Highly dispersive optical solitons and other soluions for the Radhakrishnan–Kundu–Lakshmanan equation in birefringent fibers by an efficient computational technique

In this article, we are interested to discuss the exact optical soiltons and other solutions in birefringent fibers modeled by Radhakrishnan–Kundu–Lakshmanan equation in two component form for vector solitons. We extract the solutions in the form of hyperbolic, trigonometric and exponential functions including solitary wave solutions like multiple-optical soliton, mixed complex soliton solutions. The strategy that is used to explain the dynamics of soliton is known as generalized exponential rational function method. Moreover, singular periodic wave solutions are recovered and the constraint conditions for the existence of soliton solutions are also reported. Besides, the physical action of the solution attained are recorded in terms of 3D, 2D and contour plots for distinct parameters. The achieved outcomes show that the applied computational strategy is direct, efficient, concise and can be implemented in more complex phenomena with the assistant of symbolic computations. The primary benefit of this technique is to develop a significant relationships between NLPDEs and others simple NLODEs and we have succeeded in a single move to get and organize various types of new solutions. The obtained outcomes show that the applied method is concise, direct, elementary and can be imposed in more complex phenomena with the assistant of symbolic computations


Introduction
In the science and technology fields like engineering, circuit analysis, fluid mechanics, solid state physics, chemical physics, plasma physics, geochemistry, optical fiber, quantum field theory and biological sciences, NLPDEs are used as a governing model to explain the complexity of the physical phenomena. To know the behaviour of intricate physical phenomena, it is necessary to calculate the solutions of the governing NLPDEs. Generally, the solutions of the NLPDEs are categorized into three types as exact solutions, analytic solutions and numerical solutions. Finding the exact solutions of NLPDEs has the importance to discuss the stability of numerical solutions and also development of a broad range of new scholar to simplify the routine calculation. Exact solutions to NLPDEs play an important role in nonlinear science, since they can provide much physical information and more insight of the physical aspects of the problem and thus lead to further applications. Wave phenomena in dispersion, dissipation, diffusion, reaction and convection are very much important (Biswas et al. 2018a, c;Ozkan et al. 2020aOzkan et al. , 2021Celik et al. 2021;Ali et al. 2018a;Nighat et al. 2020;Seadawy et al. 2018;Cheemaa et al. 2018;Khater et al. 2000) Furthermore, various specialists and mainstream researchers are giving more consideration to create and improve the optical transmission frameworks through optical fibers instead of birefringent fibers. However, the propagation of soliton through optical fibers governs next generation technology but there are several factors in birefringent fibers as well that are used to produce the soliton propagations Ali et al. 2018b;Arshad et al. 2017;Ghanbari et al. 2018;Seadawy et al. 2019b). It is polarization of light that prompts bunch speed mismatch,which is at last liable for differential gathering delay and numerous other negative impacts and consequently the examination of optical soliton is one of the most intriguing and interesting zones of exploration in nonlinear optics. There are well known computational and powerful techniques to find the optical exact soliton solutions of the differential equations Seadawy et al. 2019aSeadawy et al. , 2020aYounis et al. 2017Younis et al. , 2020Donne et al. 2020;Rehman et al. 2019a;Iqbal et al. 2019aIqbal et al. , b, 2020Cheemaa et al. 2019;Rizvi et al. 2020;Lu et al. 2019;Seadawy and Abdullah 2019).
Therefore, in this study we focus on constructing the exact optical soliton solutions in different structures to the RKL equation for birefringent fibers without 4WM terms in media with Kerr-law nonlinearity which is known as the basic case of fiber nonlinearity. Most optical fibers which has been quite popular recently comply with this law nonlinearity. Also, this media indicates itself as self-phase modulation, a self-induced phase-and frequency-shift of a pulse of light when it moves along with any fiber nonlinearity. In case of birefringent fibers the pulses are polarized. Naturally, vector solitons are studied in birefringent fibers. We apply an efficent compuational approach known as generlized exponential rational function method (GERFM) to find the various kinds of soliton solutions. The restraint relations are also observed during the mathematical analysis.
The RKL with Kerr law nonlinearity is given below (Biswas et al. 2018b;Ozkan et al. 2020b;Rehman et al. 2019b;Sulaiman et al. 2018;Seadawy et al. 2020b) where the dependent variable Θ(x, t) is complex-valued wave profile with two independent variables of x and t that represents spatio-temporal component, respectively. The first term characterizes temporal evolution whereas the parameters denotes group velocity dispersion (GVD) while, the coefficients is Kerr nonlinearity. Moreover, on right hand side the coefficients and sequentially accounts the third order dispersion (3OD) which induces soliton radiation and the effect of self-steepening to eliminate the formulation of shock waves. Thus, these compensatory effects of dispersion and nonlinearity provide the necessary balance to sustain soliton propagation.
Upon splitting the Eq. (1) for birefringent fibers into two components without 4WM (Jhangeera et al. 2020), we arrive at: The complex valued functions Ψ(x, t) and (x, t) represent the wave profiles and the Eqs. (2) and (3) represent the governing model for soliton transmission through birefringent fibers without 4WM. In the above coupled system, the coefficient j accounts for self-phase modulation (SPM) and the coefficient j is the cross-phase modulation terms respectively, for j = 1, 2 . The coefficients j and j correspond to self-steepening terms, while the the four-wave mixing effect is discarded.
This piece of article is discussed as sequence: In Sect. 2, the summary of the GERFM. In Sect. 3, optical solitons. In Sect. 4, results and discussion and finally paper comes at conclusions in Sect. 5.

