New Perception of the Optical Solutions to the Fokas-Lenells Equation According to Two Different Techniques

In this article, the perturbed Fokas- Lenells equation (FLE)” which plays a vital role in modern asocial media and electronic communication” is employed. Two important different meth ods are invited to demonstrating new accurate solutions of this equation. The first method is the modified simple equation method (MSEM) that reduces large volume of calculations and realizes closed form solution. While the second is the modified extended tanh-function method (METFM) “ which controlled by the auxiliary Ricatti equa tion” and used effectively to obtain accurate solutions Furthermore, few of the realized results are compatible with that obtained by previous authors elsewhere the others remains new. In addition to the varitional iteration method (VIM )is applied perfectly to achieved the numerical solution corresponding to the exact solution realized by each one of these methods individually.


Introduction
The breakthrough of the soliton technology is clearly manifested in recent asocial media and modern electronic communications namely the internet blogs, Facebook communication, twitter comments". The suggested equation [1][2][3][4][5] represent this phenomenon can be written as: 2  The first term of Eq. (1.1) denotes for temporal development of the pulses, whereas the two independent variables x and t in the complex-valued function p(x, t) represent the spatial and temporal components respectively. The variables in the left hand side are distinguished as  is self -phase modulation,  revise to the nonlinear dispersal, while 12 , aa are the group velocity dispersal and the spatial-temporal dispersals respectively. On the right hand side  is the influence of inter-modal dispersal that is in addition to chromatic dispersal and  is the self-deepening effect while the coefficient m gives the effect of nonlinear dispersal with full nonlinearity. Finely the parameter n is the full nonlinearity parameter. The accurate solution to the NLPDEs which plays a crucial role in big varieties of biological, chemical and physical phenomena is one of the most exciting advances for looking it. The NLPDEs can introduce much physical information and more insight into the physical aspects of these phenomenon's and its applications. Recently, with the aid the swift advancement of The corresponding author Email address:e_h_zahran@hotmail.com computer algebraic system the analytical solutions for the nonlinear partial differential equations (NPDEs) have been realized [6][7][8][9][10][11][12][13][14]. There are three principal axes to get the exact solutions to NLPDEs namely the reduction methods, Lie symmetry group and the ansatze approaches methods. The famous ansatze methods are demonstrated through [6][7][8][9][10][11][12][13][14][15][16]. The most famous and powerful one of these methods is the MSEM which significant by its concentrated and accuracy to describe briefly the closed form solutions for the nonlinear problems. There are some tries to study the soliton dynamics by solving FLE listed at .
There are a variety of mathematical procedures that make the study of soliton dynamics possible [31][32][33][34][35] proposed a new methods to address accurate solutions and present conservation laws to this model. Also, A.H. Kara,and others [35] have obtained the Optical soliton solution to this equation using the jacobi elliptic function and (G′/G)-expansion method. Also the varitional iteration method [36] has been used effectively to obtain the numerical solution corresponding to the exact solution to satisfied by each one of these methods. Our target going to apply the MSEM and the (METFM) as a two new techniques to get exact and hence the solitary form solutions for the equation (1.1) mentioned above as well as the numerical solution using the VIM. 2. Closed form solution using the MSEM [23][24]: This section is pending on the MSEM arising in [23][24] to get the closed form solution which contain some variables. If one give these variables definite values he will realize the solitary solution.

Description of the method
To propose the general forlasim of the nonlinear evolution equation, let us introduce R as a function of h(x,t) and its partial derivatives as,

Application
First of all, we introduce the following wave transformation, with these restrictions Solving this system of algebraic equations as: (i) Firstly from the first and the fourth part of Eq. (3.4) by dividing we obtain, (ii) Secondly, from the second and third part of Eq. (3.4) by dividing, we get, Integrating once, we obtain: Integrating another once, we obtain; Thus, substitute at Eq.
and hence, The corresponding author Email address:e_h_zahran@hotmail.com

Accurate optical solutions using the modified extended tanh-function method [27-28]
This section is composed on the modified extended tanh-function method [27][28] which is another technique (which depend on the balance rule) arising to get the exact solution in terms of some parameters. If these parameters take definite values the solitary wave solution is realized.

