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.

Nonlinear PDEs play a vital role in characterizing a wide range of physical, biological, and engineering phenomena.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 GG-expansion method,6 the GG+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
ιΨt+ιβΨxαΨxx+|Ψ|2+γ|Φ|2Ψ=0,ιΦt+αΦxx+|Φ|2+γ|Ψ|2Φ=0,
(1)
where Ψ(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.
The initial conditions are as follows:
Ψ(x,0)=G1(x),Φ(x,0)=G2(x),
(2)
the following are the boundary conditions:
Ψx(x,t)=0,Φx(x,t)=0,atx=xl,xr.
(3)
We express the complex functions Ψ and Φ as a combination of their real and imaginary components,
Ψ(x,t)=u1(x,t)+ιu2(x,t),Φ(x,t)=v1(x,t)+ιv2(x,t).
(4)
Substituting Eq. (4) into system (1), we obtain the following system, where uj and vj (with j taking values from the set {1, 2}) represent real-valued functions,
u1t+βu1xα2u2x2+(u12+u22)+γ(v12+v22)u2=0,u2t+βu1x+α2u1x2(u12+u22)+γ(v12+v22)u1=0,v1t+α2v2x2+(v12+v22)+γ(u12+u22)v2=0,v2tα2v1x2(v12+v22)+γ(u12+u22)v1=0.
(5)
Coupled nonlinear Schrödinger equations offer a strong foundation for comprehending and simulating the characteristics of coherent optical pulses within nonlinear materials. These equations are crucial for exploring soliton dynamics, pulse compression, and interaction between different modes in diverse optical systems.

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.

