NUMERICAL SOLUTION OF CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION

Kazheen H. Omar ^{1, *}, Fadhil H. Easif ¹

¹ Department of Mathematics, College of Science, University of Zakho, Kurdistan Region, Iraq.

* Corresponding author email: kazheen.omar@uoz.edu.krd

ABSTRACT:

This paper is devoted to investigating and comparing the variational iteration method (VIM) and the residual power series method (RPSM) for solving the cubic-quintic nonlinear Schrödinger equation (CQNLSE) initially developed to elucidate the propagation of pulses in optical fibers. Next, we use the initial conditions to get the numerical solutions of the CQNLSE. We compared the known exact solutions with the approximate results obtained using both the VIM and RPSM. The exact solution and the results from RPSM are evaluated against those from VIM. The findings demonstrated that VIM outperformed RPSM in terms of accuracy, efficiency, and ease of implementation for solving the CQNLSE. In addition, the current results are shown graphically and in the table.

1.        INTRODUCTION

        The CQNLSE is a universal mathematical model describing various physical applications and approximating more complex systems, such as Bose-Einstein condensates (BECs), nonlinear optics, a range of interaction phenomena in plasmas, including plasma physics, condensed matter physics, and nuclear physics (Tang & Shukla, 2007,Seadawy et al., 2022, Peleg & Chakraborty, 2023). Notably in nonlinear optics and BECs. In optics, it models the propagation of light beams through layered media, where the nonlinear response of the material changes with position or time. In BECs, it describes atomic interactions influenced by Feshbach resonances. Even in the absence of external potentials, the CQNLSE with nonlinearity management is essential for regulating soliton dynamics, particularly when nonlinearities fluctuate spatially or temporally. This enables the formation of stable soliton structures and multi-soliton bound states (Luo, 2022). The CQNLSE is used in modelling light propagation through various optical media, including non-Kerr crystals, chalcogenide glasses, organic materials, colloids, dye solutions, and ferroelectrics (Seadawy & Sayed, 2017). Studies on the cubic-quintic nonlinear Schrödinger equation also extend to optical fiber communications and nuclear hydrodynamics.

        Researchers have extensively studied the CQNLSE using various solution methods. For example, Hafez used the approach in his work to extract both singular and periodic solutions (Golam Hafez et al., 2014). By assuming an ansatz solution, Serkin (Serkin et al., 2001) discovered the novel stable bright and dark soliton management regimes for the CQNLSE model. Hao (Hao et al., 2004) published solitary wave analytical solutions. He found the explicit spatial self-similar, bright, and dark soliton solutions of the CQNLSE using distributed coefficients and an external potential (He et al., 2014). Caplan addressed the problems of solitary vortex existence, interactions, and stability in the two-dimensional CQNLSE using both analytical and numerical techniques (Caplan et al., 2009). Numerous researchers have focused on applying various techniques to identify numerical analysis solutions in recent years. Among these are the Adomian and Adomian-Padé Techniques (Sabali, Manaa, & Easif, 2018), Variational Iteration Method (Easif et al., 2015), Sumudu-Decomposition Method (Azzo & Manaa, 2022), Successive Approximation Method (Sabali et al., 2021), Residual Power Series Method (Manaa et al., 2021), and others.

        Inokuti (1978) was the first to propose a generic Lagrange multiplier approach for solving quantum mechanical problems. This method allowed for the solution of nonlinear problems. In 2006 and 2007, Chinese mathematician Ji-Huan He, a professor at Donghua University, transformed the Lagrange multiplier method into an iterative technique known as the VIM (Al-Saif et al., 2011). For a wide range of applications in physics, chemistry, biology, and engineering, homogeneous or inhomogeneous equations, including linear or nonlinear (NL), and systems of equations, the VIM approach provides a reliable and efficient process. Numerous writers have demonstrated (Faradilla et al., 2021) that this approach offers advantages over current numerical methods. This method offers quick converging successive approximations of the precise answer if one exists; if not, a few approximations may be used for numerical purposes (Shihab et al., 2023). The VIM does not need special treatments like the Adomian method (Odibat, 2010), perturbation methods, etc, it just employs the starting conditions that are specified.

        The RPSM, a relatively new methodology based on the generalized Taylor series, is explained in depth (Alquran, 2014). The RPSM (Korpinar, 2019) is a helpful technique for determining the coefficient values in the solution of fuzzy differential equations using power series. The repeating algorithm is what makes up the RPSM. The RPSM offers several significant advantages for solving both linear and nonlinear differential equations (El-Ajou et al., 2015). In contrast to conventional approaches, RPSM can effectively and immediately tackle severely nonlinear situations since it does not need linearization, perturbation techniques, or discretization (Inc et al., 2016). With each term calculated using basic chains of linear equations, it offers an analytical solution in the form of a convergent power series, which is both computationally simple and extremely precise. Furthermore, RPSM may be implemented flexibly given a suitable starting estimate, and does not require reformulation when the solution order is increased. This makes it a practical and adaptable tool for a variety of issues, such as differential and integral equation systems (Moaddy et al., 2015).

