New private types for the cubic-quartic optical solitons in birefringent fibers in its four forms of nonlinear refractive index

From the point of view of the extended simple equation method, multiple new private distinct types for the cubic-quartic optical soliton birefringent fibers with four forms nonlinear refractive index have been extracted. The suggested model has vital effective effect in all modern telecommunications process. The suggested method which has invited for this purpose examined previously for many other nonlinear problems arising in various branches of science and continuously gives good results. We will implement this method to extracting multiple new private types for the cubic-quartic soliton with its four different forms of the NLSE which are, the cubic-quartic in polarization-preserving fibers with the kerr-low nonlinearity, quadratic-cubic law nonlinearity, parabolic law nonlinearity and non-local law nonlinearity. Actual comparison between our achieved results and that realized previously by other authors has been established.


Introduction
The propagations of waves in optical fiber is one of important operations in all telecommunications process, specially the propagations in birefringence fiber and improve this process by develop the model governed this process will give future studies to this phenomena. Some studies have been established in the last few decays through good efforts via little number of authors to discuss the cubic-quartic of the birefringence for which the optical property of a material having a refractive index that depends on the polarization and propagation direction. Recently, it is important to study the transition of light through optical fibers which responsible the propagation of waves in all modern telecommunications, 1 3 680 Page 2 of 32 audio waves and asocial media processes. The birefringence fiber is a material which possesses refractive index related to the polarization and propagation direction of light. This optical property is responsible for the phenomenon of double refraction whereby a ray of light, when incident upon a birefringent material, is split by polarization into two rays taking slightly different paths. Refractive indices due to orthogonally polarized materials vary effectively in the case of fiber birefringence in optical modes. Moreover, it often quantified as the maximum difference between refractive indices exhibited by the material in general Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress. In related subject Birefringence is used in many optical devices. Liquid-crystal displays, which are used utilized in medical diagnostics. In fact the double refraction phenomenon depends on Birefringence, because when a light ray incident upon a birefringent material it will split into two rays which have slightly various paths under the polarization property. In the literature this effect has been observed in calcite, a crystal having one of the strongest birefringence by the Danish scientist Rasmus Bartholin (1669). Moreover, Augustin-Jean Fresnel at 19th described the phenomenon in terms of polarization he can understand that the light can be considered as a wave with field components in transverse polarization. In addition the birefringence for which the optical property of a material having a refractive index that depends on the polarization and propagation direction of light has been discussed by Neves (1998) ;Amos 2013). Some studies have been demonstrated to study various forms of the fractional and non-fractional NLSE which describes important nonlinear problems in different branches of sciences see for example Cevikel et al. (2014) who applied the functional variable method for finding the periodic wave and solitary wave solutions of the generalized Zakharov equation and higher-order nonlinear Schrödinger equations, Cevikel and Bekir (2013) who constructed periodic and soliton solutions for the (2 + 1)-dimensional Davey-Stewartson (DS) equations by using the sine-cosine, tanh-coth, and exp-function method, Bekir et al. (2014) who obtained the 1-soliton solutions of the (2 + 1)-dimensional Boussinesq equation and the Camassa-Holm-KP equation by using a solitary wave ansatz method, Güner, et al. (2013) who extracted the dark (topological) soliton solutions of the Kudryashov-Sinelshchikov and Jimbo-Miwa equations by using the sine-cosine method, Cevikel (2018) who extracted new exact solutions of the space-time fractional KDV Burgers and nonlinear fractional Foam-Drainage equation by using the fractional sub-equation method and the first integral method, Bekir et al. (2017) who applied the Exp-function method to extract the exact solutions of the time fractional Fitzhugh-Nagumo equation, the time fractional KDV equation, Bekir et al. (2017) who established new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative by using (G'/G)-expansion method, Aksoy et al. (2019) who applied the Kudryashov method to find exact solutions he space-time fractional Symmetric Regularized Long wave equation and the space-time fractional generalized Hirota-Satsuma coupled KdV equation, Cevikel and Aksoy (2021) who applied the generalized Kudryashov method extract a certain type of soliton solutions to the time-fractional Chan-Allen equation, the space-time fractional Klein-Gordon equation and the space-time fractional ZK-BBM equation, Kudryashov (2020) who achieved highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations, Shehata et al. (2020) who extracted the optical solitons to a perturbed Gerdjikov-Ivanov equation using two different techniques and Bekir and Zahran (2021a) Biswas and Arshed 2018;Das et al. 2019;Gonzalez-Gaxiola et al. 2020;Kohl et al. 2019). Relevant studies to other various forms of Schrödinger equation have been demonstrated see for example Hosseini et al. (2021a) who applied the modified Kudryashov method and symbolic computations are adopted to successfully retrieve optical solitons of a high-order nonlinear Schrödinger equation in a non-Kerr law media with the weak non-local nonlinearity and extracting some real and imaginary parts of the model using a wave variable transformation, Hosseini et al. (2021b) who studies a nonlinear (2 + 1)-dimensional evolution model describing the propagation of nonlinear waves in Heisenberg ferromagnetic spin chain system and a number of solitons and Jacobi elliptic function solutions to the Heisenberg ferromagnetic spin chain equation are formally derived, Hosseini et al. (2021c) who studied the high-order nonlinear Schrödinger equations in non-Kerr law media with different laws of nonlinearities using new Kudryashov method and extracted the soliton solutions of this model. The propagations of waves in optical fiber is one of important operations in all telecommunications process, specially the propagations in birefringence fiber and improve this process by develop the model governed this process will give future studies to this phenomena. The propagations of waves in optical fiber is one of important operations in all telecommunications process, specially the propagations in birefringence fiber and improve this process by develop the model governed this process will give future studies to this phenomena. Some studies have been established in the last few decays through good efforts via little number of authors to discuss the cubic-quartic of the birefringence for which the optical property of a material having a refractive index that depends on the polarization and propagation direction. The recent articles which investigate the Kerr-law, polynomial law, parabolic law, power law, dual-power law, Quadratic Law, Non-local law of this model have been demonstrated (Biswas et al. 2017a(Biswas et al. , 2017dHosseini et al. 2021c;Amsa et al. 2018;Wazwaz and Xu 2020;Yildirim et al. 2020a).
Our aimed apply the ESEM which is the famous ansatze method that governed by the auxiliary equation � ( ) = a 0 + a 1 + a 2 2 to achieve new types of optical solitons to cubic-quartic NLSE with its special forms which are mentioned in the abstract. The suggested method has been examined previously for many other NLPDE and continuously gives good result see for example (Shehata et al. 2020;Kohl et al. 2019;Bekir and Zahran 2020;Yildirim et al. 2020b;Lu et al. 2017;Zhao et al. 2013;Bekir and Zahran 2021a, b).
This article is organized as follow, the introduction section which gives short survey of the NLSE forms as well as the methods that used to solve it, in section two we will give the description of the ESEM and its application to find the soliton solution of the suggested model. In section three the application of the ESEM schema to extract the optical solitons of the cubic-quartic for the Kerr-Low NLSE in polarization-preserving fibers. In section four, the application the ESEM schema to extract the optical solitons of the cubicquartic for the quadratic-low NLSE in polarization-preserving fibers. In section five, the application the ESEM schema to extract the optical solitons of the cubic-quartic for the parabolic-low NLSE in polarization-preserving fibers. In section six the application the ESEM schema to extract optical solitons of the cubic-quartic for the Non-local-low NLSE in polarization-preserving fibers. In section seven brief conclusion of our work has been established. where R is a function of (x, t) and its partial derivatives that involves the highest order derivatives and nonlinear terms, according to the transformation (x, t) = ( ), = x − C 0 t Eq.
(1) can be reduced to the following ODE: where S is a function in ( ) and its total derivatives, while � = d d . The constructed solution according to this method is where the positive integer N in Eq. (3) can be located by balancing the highest order derivative term and the nonlinear term, while the arbitrary constants A i could be calculated later, the function ( ) satisfies the following new ansatz equation where a 0 , a 1 and a 2 other arbitrary constants which admit these two cases; (1) If a 1 = a 3 = 0 it will transform to the Riccati equation (Biswas et al. 2017b(Biswas et al. , 2017c, which has the following solutions; (2) If a 0 = a 3 = 0, it will transform the Bernoulli equation (Biswas et al. 2017c), which has the following solutions; And the general solution to ansatz Eq. (4) is as follows: (1) R( , x , t , xx , tt , ......) = 0.
(2) S( , � , �� , ...........) = 0 where 0 denotes to integration constancy. Finally, via inserting Eq. (3) into Eq. (4) and equating the coefficients of different powers of i to zero, we get a system of algebraic equations, by solving it we obtain the values of the unknown variable mentioned in these equations. Moreover, inserting these achieved variables into Eq. (4) then the aimed solutions have been extracted.