We will introduce the improved and modified extended tanh-function technique and soliton solutions for nonlinear evolution equations (NLEEs) in this section. We assume that the given NLEEs are as follows:
Ω1Ψt,Ψx,Ψxx,Φt,Φx,Φxx,=0,
(6)
where Ψ = Ψ(x, t), Φ = Φ(x, t).
The representation of wave transformation can be expressed as
Ψ(x,t)=u(ξ)eι(xσt+ϵ),Φ(x,t)=v(ξ)eι(xσt+ϵ),ξ=xwt.
(7)
By replacing the equation given in (7) with the one in (6), we derive the subsequent set of ordinary differential equations (ODEs),
Ω2u,u,v,v,=0.
(8)
The form of the traveling wave solution for Eq. (8), as obtained using the improved, modified, extended tanh-function method introduced in this work,41 can be expressed as follows:
u(ξ)=j=0N1ajϕ(ξ)j+j=1N1âjϕ(ξ)j,v(ξ)=j=0N2bjϕ(ξ)j+j=1N2b̂jϕ(ξ)j.
(9)
The values of aj, âj, bj, and b̂j will be determined later. To obtain the values of N1 and N2, 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:
ϕ(ξ)=ẑc0+c1ϕ+c2ϕ2+c3ϕ3+c4ϕ4.
(10)
We possess a set of constants represented as cl for l from the set {0, 1, 2, 3, 4}, each subject to specific limitations. Furthermore, we have ẑ, which can take on values of either 1 or −1. We aim to ascertain the appropriate values for N1 and N2 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 aj, âj, bj, and b̂j by gathering the coefficients associated with the same power of ϕ(ξ) and solving the system.
We derive new solutions that propagate as traveling waves for the system described in Eq. (1) by inserting Eq. (7) into it. We then gather the coefficients and set them equal to zero for both the real and imaginary components, leading to the structure of certain relations:
2α+βwu=0,αu+(σβ+α)u++γuv2+u3=0,2αwv=0,αv+(σα)v++γvu2+v3=0.
(11)
We acquire w and β as the following:
w=2α,β=4α.
(12)
When Eq. (12) is employed, Eq. (11) transforms into
αu+(σ3α)u+γuv2+u3=0,αv+(σα)v+γvu2+v3=0.
(13)
When we apply the balancing technique to equate the terms involving u″ and u3 in the first equation, and similarly, to v″ and v3 in the second equation, we obtain the result that N1 and N2 both equal 1, which results in Eq. (9) adopting a particular form
u=a0+a1ϕ+â1ϕ1,v=b0+b1ϕ+b̂1ϕ1.
(14)
To determine the values of aj, bj, â1, b̂1, 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.
Ifc1=c3=0,c0=c224c4, then
a1=12±sc0+4αc4ẑ2,b1=±12±sc04αc4ẑ2,â1=±s±sc0+4αc4ẑ22αc4c2ẑ2+1,b̂1=ss4αc0c4ẑ2c02αc4c2ẑ2+1,σ=2α+αc2ẑ2sẑ2c2ẑ2+1,s=c0c4α+αc2ẑ22,κ=α2c22c2+12,γ=1,a0=b0=0.
Hence, the expressions for the hyperbolic function solutions of system (1) can be represented as follows:
Ψ1(x,t)=c2c4c42αc22+κc22tanhc2(x2αt)2αc22+αc2κcoth2c2(x2αt)22αc2c2+1×expιc2(2αt+x+ϵ)2αt+κt+x+ϵc2+1,
(15)
Φ1(x,t)=c2c4c4κ2αc22c22tanhc2(x2αt)2αc22+αc2+κcoth2c2(x2αt)22αc2c2+1×expιc2(2αt+x+ϵ)2αt+κt+x+ϵc2+1.
(16)
Subsequently, the solutions in terms of the trigonometric functions for system (1) can be expressed as follows:
Ψ2(x,t)=c2c4c42αc22+κc22tanc2(x2αt)2αc22+αc2±κcot2c2(x2αt)22αc2c2+1×expιc2(2αt+x+ϵ)2αt+κt+x+ϵc2+1,
(17)
Φ2(x,t)=±c2c4c4κ2αc22c22tanc2(x2αt)2αc22+αc2κcot2c2(x2αt)22αc2c2+1×expιc2(2αt+x+ϵ)2αt+κt+x+ϵc2+1.
(18)
