In our research paper, we explore the application of mathematical techniques, both analytical and numerical, to solve the coupled nonlinear Schrödinger equation. To obtain accurate solutions, we use the improved, modified, extended tanh-function method. By breaking down the Schrödinger equation into real and imaginary components, we derive four interconnected equations. We analyze these equations using the generalized tanh method to find precise solutions. This set of equations is of great importance in quantum mechanics and helps us understand the behavior of quantum systems. We provide an analytical and numerical solution using the implicit finite difference. Our method is second-order in both space and time, and we have verified its stability through von Neumann’s stability analysis.

## I. INTRODUCTION

^{1,2}They serve as indispensable instruments for comprehending intricate systems that cannot be sufficiently represented through linear equations. Analytical techniques strive to discover exact, mathematical solutions to partial differential equations (PDEs). Nonetheless, nonlinear PDEs, because of their complexity, frequently require simpler analytical solutions. Traveling wave methods involve employing specific strategies to identify exact solutions for particular PDEs by focusing on solutions that display distinct traveling wave characteristics, such as the Kudryashov approach,

^{3,4}the improved $Q$-expansion strategy,

^{5}the $G\u2032G$-expansion method,

^{6}the $G\u2032G\u2032+G+A$-expansion technique,

^{7}and more.

^{8–28}These techniques help convert a given partial differential equation (PDE) into a more straightforward ordinary differential equation (ODE) that can be easily solved. Moreover, when it is not possible to find analytical solutions, numerical methods become essential for addressing nonlinear partial differential equations (PDEs). Various numerical techniques are frequently employed for this purpose, including the Galerkin finite element method (GFEM),

^{29}the finite difference method (FDM),

^{30–33}the compact finite-difference method,

^{34}and the adaptive mesh refinement method.

^{35–40}The adaptive mesh refinement method, in particular, stands out as a potent approach for efficiently tackling PDEs with a high degree of accuracy, making it especially valuable for simulating intricate and dynamic phenomena. For nonlinear PDEs, iteration techniques like the Newton–Raphson method, Picard iteration, or fixed-point iteration are commonly used to handle nonlinear terms. The specific method chosen depends on the problem’s nature, the desired accuracy, computational resources, and available software libraries. Additionally, it is essential to consider stability, convergence, and efficiency when selecting a numerical method for solving nonlinear PDEs. The nonlinear Schrödinger (NLS) equation plays a crucial role in investigating how waves travel in a wide range of physical systems, including areas like optics, fluid dynamics, and plasma physics. Specifically in optics, the NLSE explains how optical pulses move through nonlinear materials, like optical fibers and waveguides. When dealing with the coherent propagation of optical pulses, the NLSE becomes a fundamental concept, helping us understand phenomena such as solitons, self-phase modulation, and pulse compression. Now, let us focus on the second-order coupled nonlinear Schrödinger (C-NLS) equation

*x*,

*t*) and Φ(

*x*,

*t*) represent unfamiliar complex functions, and

*β*and

*α*characterize the dispersal within the optical fiber, while

*γ*denotes the parameter for self-phase modulation.

*u*

_{j}and

*v*

_{j}(with

*j*taking values from the set {1, 2}) represent real-valued functions,

This paper is organized as follows: In Sec. II, we discuss the improved, modified extended tanh-function method, which is employed to represent the exact solutions for traveling waves. We then utilize this method to derive solutions for system (1). Section III presents an analytical solution for system (5) through the generalized Tanh method. Section IV focuses on solving system (5) by applying the implicit finite difference method. The results and their discussion are presented in Secs. V and VI, which highlight the critical results uncovered in this research article.

## II. IMPROVED MODIFIED EXTENDED TANH-FUNCTION TECHNIQUE

*x*,

*t*), Φ = Φ(

*x*,

*t*).