The Cubic-Quartic NLSE for the Kerr-Low nonlinearity using the ESEM
In the framework of Hosseini et al. (2021a, b) the cubic-quartic for the Kerr-low NLSE in polarization-preserving fibers can be proposed in the form, where a, b and c denote respectively to the 3-th order dispersion index, the 4-th order dispersion index, Kerr-Low refractive coefficient while i = √ −1. The cubic-quartic NLSE in birefringent fiber for Kerr-Low are In the framework of the ESEM the solution is Hence, Via inserting the relations (13) and (15)(16)(17)(18)(19)(20) into the first part of Eq. (12) and the relations (14) and (21-26) into the second part of Eq. (12), the following real and imaginary parts must be emerged respectively, Equation (28) implies a j = −4b j q j , w j = 3k j a j q 2 j + 4k j b j q 3 j , hence under this constrain Eq. (27) will be converted to, Now, via integrating Eq. (29) with respect to it becomes, Equation (30) represents either the first or the second parts of Eq. (12) which are the same, for similarity we will implement the solution for the first part j = 1 which is Via balancing ′′′ 1 , 4 1 at Eq. (31) leads to 4N = N + 3 which implies that N = 1 , hence the solution is Page 7 of 32 680 where � = a 0 + a 1 + a 2 2 + a 3 3 . Case 1: The first family, in which a 1 = a 3 = 0 ⇒ � = a 0 + a 2 2 , consequently Via inserting ′′′ 1 , ′ 1 , 2 1 and 4 1 into Eq. (31), collecting and equating the coefficients of various powers of i to zero leads to system of equations, by solving it we get these results, These 4-differents results will achieve 4-various types of solutions, from which we will plot the extracted solutions for the first and the third results.
Firstly, for the first result Eq. (35) which is This result can be simplified to be, From the point of view of the proposed method the solution is

Secondly, for the third result Eq. (36) which is
This result can be simplified to be, From the point of view of the proposed method the solution is, Via the same manner, we can plot the other two cases. Case 2: The second family, in which a 0 = a 3 = 0 ⇒ � = a 1 + a 2 2 , consequently (39) Via inserting ′′′ 1 , ′ 1 2 1 and 4 1 into Eq. (31),collecting and equating the coefficients of various powers of i to zero implies a system of equations, by solving it many solutions will be achieved from which only 8-results are suitable while the remaining are refused, From which we can get 8-various solutions, we will plot some cases say, the first, the fifth and the eight results.
For the 1-st result which is This result can be simplified to be, From the point of view of the suggested method the solution is, In the same manner, we can plot the other cases.