We are providing a basic overview of the generalized tanh method, which involves outlining the key steps. Let us examine the general structure of nonlinear partial differential equations (NPDEs), which can be expressed as follows:
Ω1u1,t,u1,x,u1,xx,u2,t,u2,x,u2,xx,v1,t,v1,x,×v1,xx,v2,t,v2,x,v2,xx,=0,
(19)
where uj=uj(x,t),vj=vj(x,t),j=1,2. Next, we utilize the following wave transformation:
u1=U1(ξ),u2=U2(ξ),v1=V1(ξ),v2=V2(ξ),ξ=xwt.
(20)
By inserting Eq. (20) into Eq. (19), we derive the subsequent set of ordinary differential equations (ODEs)
Ω2U1,U1,U2,U2,V1,V1,V2,V2,=0.
(21)
We propose the subsequent series expansion as a solution for Eq. (21),
U1=a0+j=1N1ajϒj,U2=c0+j=1N2cjϒj,V1=b0+j=1N3bjϒj,V2=d0+j=1N4djϒj.
(22)
Given that ϒ is a function of ξ, it fulfills the following ordinary differential equation:
ϒ=1+ϒ2.
(23)
We can find the positive integers Nl 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 a0, aj, b0, and bj. By utilizing the general solution of Eq. (23), we can achieve this
ϒ=tanhξ.
(24)
To discover the solitary wave solutions of system (5), we employ the wave conversion (20). Consequently, system (5) can be transformed into the subsequent ordinary differential equations (ODEs)
wU1+βU1αU2+U12+U22+γV12+V22U2=0,wU2+βU2+αU1+U12+U22+γV12+V22U1=0,wV1+αV2+V12+V22+γU12+U22V2=0,wV2αV1+V12+V22+γU12+U22V1=0.
(25)
When we consider the system of Eq. (25) and balance the terms involving the highest derivative and nonlinearity for each equation, we find that for all l values in the set {1, 2, 3, 4}, we have Nl = 1. Therefore, following Eq. (22), we assume that
U1=a0+a1Φ,U2=c0+c1Φ,V1=b0+b1Φ,V2=d0+d1Φ.
(26)
By combining Eq. (26) and its corresponding derivatives into system (25) and utilizing Eq. (24), along with the application of trigonometric identities, we can gather the coefficients of similar powers of Φ. We then equate these coefficients, containing independent combinations, to zero. This process yields the following cases and their respective solutions:
Family1:a0=d022α,b1=d022α,c0=d022α,d1=d022α,w=±2αd0d022α,β=±4αd02αd0d022α,γ=1,a1=±d0,c1=d0,b0=d0.
Hence, the newly found exact solutions for the (C-NLS) can be recognized as
u1(x,t)=d022αd0tanhx2αd0td022α,u2(x,t)=d022αd0tanhx2αd0td022α,v1(x,t)=±d022αtanhx2αd0td022αd0,v2(x,t)=±d022αtanhx2αd0td022α+d0.
(27)
Therefore,
Ψ3(x,t)=d022αd0tanhx2αd0td022α+ιd022αd0tanhx2αd0td022α,Φ3(x,t)=d0±d022αtanhx2αd0td022α+ιd0±d022αtanhx2αd0td022α,
(28)
Family2:a0=2α,a1=±2α,b0=2α,b1=±2α,c0=2α,c1=2α,d0=2α,d1=2α,w=±2α,β=±4α,γ=1.
Using Family 2, we can specify exact solutions for the (C-NLS) equation
u1(x,t)=2αtanhx2αt1,u2(x,t)=±2αtanhx2αt±1,v1(x,t)=2αtanhx2αt1,v2(x,t)=±2αtanhx2αt±1.
(29)
Hence,
Ψ4(x,t)=2αtanhx2αt1±2αιtanhx2αt±1,Φ4(x,t)=2αtanhx2αt1±2αιtanhx2αt±1.
(30)
We utilize the implicit finite-difference technique to obtain the numerical outcomes of system (5) within a physical range of values from xl to xr. The range is divided into Nx smaller intervals, denoted as [xm, xm+1], such that
xm=(m1)Δx,xm[xl,xr],m=1,2,,Nx+1.