Description of the method
According to the general forlasim of the nonlinear evolution equation mentioned above in (2.1), (2.2) the constructed solution to METFM is: The balance rule mentioned in [18], is used to evaluate the integer m arise in Eq.
Let the coefficients of different powers of i  in these solutions equal to zero, we can generate The corresponding author Email address:e_h_zahran@hotmail.com an algebraic system of equations that solving by any computer program to get the values of the required constants.

Application:
We will apply the modified extended tanh-function method for the equation (3.2) mentioned above as follow, Now, we substitute about 3 , F F and F  at equation (3.2) and then equating the coefficients of different power of F to zero we get this system of algebraic equations, 3 3 1 We can easily solve this system of algebraic equations analytically as follow, equation (5.2) lead to 1 0 A  consequently equations (5.9) and (5.12) implies, Dividing equation (5.6) by equation (5.5) and integrating once we obtain, According to these obtained results we take only case (3), hence the solutions is, The corresponding author Email address:e_h_zahran@hotmail.com  Figure 4. the plot of imaginary part Eq.(5.14) in 2D and 3D with values: The following two figures represent the real equation ( The following two figures represent the imaginary part equation ( Figure 6. the plot of real part Eq.(5.15) in 2D and 3D with values: The corresponding author Email address:e_h_zahran@hotmail.com

The Variational iteration method [36]
Consider the differential equation with inhomogeneous term () f  and L, N the linear and the nonlinear operators respectively as; ( ), The VIM proposes a correction functional for equation (6.1) to be; where  is a general Lagrange's multiplier, which can be identified optimally via the The 2-nd order ODE in the form, we find that tx   , and the correction function give the iteration formula;

The corresponding author Email address:e_h_zahran@hotmail.com
The 3-th order ODE in the form, We find that The lagrange multiplier  take the general form Furthermore the zeros approximation 0 () h  can be selected perfectly to be, . where m is the order of the ODE

(6-a) Numerical solution corresponding to the exact solution realized using the MSEM
For the second order differential equation (3.2) mentioned above, With the initial condition The corresponding author Email address:e_h_zahran@hotmail.com  According to the variational iteration method the first iterations is,

(6-b) Numerical solution corresponding to the exact solution realized using the METFM
For the second order differential equation (3.2) mentioned above, In addition to, the successive iteration for the all cases mentioned above can be calculated as, Using the fact that the exact solution is obtained by using

Captions, citations and physical meaning of the figures
We can briefly give the captions and citations for the obtained figures instead of the corresponding physical meaning. Let us start from the results achieved by MSEM and the corresponding Figures (1, 2) which represents the periodic singular dark soliton solution to the equations (3.10),(3.11) respectively in two and three dimensions. Furthermore,the results realized by the METFM and the corresponding Figures (3-6) in two and three dimensions which are periodic bright and dark soliton solution. Most of achieved solutions using these two signfinacint methods weren't satisfied before. While Figures (7-10) represents the numerical solution obtained using VIT which derived its initial conditions from the realized exact solutions for each one of the proposed methods individually which are hyperbolic soliton solution.
The corresponding author Email address:e_h_zahran@hotmail.com

Results and Discussion
The main target of this section is to give some graphical representations of the solutions achieved for the FLE. in 2D and 3D with values: Figures (1-2).
While Figures (3-6) give the graphical representations of the solutions achieved using Also, Figures (7-9) give the graphical representations of the achieved solutions using VIM with the values, 1 2 0.1,

Conclusion
In this work, the modified simple equation method has been applied effectively to realize the closed form solution for the first time to the perturbed Fokas-Lenells equation (FLE) Figure 1, Figure 2 which is positive forward future studies. Also, the modified extended tanh-function method has been applied successfully to obtain new accurate optical solutions Figures (3)(4)(5)(6) for this model. In addition to VIT which derived its initial conditions from the realized exact solutions for each one of the proposed methods individually has been success to obtained numerical solution corresponding to the obtained exact solutions figures (7)(8)(9)(10). We also conclude that these constructed techniques admit new accurate effective solutions compared with that obtained by the previous work [33] and can be applied to other NLPDEs. Also the achieved numerical solution is more accurate than that obtained previously by other numerical methods.

List of Abbreviations
The modified simple equation method MSEM 3 The modified extended tanh-function method METFM 4 The varitional iteration method VIM