A reliable analytic technique and physical interpretation for the two-dimensional nonlinear Schrödinger equations

Mahmoud A. E. Abdelrahman; H. S. Alayachi; Mahmoud A. E. Abdelrahman; H. S. Alayachi

doi:10.3934/math.20241185

AIMS Mathematics

2024, Volume 9, Issue 9: 24359-24371. doi: 10.3934/math.20241185

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Research article Special Issues

A reliable analytic technique and physical interpretation for the two-dimensional nonlinear Schrödinger equations

Mahmoud A. E. Abdelrahman ^{1,2
,
,},
H. S. Alayachi ¹

1.
Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

Received: 26 April 2024 Revised: 30 June 2024 Accepted: 08 July 2024 Published: 19 August 2024
MSC : 35C08, 35Q40, 35Q55

Nonlinear Schrödinger equations are a key paradigm in nonlinear research, attracting both mathematical and physical attention. This work was primarily concerned with the usage of a reliable analytic technique in order to solve two models of (2+1)-dimensional nonlinear Schrödinger equations. By applying a comprehensible wave transformation, every nonlinear model was simplified to an ordinary differential equation. A number of critical solutions were observed that correlated to various parameters. The provided approach has various advantages, including reducing difficult computations and succinctly presenting key results. Some 2D and 3D graphical representations regarding presented solitons were considered for the appropriate values of the parameters. We also showed the effect of the physical parameters on the dynamical behavior of the presented solutions. Finally, the proposed approach may be expanded to tackle increasingly complicated problems in applied science.

Keywords:

two-dimensional nonlinear Schrödinger equations,
closed-form solutions,
solitons

Citation: Mahmoud A. E. Abdelrahman, H. S. Alayachi. A reliable analytic technique and physical interpretation for the two-dimensional nonlinear Schrödinger equations[J]. AIMS Mathematics, 2024, 9(9): 24359-24371. doi: 10.3934/math.20241185

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Abstract

1. Introduction

Plasma physics is one of the most attractive branches of science where many scientists have been focusing their attention on discovering more properties of this field ^[2]. Plasma or cytoplasm is a distinct state of matter that can be described as an ionized gas in which the electrons are free and are not bound to an atom or a molecule ^[3]. If the substance is present in nature in three states: solid, liquid, and gas, then plasma can be classified as the fourth state in which the substance can exist ^[4]. Recently, investigating the heavy Langmuir turbulence's characterization becomes a very important tool for providing a good opportunity to overcome the Langmuir condensation problem ^[5,6]. Moreover, this investigation aims to raise the amount of long-wave disturbances through the condensation paradox in Langmuir ^[7]. At that condensation, the radiation can not dampen the vibration where at severe periods, the coulomb relation is unable to dampen the variations in the pulses because of their frequency ^[8]. Recently, these radiations with its distinct variations and interactions have been mathematically formulated by some nonlinear evolution equations such as KGZ model ^[9,10,11,12].

The ability of nonlinear partial differential equations with integer or fractional order for formulating different complicated phenomena in various fields including genetics, engineering, quantum mechanics, electro chemistry, chemistry, mechanical engineering, biology, mechanics, etc, makes it the ideal and direct way for discovering the indiscoverable properties of these phenomena ^{[13,14,15,16,17,18,19]}. Thus, many mathematicians and physics pay complete attention to derive computational, semi-analytical, numerical techniques for solving these equations such as the Adomian decomposition method, Elkalla expansion method, B-spline schemes, extended simplest equation method, modified Khater method, generalized Khater method, exponential expansion method, auxiliary equation method, direct algebraic expansion method, and so no ^{[20,21,22,23,24,25,26,27,28]}. These methods have been employed on several models but until now, there is no unified method can be applied to all the nonlinear evolution equation ^{[29,30,31,32]}.