The implicit finite difference method is a numerical approach to solving partial differential equations (PDEs) by breaking them down into discrete intervals in time and space. In contrast to explicit methods, which depend on information from the present time step to compute the future solution, implicit methods consider information from the current and future time steps. This characteristic grants them unconditional stability and makes them well-suited for addressing challenging PDEs, where explicit methods might not be feasible due to their stringent stability requirements. This technique ensures stability by employing a uniform subinterval width denoted as Δx, where Δx is calculated as the difference between the right and left limits of the interval divided by the number of subintervals, represented by Nx.
Assuming that the system described in Eq. (5) can be represented using matrices and vectors, it can be expressed as follows: uT=u1u2,vT=v1v2, then
Γt+βAΓxαB2Γx2+GΓΓ=0,
(31)
where ΓT=uv,A=Izzz,k1=uTu+γvTv,k2=vTv+γuTu. When I and z belong to R2×2, with I representing the identity matrix and z representing the zero matrix, respectively,
B=0110zz0110,GΓ=0k1k10zz0k2k20.
Next, employ the approximate solution Γm instead of the analytical solution Γ(xm, t) = Γ. For estimating the first and second derivatives in Eq. (31), apply the subsequent central difference formulas
Γx=12ΔxΓm+1Γm1=12ΔxδxΓm,2Γx2=1Δx2Γm+12Γm+Γm1=1Δx2δx2Γm.
(32)
When we incorporate Eq. (32) into Eq. (31), we derive the semi-discrete system
Γt|m+β2ΔxAδxΓmαΔx2Bδx2Γm+GΓmΓm=0.
(33)
Eq. (33) can be represented in a vectorized form in the following manner:
Γt|m+FΓm=0,
(34)
where FΓm=β2ΔxAδxΓmαΔx2Bδx2Γm+GΓmΓm.
We utilize the implicit midpoint rule to address the system represented by Eq. (34). In this context, Γmn signifies the discrete approximation of the exact solution Γ(xm, tn). The semi-discretization of Eq. (34) is derived as follows:
Γt|mn+FΓmn+1+Γmn2=0.
(35)
We addressed the system as mentioned earlier by employing the DDASPK solver in FORTRAN, as outlined in Brown 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(Δt2,Δx2). 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),
Γmn+1Γmn+ΔtQΓmn+1+Γmn2,
(36)
where
QΓmn=β2ΔxAδxΓmnαΔx2Bδx2Γmn+ÃΓmn,
Ã=0k̃1k̃10zz0k̃2k̃20,andk̃1=max{k1},k̃2=max{k2}.
To assess the stability of the difference scheme, we assume that
Γmn=MneιωmΔx.
(37)
To ensure stability when employing the von Neumann method, the following requirements must be met:
maxj|μj|1,wherej=1,2,3,4.
Upon inserting Eq. (37) into Eq. (36), we derive the amplification matrix through a calculation, and this can be represented as a matrix equation,
MI+ιϑ1A+ϑ2B+Δt2Ã=Iιϑ1Aϑ2BΔt2Ã,
(38)
where ϑ1=βΔt2Δx,ϑ2=2αΔtΔx2sin2ωΔx2.
The matrix M can be given explicitly as
M=I+ιϑ1A+ϑ2B+Δt2Ã1Iιϑ1Aϑ2BΔt2Ã.
(39)
We have computed the eigenvalues of the matrix M, and they display as the following:
μ1=ϑ4ϑ2+ιϑ4ϑ2+ι,μ2=ϑ4ϑ2+ιϑ4ϑ2+ι,μ3=ϑ2+ϑ3+ϑ1+ιϑ2+ϑ3+ϑ1ι,μ4=ϑ2+ϑ3+ϑ1+ιϑ2+ϑ3ϑ1+ι,
(40)
where
ϑ3=Δt2k̃1,ϑ4=Δt2k̃2.
The modulus of these eigenvalues is equal to 1
|μj|=1,j=1,2,3,4.
(41)
The von Neumann analysis confirms that the stability requirement is met, ensuring the scheme remains unconditionally stable.

