa S.P., Belic M. (2018) Optical soliton perturbation with Fokas–Lenells equation using three exotic and efficient integration schemes, Optik 165, 288–294. [NASA ADS] [CrossRef] [Google Scholar]
 Biswas A. (2018) Chirpfree bright optical soliton perturbation with Fokas–Lenells equation by traveling wave hypothesis and semiinverse variational principle, Optik 170, 431–435. [NASA ADS] [CrossRef] [Google Scholar]
 Aljohani A., ElZahar E., Ebaid A., Ekici M., Biswas A. (2018) Optical soliton perturbation with Fokas–Lenells model by Riccati equation approach, Optik 172, 741–745. [NASA ADS] [CrossRef] [Google Scholar]
 Osman M., Ghanbari B. (2018) New optical solitary wave solutions of Fokas–Lenells equation in presence of perturbation terms by a novel approach, Optik 175, 328–333. [NASA ADS] [CrossRef] [Google Scholar]
 Krishnan E., Biswas A., Zhou Q., Alfiras M. (2019) Optical soliton perturbation with Fokas–Lenells equation by mapping methods, Optik 178, 104–110. [NASA ADS] [CrossRef] [Google Scholar]
 Arshad M., Lu D., Rehman M.U., Ahmed I., Sultan A.M. (2019) Optical solitary wave and elliptic function solutions of Fokas–Lenells equation in presence of perturbation terms and its modulation instability, Phys. Scr. 94, 105202. [NASA ADS] [CrossRef] [Google Scholar]
 GonzálezGaxiola O., Biswas A., Belic M.R. (2019) Optical soliton perturbation of Fokas–Lenells equation by the LaplaceAdomian decomposition algorithm, J. Eur. Opt. Soc. Rapid Publ. 15, 13. [CrossRef] [Google Scholar]
 AlGhafri K., Krishnan E., Biswas A. (2020) Chirped optical soliton perturbation of Fokas–Lenells equation with full nonlinearity, Adv. Differ. Equ. 2020, 1–12. [NASA ADS] [CrossRef] [Google Scholar]
 GonzálezGaxiola O., Biswas A., Mallawi F., Belic M.R. (2020) Cubicquartic bright optical solitons with improved Adomian decomposition method, J. Adv. Res. 21, 161–167. [CrossRef] [Google Scholar]
 Genc G., Ekici M., Biswas A., Belic M.R. (2020) Cubicquartic optical solitons with Kudryashov’s law of refractive index by Fexpansions schemes, Results Phys 18, 103273. [CrossRef] [Google Scholar]
 Yldrm Y., Biswas A., Kara A.H., Ekici M., Alzahrani A.K., Belic M.R. (2021) Cubic–quartic optical soliton perturbation and conservation laws with generalized Kudryashov’s form of refractive index, J. Opt. 50, 354–360. [CrossRef] [Google Scholar]
 Zayed E.M., Nofal T.A., Alngar M.E., ElHorbaty M.M. (2021) Cubicquartic optical soliton perturbation in polarizationpreserving fibers with complex GinzburgLandau equation having five nonlinear refractive index structures, Optik 231, 166381. [NASA ADS] [CrossRef] [Google Scholar]
 Kumar S., Malik S. (2021) Cubicquartic optical solitons with Kudryashov’s law of refractive index by Lie symmetry analysis, Optik 242, 167308. [NASA ADS] [CrossRef] [Google Scholar]
 Zayed E.M., Gepreel K.A., Alngar M.E., Biswas A., Dakova A., Ekici M., Alshehri H.M., Belic M.R. (2021) Cubic–quartic solitons for twincore couplers in optical metamaterials, Optik 245, 167632. [NASA ADS] [CrossRef] [Google Scholar]
 Zayed E.M., Alngar M.E., Biswas A., Yldrm Y., Khan S., Alzahrani A.K., Belic M.R. (2021) Cubic–quartic optical soliton perturbation in polarizationpreserving fibers with Fokas–Lenells equation, Optik 234, 166543. [NASA ADS] [CrossRef] [Google Scholar]
 Biswas A., Dakova A., Khan S., Ekici M., Moraru L., Belic M. (2021) Cubicquartic optical soliton perturbation with Fokas–Lenells equation by semiinverse variation, Semicond. Phys. Quantum Electron. Optoelectron. 24, 431–435. [CrossRef] [Google Scholar]
 Yıldırım Y., Biswas A., Dakova A., Khan S., Moshokoa S.P., Alzahrani A.K., Belic M.R. (2021) Cubicquartic optical soliton perturbation with Fokas–Lenells equation by sine–Gordon equation approach, Results Phys 26, 104409. [CrossRef] [Google Scholar]
 AlGhafri K., Krishnan E., Biswas A., Ekici M. (2018) Optical solitons having anticubic nonlinearity with a couple of exotic integration schemes, Optik 172, 794–800. [NASA ADS] [CrossRef] [Google Scholar]
 AlKalbani K.K., AlGhafri K., Krishnan E., Biswas A. (2021) Solitons and modulation instability of the perturbed GerdjikovIvanov equation with spatiotemporal dispersion, Chaos Solitons Fractals 153. [Google Scholar]
All Figures
Figure 1 The dispersion relation Ω = Ω(K) between frequency and wave number K given in (64). 

In the text 
Figure 2 The dynamical behavior of solution (21) with the unity value for all parameters except A = 2. 

In the text 
Figure 3 The dynamical behavior of solution (23) with the unity value for all parameters except b = −1, A = 2, B = i. 

In the text 
Figure 4 The dynamical behavior of solution (25) with the unity value for all parameters except b = −1, B = 2. 

In the text 
Figure 5 The dynamical behavior of solution (28) with the unity value for all parameters except A = i, B = 2i. 

In the text 
Figure 6 The dynamical behavior of solution (31) with the unity value for all parameters except b = −1, B = 0. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.