The cubic-quartic nlse for the quadratic-low nonlinearity using the ESEM
The cubic-quartic NLSE in polarization-preserving fibers with quadratic-cubic law can be proposed in the form, The cubic-quartic NLSE in birefringent fiber for quadratic-cubic law is Via inserting the relations (13-26) into Eq. (76) and (77) will transform them to the following real and imaginary equations respectively: . Case 1: The first family, in which a 1 = a 3 = 0 ⇒ � = a 0 + a 2 2 , consequently Via inserting ′′′ 1 , ′ 1 , 2 1 and 4 1 into Eq. (82), collecting and equating the coefficients of various powers of i to zero leads to system of equations from which we construct that there are no solutions achieved because either a 0 = 0 or a 2 = 0 or both or A −1 = A 1 = 0.
Case 2: The second family, in which a 0 = a 3 = 0 ⇒ � = a 1 + a 2 2 , consequently Via inserting ′′′ 1 , ′ 1 , 2 1 , 3 1 and 4 1 into Eq. (82), collecting and equating the coefficients of various powers of i to zero leads to system of equations which proposed big number of very complicated solutions from which only 12-solutions are valued and the other achieved solutions are refused because either a 1 = 0 or A −1 = A 1 = 0 , we will choose only 8-solutions of them where the remaining are very long which are For simplicity we will choose only one of these results say the first one, to extract its own solution and plotted it.
For the 1-th result which is This result will simplified to be, Via these values of constants the solution in the framework of the proposed method is

The Cubic-Quartic NLSE for the Parabolic-Low nonlinearity using the ESEM
The cubic-quartic NLSE in polarization-preserving fibers with parabolic law can be proposed in the form, The cubic-quartic NLSE in birefringent fiber for parabolic-law is, Via inserting the relations (13-26) into the Eqs. (101) and (102) will transform them to the following real and imaginary equations respectively Equation (104) implies a j = −4b j q j , w j = 3k j a j q 2 j + 4k j b j q 3 j , hence under this constrain Eq. (103) will be converted to, Equation (105) represents either the first part Eq. (101) or the second part Eq. (102) which is the same, for similarity we will implement the solution for the first part j = 1 which is Via balancing ′′′′ 1 , 5 1 at Eq. (106) leads to 5N = N + 4 which implies that N = 1 , hence the solution is 1 Now we will extract the different solutions corresponding to these 9-acceptable results, for simplicity we will choose only one result say the first one.
For the 1-st result which is This result will generate 4-sub results which are For simplicity we will take only one of these results say the first, and extracting the corresponding solution according to the point of view of the suggested method which is Via the same manner, we can plot the other solutions. Case 2: The second family, in which a 0 = a 3 = 0 ⇒ � = a 1 + a 2 2 , consequently (118) Via inserting ′′′′ 1 , ′′ 1 , 1 , 3 1 and 5 1 into Eq. (106), collecting and equating the coefficients of various powers of i to zero leads to system of equations in which the coefficients of −5 , −4 imply that e + g + h = 0 and the coefficients of −3 , −2 imply c 1 = −d 1 and by using these new constrain at the other equations of this system it will be reduced to the following system, The solution of the reduced system (129) gives the following three results, We will extract the solutions in the framework of the suggested method for the first and the third result.
For the first result which is, This result can be simplified to be, From the point of view of the suggested method the solution is (1)A 1 = 0, 1 = −b 1 k 4 1 q 4 1 12 , q 1 = −ia 1 k 1 √ 6 (2)A 1 = 0, 1 = −b 1 k 4 1 q 4 For the third result which is,

This result split into four sub results which are
We will plot only one say the first, from the point of view of the suggested method the solution is By the same manner we can plot the other remained cases.