The summary of GERFM
Here, we give a brief description of GERFM . Let us consider a nonlinear partial differential equation (PDE) where Ξ is a polynomial in its arguments.
The essence of GERFM can be presented in the following steps Step 1. We introduce traveling wave transformation as: where B and c represent the amplitude component and velocity respectively. After substituting this transformation into Eq. (4), we get nonlinear ODE in the following form.
where Υ is in general a polynomial function of its arguments and ′ denotes the derivative w.r.t .
Step 2. Suppose that the solution of Eq. (5) can be expressed as follows The unknown coefficients d 0 , d k , f k (1 ≤ k ≤ n) and constants r i , s i (1 ≤ i ≤ 4) are determined and the value of n will be evaluated by using homogeneous balance principle.
Step 3. After putting Eq. (6) into Eq. (5) we get an algebraic equation R( , e s 1 , e s 2 , e s 3 , e s 4 ) = 0 . Setting each coefficient of R equal to zero, we get a system of nonlinear equations in the form of d 0 , d k , f k (1 ≤ k ≤ n) and r i , s i (1 ≤ i ≤ 4) is yielded.
Step 4. On solving the above system of equations we will get the values of d 0 , d k , f k (1 ≤ k ≤ n) and r i , s i (1 ≤ i ≤ 4) with the aid of any symbolic computation packages . Substituting these values in Eq. (6) we attain the soliton solutions of Eq. (4).

Optical solitons
For solving the above couple system of Eqs.
(2)-(3), we suppose the traveling wave transformation as follows: where And Q j ( ) for ( j = 1, 2 ), B, , 0 , and k represent the amplitude component of the soliton, velocity of soliton, phase constant, soliton wave number and soliton frequency respectively. Substituting above transformations of Eqs. (8)-(10) into Eqs. (2) and (3), we get real and imaginary parts, respectively of the form In order to use the balancing rule, the important results emerged from Eqs. (11)-(12) by using of Q j = Q j are We can obtain (7) Ω( ) = r 1 e s 1 + r 2 e s 2 r 3 e s 3 + r 4 e s 4 .

Case 1
Inserting these values in Eqs. (18) and (19), then we obtain The optical dark soliton solution as The graphical representations of the solutions are shown for different values of parameters.

Case 2
Substituting these values in Eqs. (18) and (19), we attain The singular optical soliton solution as The graphical representations of the solutions are shown for different values of parameters.
Case 3 Imposing these values in Eqs. (18) and (19), then we derive The combined optical soliton solution as The graphical representations of the solutions are shown for different values of parameters. Family The following trigonometric and combined trigonometric traveling wave solutions are Case 1 Substituting these values in Eqs. (18) and (26), we attain Replacing these values in Eqs. (18) and (26), we get Here ( k 3 j + k 2 j + )( j + 3k j ) > 0 and j ≠ 0 with j = 1, 2 for valid solution.

Case 1
Inserting these values in Eqs. (18) and (33), then The exponential function solution can be expressed as Here (k 2 − j + k j − )(− j − j + k j + j ) > 0 and j ≠ 0 with j = 1, 2 for valid solution. The graphical representations of the solutions are shown for different values of parameters.

Case 1
Imposing these values in Eqs. (18) and (38), then The singular periodic wave solution can be expressed as Here (k 2 j + k j + )( j + 3k j ) > 0 and j ≠ 0 with j = 1, 2 for valid solution.

Results and discussion
The results of this paper will be beneficial for learners to analyze the most attractive applications of the RKL equation, which describes the prorogation of waves without 4WM in birefringent fibers. Figures 1, 2 , 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 clearly demonstrates the surfaces of the solution obtained for 3D, 2D and the contour plots, with choices of parameters for the RKL model. We can capture the behaviour of the of the acquired solution with the assistant of contour plots. In the same way, 3D figures tell us to model and demonstrate accurate physical behaviour. Through this study, we consider the exact optical soliton solutions to the nonlinear RKL model using generalized exponential rational function approach. The authors proposed different analytic approach in newly issued article and reported some fascinating findings. We can grasp from all the graphs that the GERFM is very efficient and more precise in evaluating the equation under consideration.

Conclusions
This paper extracted the dynamics of optical solitons in the RKL equation without 4WM in birefringent fibers. The solutions are achieved in the shape of exponential, trigonometric and hyperbolic functions as well as exact optical soliton solutions by the mechanism of GERFM under different constraint conditions which provide the guaranty and validity of solutions are also listed. In addition, we attained the singular periodic wave solutions. The model must be extended to govern DWDM networks so that parallel communication of soliton dynamics can be addressed. These new families of solutions are shown the power, effectiveness and fruitfulness of this method. This article shelters the application of optical fibers. Also, these fresh solutions have many applications in physics and other branches of physical sciences.