Obtaining optical soliton solutions of the cubic–quartic Fokas–Lenells equation via three different analytical methods

In this study, we have focused on finding soliton solutions of the cubic–quartic Fokas–Lenells equation, which models the nonlinear pulse transmission through optical fiber, is a pretty new and updated model. The main motivation of this study is to produce new solutions with previously unused methods for data transmission models in fiber optic cables together with the developing technology, which has been used very frequently today. We have used three different efficient analytical methods, namely, the Sinh–Gordon expansion method, enhanced modified extended tanh expansion method and unified Riccati equation expansion method. By modifying our proposed method and using it effectively, we have obtained much more optical solutions. We have supported the results with the 3D surface, 2D and contour plots of the soliton solutions, such as singular, periodic, periodic-singular, dark and M-shaped solitons. The originality of the method we propose is that it has never been used before. With the innovation of the method, we have obtained M-shaped solitons, which are very rarely obtainable in soliton waves. On the other hand, enhanced modified extended tanh expansion method have provided us more extra solutions. As the last and most important part, in order to examine physical behaviors, we have investigated the effect of the coefficient of the fourth order dispersion parameter on wave propagation by presenting the graphical representation.

where = (x, t) is soliton profile, a, b, c, d, , are the coefficients of 3OD, 4OD, Kerr law nonlinearity, nonlinear dispersion, inter-modal dispersion (IMD), self-steepening and higher order dispersion terms, respectively. In addition, m is the degree of full nonlinearity term. CQFLE is a quite new model thus there exist a few studies in the literature. Obtaining optical soliton by using sine-Gordon method ), using modified Kudryashov scheme and unified Riccati equation expansion method (Zayed et al. 2021), using semiinverse variational principle , modulation instability analysis (Kumar 2022) and numerical simulation of CQFLE with improved Adomian decomposition method (Al-Qarni et al. 2022) can be listed as some important studies in the literature. The organization of the paper is as follows: Sect. 2 contains obtaining the nonlinear ordinary differential equation (NODE) form of CQFLE and the application of the proposed methods to the considered equation. Section 3 includes 3D surface figures and 2D and contour plots of the obtained soliton solutions. Lastly, Sect. 4 is the conclusion part.

Obtaining nonlinear ordinary differential form of CQFL equation
Let us consider the following wave transformation; where v is soliton velocity, k is frequency of soliton, w is wave number, is phase component and 0 is phase constant. Substituting the Eq. (2) into the Eq. (1) and decomposing the obtained equation into real and imaginary parts, the following equations can be obtained, respectively.

Application of Sinh-Gordon expansion method to the CQFL equation
Let us suppose that Eq. (3) has a solution in general form as follows: where H = H( ) , N is balancing constant, A 0 , A i and B i , (i = 1, 2, … N) are real values to be determinated later and H( ) satisfies the following equation: where C is integration constant and Eq. (7) gives the following solutions: where C = 0 . Taking into account that N = 2 , the Eq. (6) turns into following form: Substituting the Eq. (10) and its related derivatives to Eq. (3) by considering Eq. (7), we have a polynomial. Then, equating the coefficients of cosh i (H)sinh j (H), i = (0...6), j = (0, 1) to zero.
The above system produces a many solution sets but we present some of them. One can easily get the others.

Application of eMETEM to the CQFL equation
According to eMETEM, we seek the solution of Eq. (3) in the following form: where N is the balancing constant which has been computed as N = 2 , A 0 , A i , B i , (i = 1, 2, … N) are real numbers to be determinated later and A N , B N should not be zero, simultaneously. Moreover, R( ) satisfies the following Riccati differential equation: where R( ) admits the solutions given in Table 1 which covers the more solutions of Eq. Ozisik et al. (2022). where 0 , a c , b c and are free real parameters and = ±1. Considering that N = 2 , the proposed solution in Eq. (17) converts into the following form: (11) SET 1,1 : SET 1,2 : Substituting Eqs. (19) and (18) into Eq.
(3), we get a polynomial in power of R( ) . Then equating the coefficients of R( ) j , (j = −6, .., 0, ..6) to zero, we construct the following algebraic system: Solution of the above system, gives the following sets: Remark 1 For simplicity, the following abbreviations are used in Eqs. (24)-(38).
(2), we can get the solutions of Eq. (1). One can easily get the any solution functions by inserting the sets in Eqs. (20)-(23) into resultant function. But in order not to take up much volume, we give the solutions of Eq. (1) in general form as in the following forms: where a c , b c are real constants and v = − − 8bk 3 .