The Cubic-Quartic NLSE for the Non-Local Low nonlinearity using the ESEM
The cubic-quartic NLSE in polarization-preserving fibers with non-local low be proposed in the form, The cubic-quartic NLSE in birefringent fiber non-local low is, Via inserting the relations (13-26) into Eq. (142) and (143) will transform them to the following real and imaginary equations respectively: Equation (145) implies a j = −4b j q j , w j = 3k j a j q 2 j + 4k j b j q 3 j , hence under this constrain Eq. (144) will be converted to, Equation (146) represents either the first part Eq. (142) or the second part Eq. (143) which is the same, for similarity we will implement the solution for the first part j = 1 which is, Via balancing ′′′′ j , ′′ 2 j at Eq. (147) leads to 4N = 2N + N + 2 which implies that N = 2 , hence the solution is where � = a 0 + a 1 + a 2 2 + a 3 3 . Case 1: The first family, in which a 1 = a 3 = 0 ⇒ � = a 0 + a 2 2 , consequently 2a 0 a 2 A 1 + +8a 0 a 2 A 2 2 + 2a 2 2 A 1 3 + 6a 2 2 A 2 4 .
Via inserting ′′′′ 1 , 2 1 ′′ 1 , ′′ 1 , 1 ′2 1 and 3 1 into Eq. (147), collecting and equating the coefficients of various powers of i to zero leads to system of equations, by solving this system rejected results will be achieved because in all these results either a 0 = 0 or a 2 = 0 or A −2 = A −1 = A 1 = A 2 = 0 , hence all these results will be refused.
For the first result which is This result can be simplified to be, The solution in the framework of the suggested method iswhere, For the six result which is, This result can be simplified to be, The solution in the framework of the suggested method is, By the same manner we can plot the other results.

Conclusion
In this paper, we already extracted new multiple accurate soliton types to the cubic-quartic NLSE in polarization-preserving fibers with its four various forms via the ESEM. Firstly, through overview the plotted Figs. 1, 2 , 3, 4, 5, 6, 7, 8, 9 and 10) it is clear that the ability of the suggested method to gives good new description of soliton types which never Fig. 4 The cubic-quartic soliton of the Im. part of kerr-low Eq. (48) in 2D and 3D with values:

Fig. 5
The cubic-quartic soliton of the Re. part of kerr-low Eq. (62) in 2D and 3D with values:w 1 = −8 A −1 = 0, A 0 = 3, A 1 = 1, a 1 = 3.5, a 2 = 0.6, k 1 = q 1 = d 1 = 0 = 1, c 1 = −5.6, 1 = −23.8 Fig. 6 The cubic-quartic soliton of the Im. part of kerr-low Eq. (63) in 2D and 3D with values:w 1 = −8 A −1 = 0, A 0 = 3, A 1 = 1, a 1 = 3.5, a 2 = 0.6, k 1 = q 1 = d 1 = 0 = 1, c 1 = −5.6, 1 = −23.8 achieved before to the cubic-quartic for the Kerr-Low NLSE in polarization-preserving fibers. Furthermore, through overview the plotted Figs. 11 and 12 we also already documented other new soliton types to the cubic-quartic for the Quadratic-Low NLSE in polarization-preserving fibers coupled equations by the same method. In related subject, the suggested method has been used effectively to established other new soliton types to the cubic-quartic for the Parabolic-Low NLSE in polarization-preserving fibers coupled equations which appeared clearly in the plotted Figures 13,14,15,16,17 and 18). Finally, we also achieved new types of soliton to the cubic-quartic for the Non-local Low NLSE in polarization-preserving fibers in the framework of the suggested method, we can detect it through the plotted Figs. 19, 20, 21 and 22).All the new types of soliton that we documented for each case the four forms mentioned above individually "which weren't achieved before by any other authors" denote to the novelty of these results, especially compared with that achieved previously by Bekir and Zahran (2020); Bekir and Zahran 2021b;Hosseini et al. 2021a;Hosseini et al. 2021b;Hosseini et al. 2021c) who used various manners to study these forms that emerged from this model. Consequently, new distinct and impressive visions of the solitons to the quartic-cubic NLSE for birefringent fibers of these four different forms have been constructed and the achieved solitons will add improved extended studies for all modern telecommunication engineering and all related phenomena.
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Conflict of interest
The authors declare that they have no conflict of interest.