^{41}can be expressed as follows:

*a*

_{j}, $a\u0302j$,

*b*

_{j}, and $b\u0302j$ will be determined later. To obtain the values of

*N*

_{1}and

*N*

_{2}, we need to find a homogeneous balance between the nonlinear term and the highest derivative. The function

*ϕ*(

*ξ*) solves the following Riccati differential equation:

*c*

_{l}for

*l*from the set {0, 1, 2, 3, 4}, each subject to specific limitations. Furthermore, we have $z\u0302$, which can take on values of either 1 or −1. We aim to ascertain the appropriate values for

*N*

_{1}and

*N*

_{2}to be utilized in Eq. (9). Subsequently, we can combine Eqs. (9) and (10) with Eq. (8) to formulate an algebraic equation. This process results in a system of equations where we can determine the values of

*a*

_{j}, $a\u0302j$,

*b*

_{j}, and $b\u0302j$ by gathering the coefficients associated with the same power of

*ϕ*(

*ξ*) and solving the system.

### A. Application of the method

*w*and

*β*as the following:

*u*″ and

*u*

^{3}in the first equation, and similarly, to

*v*″ and

*v*

^{3}in the second equation, we obtain the result that

*N*

_{1}and

*N*

_{2}both equal 1, which results in Eq. (9) adopting a particular form

*a*

_{j},

*b*

_{j}, $a\u03021$, $b\u03021$,

*γ*, and

*σ*, we can substitute Eqs. (14) and (10) into (13) and then group terms with the same order of

*ϕ*. This process transforms the left-hand side of (13) into a polynomial in

*ϕ*. By setting each coefficient of this resulting polynomial to zero, we derive a system of algebraic equations. These equations can be solved using Mathematica, yielding the following solutions for the given values of

*j*within the set $0,1$.

## III. GENERALIZED TANH METHOD

*ξ*, it fulfills the following ordinary differential equation:

*N*

_{l}for values of

*l*in the set {1, 2, 3, 4} by equating the highest derivative term to the nonlinear terms in Eq. (21). After substituting Eqs. (22) and (23) into Eq. (21) and then setting all coefficients of the same order of ϒ to zero, we can derive a system of algebraic equations. This system allows for explicitly determining the constants

*a*

_{0},

*a*

_{j},

*b*

_{0}, and

*b*

_{j}. By utilizing the general solution of Eq. (23), we can achieve this

### A. Application of the method

*l*values in the set {1, 2, 3, 4}, we have

*N*

_{l}= 1. Therefore, following Eq. (22), we assume that

## IV. NUMERICAL SOLUTION

*x*

_{l}to

*x*

_{r}. The range is divided into

*N*

_{x}smaller intervals, denoted as [

*x*

_{m},

*x*

_{m+1}], such that

_{x}, where Δ

_{x}is calculated as the difference between the right and left limits of the interval divided by the number of subintervals, represented by

*N*

_{x}.

*z*belong to $R2\xd72$, with I representing the identity matrix and

*z*representing the zero matrix, respectively,

_{m}instead of the analytical solution Γ(

*x*

_{m},

*t*) = Γ. For estimating the first and second derivatives in Eq. (31), apply the subsequent central difference formulas

*x*

_{m},

*t*

_{n}). The semi-discretization of Eq. (34) is derived as follows:

*et al.*’s work.

^{42}This solver employs implicit differentiation operators to estimate time derivatives, yielding numerically acceptable results. To assess the numerical scheme’s precision, we use the Taylor expansion. Upon applying and simplifying the expansion, we ascertain that the scheme exhibits second-order accuracy in both spatial and temporal dimensions, with an accuracy of $O(\Delta t2,\Delta x2)$. Subsequently, we evaluate the stability of the numerical method through a von Neumann stability analysis. It is important to note that this analysis is solely applicable to linear difference schemes and, therefore, we examine the linearized form of Eq. (35),