        Numerous writers from a variety of subjects have employed the RPSM approach, such as the Klein-Gordon Schrödinger equation (Manaa et al., 2021), fractional diffusion equations (Jumarie, 2011), fractional Burger-type equations (Zunino et al., 2008), Boussinesq–Burgers equations (Mahmood & Yousif, 2017). Systems of Fredholm integral equations (Komashynska et al., 2016), nonlinear fractional KdV–Burgers equation (He, 1999), time fractional nonlinear coupled Boussinesq–Burgers equations (Ubriaco, 2009), fuzzy differential equations (Mainardi, 2012), for linear and nonlinear Lane–Emden equations (Oldham & Spanier, 1974), higher order initial value problems (Miller & Ross, 1993), time-fractional Fokker–Planck equations (Cifani & Jakobsen, 2011).

        The following sections make up this paper: Section 2 presents the CQNLSE's mathematical model. In section 3, the basic ideas of VIM and RPSM are provided. In section 4, the derivation of VIM and RPSM are illustrated for the CQNLSE, and in section 5, a numerical example demonstrates the approaches' accuracy and efficiency.

2. MATHEMATICAL MODEL

Cubic-Quintic Nonlinear Schrödinger Equation:

Consider the CQNLSE (Lai et al., 2006).

                                                                                                (1)

where    and .

Where is the slowly changing electric field envelope,  is the distance along the velocity dispersion direction,  is the time, with respect to the group velocity dispersion,  is the second-order dispersion,  is the 3rd-order dispersion, and  is the fourth-order dispersion. For the cubic-quintic terms, the coefficients are  and .

Bright Soliton:

The bright soliton for the CQNLSE (1) is (Lai et al., 2006):

                                                                                                                                        (2)

With the initial condition

                                                                                                                                             (3)

The Method's Description:

In this section, we will display the main ideas of the VIM and RPSM.

Basic Idea of the Variational Iteration Method:

To demonstrate the fundamental idea of VIM, consider the following general NL partial differential equation (PDE), (Odibat, 2010):

                                                                                                                                   (4)

with the initial condition:

                                                                                                                                                                      (5)

When  is a known analytic function,  is the NL operator component, and  ,  is a linear operator part and is the term of the highest-order derivative.

We may create the following iteration formula using VIM:

                                             …           (6)

A generic Lagrangian multiplier, denoted by λ (Inokuti, 1978), can be optimally determined from the stationary conditions of equation (6) concerning  using the variational theory (Anjum & He, 2019), the subscript  indicates the n-th approximation, and is regarded as a confined variation, i.e. .

The Lagrange multiplier can be identified as (He and Wu, 2007).

                                                                                                                                        (7)

By applying the Laplace transform to both sides of (4), we may more readily find the Lagrange multiplier, following the methodology of Tsai and Chen. This converts the linear portion with constant coefficients into an algebraic one (Wu, 2013).

Next, the approximate solution to equation (4) is thus provided by:  

The approximation is denoted by the subscript , and  is regarded as a constrained variant (Momani & Abuasad, 2006).

Basic Idea of the Residual Power Series:

3                 METHOD

Considering the general form of an NLPDE:

                                                                                                                                (8)

with the initial condition equation (5),               

where  is the highest order partial derivative with respect to time t.  is a nonlinear term and  a linear term. The standard RPSM defines the solution  as the power series of the form:

                                                                                                                                                     (9)

Where  afterward, we may define to denote the nth truncated series of  i.e.

                                                                                                                                (10)

The zeroth residual power series approximation solution for  is

                                                                                                                            (11)

Where  is the initial condition. Now, if we replace equation (11) with equation (10), we obtain:

                                                             (12)

to complete the coefficients ,  of equation (12), the residual function for equation (8) is first defined as follows:

                                                                                (13)

and the  Residual function is of the form:

                                           (14)

We present a few RPSM findings from (Cifani & Jakobsen, 2011), which are crucial to RPSM:

· 

·  , 

·                                                                                                 (15)

As a result, we may acquire all of the necessary coefficients  of the power series of equation (8).

Illustrative Application:

This section will apply the aforementioned techniques to the CQNLSE.