In this context, this paper investigate the numerical solutions of the nonlinear KGZ model. This model is formulates as follows ^[33,34,35]:

$\begin{eqnarray} \left\lbrace \begin{array}{c} \mathcal{G}_{t\,t}-\mathcal{G}_{x\,x}+\mathcal{G}+\upsilon_{0}\,\mathcal{Q}\,\mathcal{G} = 0, \\ \\ \mathcal{Q}_{t\,t}-\mathcal{Q}_{x\,x}-\upsilon_{1}\, \left( |\mathcal{G}|^{2} \right)_{x\,x},\; \; = 0, \end{array}\right. \end{eqnarray}$

(1.1)

where $\upsilon_{0}, \, \upsilon_{1}$ are nonzero real parameters describing the consistency of the initial data of the KGZ system while $\mathcal{Q} = \mathcal{Q}(x, t), \, \mathcal{G} = \mathcal{G}(x, t)$ are receptively real and complex functions which represent the fast time scale component of the electric field raised by electrons and the derivation of ion density from its equilibrium. Eq (1.1) describes the interaction and contact between the Langmuir wave and the acoustic wave of the ions in a high frequency plasma. ^[1] have employed the generalized Khater method to Eq (1.1) and converted it into the following ordinary differential equation with the following initial and boundary conditions

$\begin{eqnarray} \left\lbrace \begin{array}{c} \mathfrak{P}''+\mathcal{L}_{1}\,\mathfrak{P}+\mathcal{L}_{2}\,\mathfrak{P}^{3} = 0,\; \; \; \; , \\ \\ \mathfrak{P}(0) = \mathcal{F}(\mathfrak{Z}),\, \mathfrak{P}_{\mathfrak{Z}}(0) = \mathcal{E}(\mathfrak{Z}). \end{array} \right. \end{eqnarray}$

(1.2)

The generalized Khater method have been constructed the values of $\mathcal{F}, \, \mathcal{E}$ under the following value of the above-mentioned parameters $\mathcal{L}_{0} = \frac{1}{2}, \, \mathcal{L}_{1} = -\frac{225}{32}$ as follows ^[1]:

$\begin{eqnarray} \left\lbrace\begin{array}{c} \mathcal{F}(\mathfrak{Z}) = \frac{1}{15} (-4) \tanh \left(\frac{\mathfrak{Z}}{2}\right), \\ \\ \mathcal{E}(\mathfrak{Z}) = -\frac{2}{15}.\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; . \end{array} \right. \end{eqnarray}$

(1.3)

This model can be used to calculate from Euler's equations for electrons and ions, with Maxwell's electromagnetic field law for ions, by disregarding the influence of magnetic fields ^[36,37]. The nonlinear KGZ model has numerically studied through some recent approximate schemes such as a finite difference method ^[38] where Chunmei Su and Wenfan Yi have investigated the numerical solutions and established the error estimates of a conservative finite difference method for the considered model with a dimensionless parameter $0 < \varepsilon \ll 1,$ which is inversely proportional to the speed of sound. While ^[39] has compared the obtained numerical solutions that have been obtained through applying Finite difference time domain (FDTD) methods, Exponential wave integrator (EWI) and Time-splitting (TS) method, Uniformly and optimally accurate (UOA) methods and Uniformly accurate (UA) methods that have been applied in ^[40,41] of the same model that give a precision, computational sophistication, and other properties are also addressed. ^[15] has employed the well-known Chebyshev Cardinal Functions for investigating the numerical solutions of the nonlinear KGZ model where operational matrices of derivatives have been used to convert partial differential equations into nonlinear algebraic equations. ^[16] has used a new conservative finite difference scheme with a parameter $\theta$ has been employed for obtaining the numerical solutions of the considered model. Moreover, Convergence of the numerical solutions has been investigated. For further information of the numerical solutions of the nonlinear KGZ model, you can see ^[17,18].

The rest sections in this manuscript is organized as follows; Section 2 applies the above-mentioned numerical schemes to the nonlinear KGZ equation for estimating the numerical solutions. Section 3 discusses the obtained numerical solutions. Section 4 gives the conclusion of the whole paper.

2. Numerical investigation

Here, we give the headline of the used methods they we give the obtained results along with these approximate schemes

2.1. Methodology

2.1.1. Trigonometric Quintic B-spline scheme

Using the trigonometric Quintic B-spline scheme supposes the solutions of Eq (1.2) is formulated as following

$\begin{eqnarray} \mathfrak{P}(\mathfrak{Z}) = \sum\limits_{j = -2}^{r+2} \mathcal{C}_{j} \, \mathcal{G}_{j}(\mathfrak{Z}), \; \; \; j = (0,1,\cdots,r), \end{eqnarray}$

(2.1)

where $\mathcal{C}_{j}$ are be determined form the collocation points $\mathfrak{Z}_{j}$ and $\mathcal{G}_{j}(\mathfrak{Z})$ satisfies the following values