Application of UREEM to the CQFL equation
According to the UREEM (Sirendaoreji 2017), we seek a solution of Eq. (3) in the following form: where N is the balancing term which has been computed N = 2 , i , (i = 1, 2 … N) are real values to be determinated later. Besides, F( ) satisfies the following equation: in which c 0 , c 1 , c 2 are real constants and F( ) admits the solutions given in Table 2. where = c 2 1 − 4c 0 c 2 . Considering N = 2 , the Eq. (43) turns into the following equation: Substituting Eqs. (45) and (44) into Eq. (3), we get a polynomial in power of F( ) . Then equating the coefficients of F( ) j , (j = 0, … 6) to zero, the following algebraic system is obtained: Solving this system gives the following set: Combination of the SET 3,1 , Eq. (2), Eq. (45) and Table 2, gives, 3 Results and discussion Figure 1 shows 3D, contour and 2D plots of 1,4 (x, t) in Eq. (16) with = 0.1, b = 1, d = 0.1, c = −1, = −1 and 0 = 1 . In Fig. 1a, we depict 3D view of | 1,4 (x, t)| which represents M-shaped soliton solution. Figure 1b, c show the 3D plots of real and imaginary parts of 1,4 (x, t) with aforementioned parameters. Figure 1d-f show the contour plots of modulus, real and imaginary parts of 1,4 (x, t). Figure 1a, g show the M-shaped soliton character as double-bright soliton which is not a common type of soliton for 1,4 (x, t) . In Fig. 1g, the shape of soliton is conserved for different t values and the soliton is traveling towards the left. But, Fig. 1h, i show behavior of Re( 1,4 (x, t)) and Im( 1,4 (x, t)) soliton, in addition, both Re( 1,4 (x, t)) and Im( 1,4 (x, t)) are traveling towards left with different t values also behaviors of solitons are changing according to the time. For example, in Fig. 1h, there are two dark solitons with different amplitudes for t = 0 value (blue line), but there are two bright solitons with different amplitudes for t = 0.2 value (red line).
| is plotted and this soliton type is known as M-shaped soliton. Figure 5b shows the contour plot of | 2,3 (x, t)| in Eq. (39) with mentioned parameters. Figure 5c shows the 2D plots of | 2,3 (x, t)| in Eq. (39) at t = 0 and t = 0.4 while the wave is traveling through to the right along the x-axis.
In Fig. 6a, 2D plots of | 1,4 (x, t)| in Eq. (16) are placed. Graphs are obtained with = 0.1, 0 = 1, B 2 = 1, d = 0.1, c = −1 and = −1 . There are four different styled plots in Fig. 6. Red stripe is obtained selecting t = 0 and b = 1 , blue stripe is obtained selecting t = 0 and b = 3 , orange dashed stripe is obtained selecting t = 0.15 and b = 3 , lastly purple dashed stripe is obtained selecting t = 0.15 and b = 3 . We can obtain two different results from the Fig. 6. Firstly  (2) is considered, w is the wave number and the wave number is inversely proportional to the wave length. So increasing wave number affects the wave length in a negative way. Considering the SET 2,3 , parameter w is affected by parameter b in a negative way so when b is increased, the wave number w is decreased, and consequently wave length increases and frequency decreases naturally. These consecutive results are shown in Fig. 6b. Figure  7 shows some plots of 3,1 (x, t) in Eq.
If we make a general comparison over the results obtained, the soliton types obtained in this study; They are M-shaped, singular, singular-periodic, periodic and dark solitons. When other CQFLE studies in the literature are examined, it is seen that they obtain M-shaped, singular and bright soliton solutions Zayed et al. 2021). Although we see that the difference is due to the methods used, when we examine the other studies in which the mentioned methods are used, we can see that different styles of soliton solutions are also obtained such as kink (Neirameh and Parvaneh 2021), dark and bright (Zayed et al. 2020). In addition to all this, we find it useful to examine some other methods that the authors can use to find optical solitons such as extended rational sine-cosine and sinh-cosh method (Mahak and Akram 2019a), G ′ G 2 method (Akram and Mahak 2018), uniform algebraic method (Mahak and Akram 2019b) and auxiliary equation method (Mahak and Akram 2020b).

Conclusion
In this paper, we handled the CQFL equation which models the nonlinear pulse transmission through optical fiber. This new model is obtained by adding the cubic and quartic nonlinearities when the chromatic dispersion is low due to different colors of light having different velocities. The equation models nonlinear pulse transmission on the optical silica fibers. We obtained the NODE form of the model by using wave transformation in Eq.
(2). Then we applied three different analytical methods to the NODE. Thus various analytical soliton solutions of the model are received. The novelty of the paper is to obtain many solutions of considered equation, which have different kinds of solitons. Especially, we obtained singular, periodic, periodic-singular, dark and M-shaped soliton solutions such that M-shaped is a rare type of soliton. In addition, we used eMETEM therefore the obtained solutions are pretty new. We plotted the 3D surface and contour graphs of the obtained solutions. We stated the direction of the traveling wave via 2D plots and then we interpreted velocity, the height of wave envelope, wave number and wave length in case of changing parameters. As a result, we hope that this study will help future researches in the optic field of physics.