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 Ψ1(x, t) and Φ1(x, t), with Figs. 1(a) and 1(b) displaying the real and imaginary components, respectively. While examining the real and imaginary components of Φ1(x, t) in Figs. 2(a) and 2(b), we consider specific parameter values: α = −4, ϵ = 1, c2 = −2, c4 = 1, ẑ=1, Nx = 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, d0 = −1/2, Nx = 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 d0 = −2 and Nx = 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 u1(x, t) and u2(x, t), whereas it produces similar effects for v1(x, t) and v2(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 Nx = 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 Δx 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 d0 = −0.5. This approach yields dependable and robust outcomes.

FIG. 1.

(a) and (b) display representations of real and imaginary traveling wave solutions for Ψ1(x, t). The specified parameter values include α = −4, ϵ = 1, c2 = −2, c4 = 1, ẑ=1, and Nx = 1000, with the domain of the time t = 0–10 and x ranging from −5 to 3.

FIG. 1.

(a) and (b) display representations of real and imaginary traveling wave solutions for Ψ1(x, t). The specified parameter values include α = −4, ϵ = 1, c2 = −2, c4 = 1, ẑ=1, and Nx = 1000, with the domain of the time t = 0–10 and x ranging from −5 to 3.

Close modal
FIG. 2.

3D surfaces representing both real and imaginary traveling wave solutions for Φ1(x, t) are depicted in (a) and (b). The specified parameters are α = −4, ϵ = 1, c2 = −2, c4 = 1, ẑ=1, and Nx = 1000, with the range t varying from 0 to 10 and x ranging from −5 to 3.

FIG. 2.

3D surfaces representing both real and imaginary traveling wave solutions for Φ1(x, t) are depicted in (a) and (b). The specified parameters are α = −4, ϵ = 1, c2 = −2, c4 = 1, ẑ=1, and Nx = 1000, with the range t varying from 0 to 10 and x ranging from −5 to 3.