$\begin{eqnarray} \mathcal{G}_{j}(\mathfrak{Z}) = \left\lbrace \begin{array}{cc} \psi^{5}(\mathfrak{Z}_{j-3}),& \; \; \; \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j-3},\mathfrak{Z}_{j-2}] \\ \\ \psi^{4}(\mathfrak{Z}_{j-3})\,\Psi(\mathfrak{Z}_{j-1})+\cdots+\Psi(\mathfrak{Z}_{j+3})\,\psi^{4}(\mathfrak{Z}_{j-2})& \; \; \; \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j-2},\mathfrak{Z}_{j-1}]\\ \\ \psi^{3}(\mathfrak{Z}_{j-3})\,\Psi^{2}(\mathfrak{Z}_{j})+\cdots+\Psi^{2}(\mathfrak{Z}_{j+3})\, \psi^{3}(\mathfrak{Z}_{j-1}) & \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j-1},\mathfrak{Z}_{j}] \\ \\ \psi^{2}(\mathfrak{Z}_{j-3})\,\Psi^{3}(\mathfrak{Z}_{j+1})+\cdots+\Psi^{3}(\mathfrak{Z}_{j+3})\,\psi^{2}(\mathfrak{Z}_{j}) & \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j},\mathfrak{Z}_{j+1}]\\ \\ \psi(\mathfrak{Z}_{j-1})\,\Psi^{2}(j+2)+\cdots+\Psi^{4}(\mathfrak{Z}_{j+3})\,\psi(\mathfrak{Z}_{j+1}), & \; \; \; \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j+1},\mathfrak{Z}_{j+2}]\\ \\ \Psi^{5}(\mathfrak{Z}_{j+3})& \; \; \; \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j+2},\mathfrak{Z}_{j+3}]\\ \\ 0, & \; \; \; \text{otherwise} \end{array} \right. \end{eqnarray}$

(2.2)

where $\psi(\mathfrak{Z}_{j}) = sin\left(\frac{\mathfrak{Z}-\mathfrak{Z}_{j}}{2}\right), \, \Psi(\mathfrak{Z}_{j}) = sin\left(\frac{\mathfrak{Z}_{j}-\mathfrak{Z}}{2}\right)$ . Consequently, we can find the values of $\mathcal{G}_{j}(\mathfrak{Z})$ as shown in the next where the values of $\mathcal{P}_{\mathcal{L}}, \, \mathcal{L} = 1, \cdots, 13.$

$\begin{eqnarray} \mathcal{P}_{1} = \frac{\sin ^5\left(\frac{h}{2}\right)}{\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)}, \end{eqnarray}$

(2.3)

$\begin{eqnarray} \mathcal{P}_{2} = \frac{2 \sin ^5\left(\frac{h}{2}\right) \cos \left(\frac{h}{2}\right) \left(16 \cos ^2\left(\frac{h}{2}\right)-3\right)}{\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)}, \end{eqnarray}$

(2.4)

$\begin{eqnarray} \mathcal{P}_{3} = \frac{2 \sin ^5\left(\frac{h}{2}\right) \left(48 \cos ^4\left(\frac{h}{2}\right)-16 \cos ^2\left(\frac{h}{2}\right)+1\right)}{\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)}, \end{eqnarray}$

(2.5)

$\begin{eqnarray} \mathcal{P}_{4} = \frac{5 \sin ^4\left(\frac{h}{2}\right) \cos \left(\frac{h}{2}\right)}{2 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$

(2.6)

$\begin{eqnarray} \mathcal{P}_{5} = \frac{5 \sin ^4\left(\frac{h}{2}\right) \cos ^2\left(\frac{h}{2}\right) \left(8 \cos ^2\left(\frac{h}{2}\right)-3\right)}{\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)}, \end{eqnarray}$

(2.7)

$\begin{eqnarray} \mathcal{P}_{6} = \frac{5 \sin ^3\left(\frac{h}{2}\right) \left(5 \cos ^2\left(\frac{h}{2}\right)-1\right)}{4 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$

(2.8)

$\begin{eqnarray} \mathcal{P}_{7} = \frac{5 \sin ^3\left(\frac{h}{2}\right) \cos \left(\frac{h}{2}\right) \left(16 \cos ^4\left(\frac{h}{2}\right)-15 \cos ^2\left(\frac{h}{2}\right)+3\right)}{2 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$

(2.9)