**M**can be given explicitly as

**M**, and they display as the following:

## V. RESULTS AND DISCUSSION

Using an improved variation of the modified, extended hyperbolic tangent function technique, we have successfully derived exact analytical solutions for the second-order coupled nonlinear Schrödinger (C-NLS) equation as described in Eq. (1). In Figs. 1 and 2, we can observe the exact solutions for the system represented as Ψ_{1}(*x*, *t*) and Φ_{1}(*x*, *t*), with Figs. 1(a) and 1(b) displaying the real and imaginary components, respectively. While examining the real and imaginary components of Φ_{1}(*x*, *t*) in Figs. 2(a) and 2(b), we consider specific parameter values: *α* = −4, *ϵ* = 1, *c*_{2} = −2, *c*_{4} = 1, $z\u0302=1$, *N*_{x} = 1000. Our analysis covers the range of *t* from 0 to 10 and *x* from −5 to 3. We transform the coupled Nonlinear Schrödinger (NLS) equation into a system of real and imaginary equations described in (5). We then explore the analytical solutions of this system using the generalized tanh method. In Figs. 3 and 4, we can see a 3D representation of the surface defined by Eq. (27), where the parameters have specific values: *α* = −1/8, *d*_{0} = −1/2, *N*_{x} = 1000, and the ranges for *t* and *x* are *t* = 0 → 10 and *x* = −10 → 10. Figure 5, on the other hand, illustrates how the analytical solutions described by Eq. (27) behave as we vary the parameter *α* while keeping the other parameters constant at *d*_{0} = −2 and *N*_{x} = 1000 and with *t* = 1 and *x* ranging from −10 to 10. Consequently, it becomes apparent that modifying the parameter *α* yields distinct outcomes for *u*_{1}(*x*, *t*) and *u*_{2}(*x*, *t*), whereas it produces similar effects for *v*_{1}(*x*, *t*) and *v*_{2}(*x*, *t*). We investigate numerical solutions by employing the implicit finite difference method to transform the fundamental problems into a system of ordinary differential equations (ODEs) while preserving continuous time derivatives. Figures 6 and 7 and Table I display the critical aspects of the solutions, facilitating a direct comparison between the traveling wave solutions obtained using the proposed exact methods with *N*_{x} = 2000 and the numerical solutions obtained using the finite difference method with various mesh numbers in the *x* direction. This comparison is feasible due to the presentation of Figs. 6 and 7 and Table I. The numerical results exhibit a significant level of comparability. As the value of Δ_{x} approaches zero, the mean error also approaches zero (Fig. 8. The numerical methods remain stable when the parameter values are configured as *α* = −0.125 and *d*_{0} = −0.5. This approach yields dependable and robust outcomes.

N_{x}
. | The relative error . | CPU (s) . |
---|---|---|

100 | 6.50 × 10^{−1} | 0.043 × 10^{3} |

200 | 1.30 × 10^{−2} | 0.16 × 10^{3} |

400 | 5.50 × 10^{−3} | 0.41 × 10^{3} |

800 | 2.20 × 10^{−4} | 0.91 × 10^{3} |

1000 | 4.20 × 10^{−5} | 2.81 × 10^{3} |

2000 | 1.32 × 10^{−5} | 5.50 × 10^{3} |

N_{x}
. | The relative error . | CPU (s) . |
---|---|---|

100 | 6.50 × 10^{−1} | 0.043 × 10^{3} |

200 | 1.30 × 10^{−2} | 0.16 × 10^{3} |

400 | 5.50 × 10^{−3} | 0.41 × 10^{3} |

800 | 2.20 × 10^{−4} | 0.91 × 10^{3} |

1000 | 4.20 × 10^{−5} | 2.81 × 10^{3} |

2000 | 1.32 × 10^{−5} | 5.50 × 10^{3} |

## VI. CONCLUSIONS

In this study, we have achieved exact solutions for propagating waves in the corresponding nonlinear Schrödinger Eq. (1) using an improved and adapted tanh-function approach. Furthermore, we have examined both exact and numerical solutions for system (5) by utilizing the generalized tanh method and an implicit finite difference approach, respectively. We have employed Matlab software to create 3D plots that accurately depict the results for specific parameter values. Additionally, in Fig. 5, we have included 2D plots to illustrate how the solution behaves as the parameter *α* varies while keeping other variables constant. The techniques employed in this research could potentially find applications in other nonlinear partial differential equations within the field of natural sciences.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Taghread Ghannam Alharbi**: Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). **Abdulghani Alharbi**: Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

^{∞}convergence of a difference scheme for coupled nonlinear Schrödinger equations