Close modal
FIG. 3.

3D surface show of the exact solutions in Eq. (27) is depicted in the following manner: In Figure (a), the exact solution for u1(x, t) is displayed, and in Figure (b), the exact solution for u2(x, t) is presented. The parameters employed for this analysis include α = −1/8, d0 = −1/2, and Nx = 1000, with a range of t spanning from 0 to 10 and x ranging from −10 to 10.

FIG. 3.

3D surface show of the exact solutions in Eq. (27) is depicted in the following manner: In Figure (a), the exact solution for u1(x, t) is displayed, and in Figure (b), the exact solution for u2(x, t) is presented. The parameters employed for this analysis include α = −1/8, d0 = −1/2, and Nx = 1000, with a range of t spanning from 0 to 10 and x ranging from −10 to 10.

Close modal
FIG. 4.

3D figures illustrate the exact solutions in Eq. (27). Graphs (a) and (b) depict 3D representations of the exact solutions for v1(x, t) and v2(x, t), correspondingly. The parameters utilized are α = −1/8, d0 = −1/2, and Nx = 1000, with t ranging from 0 to 10 and x ranging from −10 to 10.

FIG. 4.

3D figures illustrate the exact solutions in Eq. (27). Graphs (a) and (b) depict 3D representations of the exact solutions for v1(x, t) and v2(x, t), correspondingly. The parameters utilized are α = −1/8, d0 = −1/2, and Nx = 1000, with t ranging from 0 to 10 and x ranging from −10 to 10.

Close modal
FIG. 5.

These illustrations depict how altering a parameter in Eq. (27) affects wave behavior while keeping the other parameters constant. In figures (a) and (b), we can observe the exact solutions for u1(x, t) and u2(x, t), respectively, whereas in figures (c) and (d), we can observe the exact solutions for v1(x, t) and v2(x, t). These figures illustrate how the behavior changes as α increases, with γ held constant at −1.

FIG. 5.

These illustrations depict how altering a parameter in Eq. (27) affects wave behavior while keeping the other parameters constant. In figures (a) and (b), we can observe the exact solutions for u1(x, t) and u2(x, t), respectively, whereas in figures (c) and (d), we can observe the exact solutions for v1(x, t) and v2(x, t). These figures illustrate how the behavior changes as α increases, with γ held constant at −1.

Close modal
FIG. 6.

In (a) and (c), we see depictions of the real and imaginary aspects of Ψ3(x, t)’s traveling wave solutions. (b) and (d) show numerical solutions for the real and imaginary parts of Ψ3(x, t). The parameter values used were α = −0.125, d0 = −0.5, and Nx = 1000 with t = 0 → 10 and x = −40 → 40.

FIG. 6.

In (a) and (c), we see depictions of the real and imaginary aspects of Ψ3(x, t)’s traveling wave solutions. (b) and (d) show numerical solutions for the real and imaginary parts of Ψ3(x, t). The parameter values used were α = −0.125, d0 = −0.5, and Nx = 1000 with t = 0 → 10 and x = −40 → 40.

Close modal
FIG. 7.

Figures (a) and (c) display representations of the real and imaginary components of the traveling wave solutions for Ψ3(x, t). Meanwhile, figures (b) and (d) depict computed solutions for the real and imaginary segments of Ψ3(x, t). These solutions were obtained using specific parameter values: α = −0.125, d0 = −0.5, and Nx = 1000, while varying the time from t = 0–10 and the position from x = −40 to 40.

FIG. 7.

Figures (a) and (c) display representations of the real and imaginary components of the traveling wave solutions for Ψ3(x, t). Meanwhile, figures (b) and (d) depict computed solutions for the real and imaginary segments of Ψ3(x, t). These solutions were obtained using specific parameter values: α = −0.125, d0 = −0.5, and Nx = 1000, while varying the time from t = 0–10 and the position from x = −40 to 40.

Close modal
TABLE I.

The relative error with L2 norm and CPU, at the time t = 10.