Derivation of VIM for Solving Cubic-Quintic Nonlinear Schrödinger Equation:

Consider the CQNLSE (1) with the initial condition equation (3) by means of VIM, the correcting function is provided as

                                                                                                                                         (16)

where

  and   

  is the conjugate of .

In our equation,  Then by formula (7), substituting in equation (16), we get:

                                                                                                       (17)

Equation (3) provides us with the initial approximation  . The following is how we may retrieve the additional components using the iteration formula (17):

.                                                                                                    (18)

For  

                                                                                                               (19)

For

                                                                                                                                               (20)                                                                                                                                                                                                

And after similarly for

Derivation of RPSM for Solving Cubic-Quintic Nonlinear Schrödinger Equation:

Consider the CQNLS equation (1), with the initial condition  

                                                                                                                     (21)

Where  is the initial condition. Using equation (21) to apply RPSM to equation (1). Next, the following is the RPSM solution to equation (1) around the beginning point :

                                                                  .                                                 (22)

Where  then, we can define to indicate the nth truncated series of  that’s

                                                                   .                                              (23)

When we now replace equation (21) with equation (23), we obtain:    

                                                                                                    (24)

To calculate the value of the coefficients , of equation (24), . We defined the residual function for equation (1) as:

          (25)

Thus, the residual functions, are to the form

for                                                                                                  (26)

to determine , we write  in equation (26). Then, we will have:

                                                                                                                                                     (27)

where

                                                                .                                                                     (28)

We obtain the residual functions as follows by changing equation (28) into equation (27).

,                                             (29)

 

                                                                             (30)

 

,         (31)

According to , we get

.                       (32)

So

.            (33)

When  the same technique can be used to obtain a higher degree of an approximate solution.

Numerical Results:

Within this segment, we use the methods from the previous part to solve the CQNLS problem numerically. We then compare the results with exact solutions.

Solving Cubic-Quintic Nonlinear Schrödinger Equation with the use of VIM:

Consider the CQNLS equation (1), assume that

 , (Lai et al., 2006).

With the initial condition 

                                                                                                                                         (34)

The exact solution is given by equation (2), and that  and from equations (19) and (20).

                                                                      (35)

 

 

 

 

 

                                                                                        (36)

 

Where

.

By the same way, we can find  and so on.

Solving Cubic-Quintic Nonlinear Schrödinger Equation with the use of RPSM:

Subject to the initial condition equation (3), equation (33), which is obtained by applying RPSM to the CQNLSE, is obtained.

For , we get

                                                                                     (37)

For , we get

.                (38)

Table 1: Shows exact solution, VIM, RPSM, and absolute error between the exact solution and the approximate solutions by VIM and RPSM for  and

 

 

 

                   (a) The 2D graph of .                                                  (b) The zoomed 2D graph of .

 

Figure 1: Exact solution and VIM and RPSM at  and

.               (a) The 2D graph of .                                                       (b) The 2D graph of

Figure 2: The curves of exact solution, VIM, and RPSM for real and imaginary, when  and .

                                                                                     (a) The 3D graph of .

                     (b) The 3D graph of .                                                                     (c) The 3D graph of .

Figure 3: The surfaces of the exact solution, VIM, and RPSM CQNLSE, when  and .

 

(a) The 3D graph of

 

 

                   (b) The 3D graph of                                                              (c) The 3D graph of .

Figure 4: The 3D surfaces for the real part of , when  and

 

                                                                    (a) The 3D graph of .

 

 

 

 

 

 

 

             (b) The 3D graph of .                                                                           (c) The 3D graph of .

Figure 5: The 3D surfaces for the imaginary part of , when  and

 

CONCLUSION

        The VIM and RPSM are both utilized in this paper to obtain approximate analytical solutions for the cubic-quintic nonlinear Schrödinger equation. We took an example of the CQNLS equation to compare between the 2nd order of VIM and RPSM with the exact solution. The results obtained by the two methods are compared with the exact solution of the equation. Moreover, we concluded that VIM is powerful, reliable, and elegant, and it yields solutions in a rapidly converging sequence compared to RPSM. It was also found that VIM is significantly more accurate and efficient, matching the exact solution more closely than RPSM. The solutions that have been obtained with real and imaginary parts are plotted.

Acknowledgments:

        The authors gratefully acknowledge the editors' and referees' comments that helped improve this work. We want to express our sincere appreciation to Professor Dr. Saad A. Manaa for his valuable comments on this manuscript.

Author Contributions:

        K. H. O. was responsible for writing the main script and conducting the primary research. F. H. E. supervised, provided guidance, and reviewed the manuscript. Both authors contributed to the final version of the paper. 

Funding:

This research received no external funding.

Conflict of Interest:

The authors declare that they have no conflict of interest.

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