Numerical investigation of the Adomian-based methods with w-shaped optical solitons of Chen-Lee-Liu equation

The present paper computationally examines the w-shaped solitary wave solutions for an important type of nonlinear Schrödinger equation that appeared in 1979 called the Chen-Lee-Liu (CLL) equation by proposing two recursive schemes. The schemes are based on the famous Adomian’s efficient decomposition technique. We successfully simulated the two proposed schemes with the aid of mathematical software and established a comparative analysis. It is noted from the present study that the improved method performs better than the classical method at different time levels. This is in fact in conformity with most of the results in the related literature. We finally present tables and a series of figures to support the presented results.


Introduction
The study of soliton solutions corresponding to nonlinear evolution equations helps a lot in understanding certain interesting physical properties posed by the equations. The homogeneous balance involving the highest-nonlinear term and the highest-linear term arising in such evolution equations causes the shape of soliton wave pulses not to change during propagation [1,2]. Generally, the nonlinear Schrödinger equation is an important class of evolution equations that plays a vital part in the study of diverse areas of nonlinear sciences such as nonlinear optics, plasma physics and optical fibers among others. Besides, these equations explain the pulse dynamics in optical fibers [3,4]. Furthermore, various computational, semi-analytical and analytical methods have been proposed and used in the past decades to examine many classes of Schrödinger equations, see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and the references therein. However, a class of nonlinear Schrödinger equation with a great number of applications is the so-called Chen-Lee-Liu equation given in dimensionless form as [23] with u x t , ( ) designating the soliton's profile in space and time variables x and t; while and a and b are nonzero constants standing for the group velocity dispersion and self-steepening phenomena, correspondingly. Moreover, when = = a b 1, equation (1.1) reduces to a Regular Chen-Lee-Liu (RCLL) equation as extensively examined in [23]. The CLL model emanated from the extension of the classical nonlinear Schrödinger equation to accommodate more applications in the propagation and interaction of ultrashort pulses in optics. More, various numerical methods were used to study many mathematical models including the standard Adomian decomposition method [24] and the improved Adomian decomposition method introduced by [25] for CLL model with some cases bright optical solitons [19,26]. Also, this model was solved by Adomian-based Methods introduced by Wazwaz and El-Sayed [27,28] with different kinds of optical solitons [29,30]. Furthermore, the w-shaped optical soliton of CLL equation studied only by Laplace Adomian decomposition method [5]. However, in the present paper, we numerically study the w-shaped optical soliton solution of the CLL equation. We derive two recursive schemes based on the Adomian's efficient algorithm by Wazwaz [24] and the other by improved Adomian's method [25]. A comparative analysis of the two methods will be carried out by extensively Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. examining the absolute errors posed by the schemes. We further aim to submit a comprehensive conclusion at the end. Besides, it is also pertinent to mention here that there other numerical or rather semi-analytical methods that go hand-in-hand with the Adomian's method in solving vast classes of nonlinear differential and evolution equations. These methods include the homotopy perturbation method, homotopy analysis method, variational iteration method and transform-based decomposition methods to mention a few, see [31][32][33][34][35][36]. Also, the arrangement of the paper takes the following form: we recall an exact w-shaped optical soliton solution in section 2. In section 3, we give the outlines of the two methods. Also, in section 4, we discuss the obtained numerical results; while section 5 presents some concluding remarks.

The w-shaped optical solitary waves
In this section, we present the recently obtained exact solution in the form of w-shaped soliton of the CLL equation by Triki et al [9] given by where g, h and l are parameters defined by the following formulae: , , , , and are arbitrary constants; while a a , 1 2 and a 3 are constants satisfying the constraint conditions a aa a The w-shaped soliton of the CLL equation presented in equation (2.1) will be used in the subsequent section as a benchmark solution for the numerical investigation and comparison of the ADM and IADM methods, respectively. in particular, equation (2.1) gives the expressions for two dissimilar bright pulses of the CLL equation earlier given in equation (1.1). The parameter h is an important factor that determines the nature of the propagating profile for the wave in the background. If for instance we further analyze the (−) signed solution given from equation (2.1), it is easy to clearly notice that the soliton's profile looks exactly like the letter w. It is however noted that the profile maintains unaffected for significantly long distance while propagating. In the same manner, the (+) signed solution given from equation (2.1) is also a bright soliton solution of the CLL equation [9].

Numerical solutions
In this section, we derive the numerical recursive schemes for the CLL equation by employing both the ADM and IADM methods.

Adomian Decomposition Method (ADM)
We adopted the version of the ADM introduced by Wazwaz in [24]. First, consider again the CLL equation as follows Then, on using the operator notation, let = ¶ ¶ L t t with its inverse operator and accordingly obtain the following solution recursively given by Therefore, the targeting recursive scheme via the application of the ADM is thus given in equations (3.8) -(3.9). The scheme will then be simulated together with certain exact w-shaped soliton solutions in the subsequent section.

Improved Adomian Decomposition Method (IADM)
The IADM was proposed by [25] while converting the complex-valued equation of equation (1.1) type into a real-valued equation by splitting the complex-valued function u x t ,  are to be determined, and we further represent the nonlinear expression in equations (3.11) and (3.12) by Also applying the corresponding the inverse operator in equations (3.11) and (3.12) through equations (3.14) -(3.15) take the form 2 are the Adomian's polynomials [37] to be recurrently computed from equation (3.6). Therefore, the targeting recursive scheme via the application of the IADM is thus given in equations (3.18) -(3.21). The scheme will then be simulated together with certain exact w-shaped soliton solutions in the subsequent section.

Results of numerical analysis
The present section gives numerical results of the obtained schemes and establishes some comparative study. Considering one of the w-shaped soliton solutions of the CLL equation for h > 0 given in equation (2.1), that is, x vt e , s e c h , with the corresponding initial condition we are able to numerically simulate the two recursive schemes for the ADM and IADM with the help of the Maple software and present the corresponding absolute error analysis in tables 1 and 2, and their respective graphical representations in figures 1-8. From the error analysis in tables 1 and 2, it is observed that the IADM performs better than the ADM as fully documented in the literature looking at the error discrepancies both when and. Also in figures 1, 2, 5 and 6, we give the absolute error comparisons of the considered exact W-shaped soliton and the approximate solutions using the ADM and IADM, respectively; while figures 3, 4, 7 and 8 give the three-dimensional (3D) surface of the numerical solutions in the same manner. Furthermore, the graphical representations corresponding to the bright optical soliton solutions under consideration turned to look like a bell-shaped profile as in case (1) and w-shaped profile for case (2). This in fact depicts the physics behind the dynamics of each solution and will certainly give more insight while digging more information about the CLL equation with regards to interactions and propagations of pulses. We thus remarked here that the two schemes are efficient since they reveal good results with high level of exactness.
Case (1). consider the positive-signed solution (+) given by x vt e , s e c h .     Case (2). consider the negative-signed solution (−) given by x vt e , s e c h .

Conclusion
In conclusion, the present paper examines numerically the w-shaped soliton solution of the CLL equation by proposing two recursive schemes. The schemes are based on the Adomian's efficient decomposition technique for treating differential equations. We successfully derived two numerical iterative schemes using the ADM and IADM for the CLL equation and further simulated with the aid of Maple software for error analysis. It is noted from both the table and figures that the IADM performs better than the ADM using different time levels; which is in conformity with most of the results obtainable in the literature. We thus recommend the proposed schemes for investigating different evolution and Schrödinger equations numerically whenever less number of iterations and minimum errors are aimed at.