NxThe relative errorCPU (s)
100 6.50 × 10−1 0.043 × 103 
200 1.30 × 10−2 0.16 × 103 
400 5.50 × 10−3 0.41 × 103 
800 2.20 × 10−4 0.91 × 103 
1000 4.20 × 10−5 2.81 × 103 
2000 1.32 × 10−5 5.50 × 103 
NxThe relative errorCPU (s)
100 6.50 × 10−1 0.043 × 103 
200 1.30 × 10−2 0.16 × 103 
400 5.50 × 10−3 0.41 × 103 
800 2.20 × 10−4 0.91 × 103 
1000 4.20 × 10−5 2.81 × 103 
2000 1.32 × 10−5 5.50 × 103 
FIG. 8.

Utilizing the L2 norm relative error from Table I, we approximated the convergence data by varying Nx, specifically selecting t = 10 and x ranging from −40 to 40.

FIG. 8.

Utilizing the L2 norm relative error from Table I, we approximated the convergence data by varying Nx, specifically selecting t = 10 and x ranging from −40 to 40.

Close modal

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.

The authors have no conflicts to disclose.

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).

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

1.
T.
Li
,
S.
Frassu
, and
G.
Viglialoro
, “
Combining effects ensuring boundedness in an attraction–repulsion chemotaxis model with production and consumption
,”
Z. Angew. Math. Phys.
74
(
3
),
109
(
2023
).
2.
T.
Li
,
N.
Pintus
, and
G.
Viglialoro
, “
Properties of solutions to porous medium problems with different sources and boundary conditions
,”
Z. Angew. Math. Phys.
70
,
86
(
2019
).
3.
S.
Malik
,
M. S.
Hashemi
,
S.
Kumar
,
H.
Rezazadeh
,
W.
Mahmoud
, and
M. S.
Osman
, “
Application of new Kudryashov method to various nonlinear partial differential equations
,”
Opt. Quantum Electron.
55
(
1
),
8
(
2023
).
4.
S.
Kumar
,
K. S.
Nisar
, and
M.
Niwas
, “
On the dynamics of exact solutions to a (3 + 1)-dimensional YTSF equation emerging in shallow sea waves: Lie symmetry analysis and generalized Kudryashov method
,”
Res. Phys.
48
,
106432
(
2023
).
5.
M. B.
Almatrafi
and
A.
Alharbi
, “
New soliton wave solutions to a nonlinear equation arising in plasma physics
,”
Comput. Model. Eng. Sci.
137
(
1
),
827
(
2023
).
6.
M.
Djilali
and
H.
Ali
, “
(g′/g)-expansion method to seek traveling wave solutions for some fractional nonlinear PDES arising in natural sciences
,”
Adv. Theory Nonlinear Anal. Appl.
7
(
2
),
303
318
(
2023
).
7.
S.
Khaliq
,
S.
Ahmad
,
A.
Ullah
,
H.
Ahmad
,
S.
Saifullah
, and
T. A.
Nofal
, “
New waves solutions of the (2 + 1)-dimensional generalized Hirota–Satsuma–Ito equation using a novel expansion method
,”
Res. Phys.
50
,
106450
(
2023
).
8.
Y.
Qiu
,
B.
Tian
,
D.
Xian
, and
L.
Xian
, “
New exact solutions of nontraveling wave and local excitation of dynamic behavior for GGKDV equation
,”
Res. Phys.
49
,
106463
(
2023
).
9.
A. R.
Alharbi
and
M. B.
Almatrafi
, “
Exact solitary wave and numerical solutions for geophysical KdV equation
,”
J. King Saud Univ.-Sci.
34
(
6
),
102087
(
2022
).
10.
C.
Dai
and
J.
Zhang
, “
Jacobian elliptic function method for nonlinear differential-difference equations
,”
Chaos, Solitons Fractals
27
(
4
),
1042
1047
(
2006
).
11.
C.-C.
Wei
,
B.
Tian
,
D.-Y.
Yang
, and
S.-H.
Liu
, “
Jacobian-elliptic-function and rogue-periodic-wave solutions of a high-order nonlinear Schrödinger equation in an inhomogeneous optical fiber
,”
Chin. J. Phys.
81
,
354
361
(
2023
).
12.
S. A.
Khuri
, “
A complex tanh-function method applied to nonlinear equations of Schrödinger type
,”
Chaos, Solitons Fractals
20
(
5
),
1037
1040
(
2004
).
13.
A.
Alharbi
, “
Traveling-wave and numerical solutions to a Novikov-Veselov system via the modified mathematical methods
,”
AIMS Math.
8
,
1230
1250
(
2023
).
14.
A. R.
Alharbi
,
M. B.
Almatrafi
, and
M. A. E.
Abdelrahman
, “
Analytical and numerical investigation for Kadomtsev–Petviashvili equation arising in plasma physics
,”
Phys. Scr.
95
(
4
),
045215
(
2020
).
15.
M. A. E.
Abdelrahman
,
M. B.
Almatrafi
, and
A.
Alharbi
, “
Fundamental solutions for the coupled KdV system and its stability
,”
Symmetry
12
(
3
),
429
(
2020
).
16.
A.
Alharbi
,
M. B.
Almatrafi
, and
M. A. E.
Abdelrahman
, “
Constructions of the travelling wave solutions to the MRLW equation and their stability and accuracy arising in plasma physics
,”
Int. J. Appl. Comput. Math.
9
(
3
),
46
(
2023
).
17.
A. R.
Alharbi
, “
A study of traveling wave structures and numerical investigation of two-dimensional Riemann problems with their stability and accuracy
,”
Comput. Model. Eng. Sci.
134
(
3
),
2193
(
2023
).
18.
M. B.
Almatrafi
,
A.
Alharbi
,
K. H.
Lotfy
, and
A. A.
El-Bary
, “
Exact and numerical solutions for the GBBM equation using an adaptive moving mesh method
,”
Alexandria Eng. J.
60
(
5
),
4441
4450
(
2021
).
19.
M. A. E.
Abdelrahman
and
A.
Alharbi
, “
Analytical and numerical investigations of the modified Camassa–Holm equation
,”
Pramana
95
,
117
(
2021
).
20.
M. B.
Almatrafi
,
A. R.
Alharbi
, and
A. R.
Seadawy
, “
Structure of analytical and numerical wave solutions for the ito integro-differential equation arising in shallow water waves
,”
J. King Saud Univ.-Sci.
33
(
3
),
101375
(
2021
).
21.
M. B.
Almatrafi
,
A. R.
Alharbi
, and
M. A. E.
Abdelrahman
, “
New exact and numerical solutions for the KdV system arising in physical applications
,”
Arab J. Basic Appl. Sci.
28
(
1
),
113
121
(
2021
).
22.
A. R.
Alharbi
,
M. B.
Almatrafi
, and
A. R.
Seadawy
, “
Construction of the numerical and analytical wave solutions of the Joseph–Egri dynamical equation for the long waves in nonlinear dispersive systems
,”
Int. J. Mod. Phys. B
34
(
30
),
2050289
(
2020
).
23.
M. B.
Almatrafi
,
A. R.
Alharbi
, and
C.
Tunç
, “
Constructions of the soliton solutions to the good Boussinesq equation
,”
Adv. Differ. Equations
2020
(
1
),
629
.
24.
A. R.
Alharbi
,
M. B.
Almatrafi
, and
K. H.
Lotfy
, “
Constructions of solitary travelling wave solutions for ito integro-differential equation arising in plasma physics
,”
Res. Phys.
19
,
103533
(
2020
).
25.
A.
Alharbi
and
M. B.
Almatrafi
, “
Exact and numerical solitary wave structures to the variant Boussinesq system
,”
Symmetry
12
(
9
),
1473
(
2020
).
26.
K. H.
Lotfy
,
A. A.
El-Bary
,
W.
Hassan
,
A. R.
Alharbi
, and
M. B.
Almatrafi
, “
Electromagnetic and Thomson effects during photothermal transport process of a rotator semiconductor medium under hydrostatic initial stress
,”
Res. Phys.
16
,
102983
(
2020
).
27.
A. R.
Alharbi
,
M. B.
Almatrafi
, and
M. A. E.
Abdelrahman
, “
The extended Jacobian elliptic function expansion approach to the generalized fifth order KdV equation
,”
J Phys. Math
10
(
4
),
310
(
2019
).
28.
M. A. E.
Abdelrahman
,
M. A.
Sohaly
, and
A.
Alharbi
, “
The new exact solutions for the deterministic and stochastic (2 + 1)-dimensional equations in natural sciences
,”
J. Taibah Univ. Sci.
13
(
1
),
834
843
(
2019
).
29.
H.
Ali
and
M. D.
Kamrujjaman
, “
Numerical solutions of nonlinear parabolic equations with robin condition: Galerkin approach
,”
TWMS Journal of Applied and Engineering Mathematics
12
,
851
(
2022
).
30.
Z.
Sun
and
D.
Zhao
, “
On the L convergence of a difference scheme for coupled nonlinear Schrödinger equations
,”
Comput. Math. Appl.
59
(
10
),
3286
3300
(
2010
).
31.
M. S.
Ismail
and
T. R.
Taha
, “
Numerical simulation of coupled nonlinear Schrödinger equation
,”
Math. Comput. Simul.
56
(
6
),
547
562
(
2001
).
32.
A.
Yokuş
and
D.
Kaya
, “
Comparison exact and numerical simulation of the traveling wave solution in nonlinear dynamics
,”
Int. J. Mod. Phys. B
34
(
29
),
2050282
(
2020
).
33.
S. O.
Abdulla
,
S. T.
Abdulazeez
, and
M.
Modanli
, “
Comparison of third-order fractional partial differential equation based on the fractional operators using the explicit finite difference method
,”
Alexandria Eng. J.
70
,
37
44
(
2023
).
34.
H.
Ahmad
,
T. A.
Khan
,
P. S.
Stanimirovic
,
W.
Shatanawi
, and
T.
Botmart
, “
New approach on conventional solutions to nonlinear partial differential equations describing physical phenomena
,”
Res. Phys.
41
,
105936
(
2022
).
35.
K. L.
DiPietro
and
A. E.
Lindsay
, “
Monge–ampére simulation of fourth order PDES in two dimensions with application to elastic–electrostatic contact problems
,”
J. Comput. Phys.
349
,
328
350
(
2017
).
36.
A. R.
Alharbi
, “
Numerical solutions to two-dimensional fourth order parabolic thin film equations using the parabolic monge-ampere method
,”
AIMS Math.
8
(
7
),
16463
16478
(
2023
).
37.
C. J.
Budd
and
J. F.
Williams
, “
Moving mesh generation using the parabolic Monge–Ampère equation
,”
SIAM J. Sci. Comput.
31
(
5
),
3438
3465
(
2009
).
38.
A. R.
Alharbi
,
A.-M.
Al-Munawarah
, and
S.
Arabia
, “
Numerical investigation for the GRLW equation using parabolic monge ampere equation
,”
Comput. Sci.
15
,
443
462
(
2020
).
39.
A. R.
Alharbi
et al
, “
Numerical solution of thin-film flow equations using adaptive moving mesh methods
,” Ph.D. thesis,
Keele University
,
2016
.
40.
A.
Alharbi
and
S.
Naire
, “
An adaptive moving mesh method for thin film flow equations with surface tension
,”
J. Comput. Appl. Math.
319
,
365
384
(
2017
).
41.
M. B.
Almatrafi
, “
Solitary wave solutions to a fractional model using the improved modified extended tanh-function method
,”
Fractal Fractional
7
(
3
),
252
(
2023
).
42.
P. N.
Brown
,
A. C.
Hindmarsh
, and
L. R.
Petzold
, “
Using Krylov methods in the solution of large-scale differential-algebraic systems
,”
SIAM J. Sci. Comput.
15
(
6
),
1467
1488
(
1994
).