$\begin{eqnarray} \mathcal{P}_{8} = -\frac{5 \sin ^3\left(\frac{h}{2}\right) \left(16 \cos ^6\left(\frac{h}{2}\right)-5 \cos ^2\left(\frac{h}{2}\right)+1\right)}{2 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$

(2.10)

$\begin{eqnarray} \mathcal{P}_{9} = \frac{5 \sin ^2\left(\frac{h}{2}\right) \cos \left(\frac{h}{2}\right) (25 \cos (h)-1)}{16 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$

(2.11)

$\begin{eqnarray} \mathcal{P}_{10} = -\frac{5 \sin ^2(h) (-27 \cos (h)+2 \cos (2 h)+1)}{32 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$

(2.12)

$\begin{eqnarray} \mathcal{P}_{11} = \frac{5 \sin \left(\frac{h}{2}\right) (44 \cos (h)+125 \cos (2 h)+23)}{128 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$

(2.13)

$\begin{eqnarray} \mathcal{P}_{12} = -\frac{5 \sin (h) (88 \cos (h)+127 \cos (2 h)+44 \cos (3 h)+125)}{128 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$

(2.14)

$\begin{eqnarray} \mathcal{P}_{13} = \frac{5 \sin \left(\frac{h}{2}\right) (2 \cos (h)+1) (125 \cos (h)+21 \cos (2 h)+23 \cos (3 h)+23)}{64 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$

(2.15)

Table 1. Value of

$\mathcal{G}_{j}(\mathfrak{Z})$ and its principle two derivatives at the knot points.

$\mathfrak{Z}$	$\mathfrak{Z}_{j-3}$	$\mathfrak{Z}_{j-2}$	$\mathfrak{Z}_{j-1}$	$\mathfrak{Z}_{j}$	$\mathfrak{Z}_{j+1}$	$\mathfrak{Z}_{j+2}$	$\mathfrak{Z}_{j+3}$

$\mathcal{G}_{j}(\mathfrak{Z})$	0	$\mathcal{P}_{1}$	$\mathcal{P}_{2}$	$\mathcal{P}_{3}$	$\mathcal{P}_{2}$	$\mathcal{P}_{1}$	0
$\mathcal{G}'_{j}(\mathfrak{Z})$	0	$-\mathcal{P}_{4}$	$-\mathcal{P}_{5}$	0	$\mathcal{P}_{5}$	$\mathcal{P}_{4}$	0
$\mathcal{G}''_{j}(\mathfrak{Z})$	0	$\mathcal{P}_{6}$	$\mathcal{P}_{7}$	$\mathcal{P}_{8}$	$\mathcal{P}_{7}$	$\mathcal{P}_{6}$	0
$\mathcal{G}'''_{j}(\mathfrak{Z})$	0	$\mathcal{P}_{9}$	$\mathcal{P}_{10}$	0	$\mathcal{P}_{10}$	$\mathcal{P}_{9}$	0
$\mathcal{G}''''_{j}(\mathfrak{Z})$	0	$\mathcal{P}_{11}$	$\mathcal{P}_{12}$	$\mathcal{P}_{13}$	$\mathcal{P}_{12}$	$\mathcal{P}_{11}$	0

| Show Table

DownLoad: CSV

where $h = \frac{q-p}{r}, \, q > p$ such that $[p, q]$ is the problem's domain.

2.1.2. Exponential cubic B-spline schemes

Employing the exponential cubic spline technique to considered model with the above conditions, yields elicit its numerical solutions as following

$\begin{eqnarray} \mathfrak{P}(\mathfrak{Z}) = \sum\limits_{\mathfrak{T} = -1} ^{\mathfrak{M}+1} \mathfrak{C}_{\mathfrak{T}}\,\mathcal{E}_{\mathfrak{T}}, \end{eqnarray}$

(2.16)

where $\mathfrak{C}_{\mathfrak{T}}, \, \mathcal{E}_{\mathfrak{T}}$ follow the next conditions, respectively:

$\begin{eqnarray*} \label{c2} \mathfrak{L}\,\mathfrak{B}(\mathfrak{Z}) = \mathcal{F}(\mathfrak{Z}_{\mathfrak{T}},\mathfrak{B}(\mathfrak{Z}_{\mathfrak{T}}))\,\,\text{where}\,\, (\mathfrak{T} = 0,1,...,n) \end{eqnarray*}$

and

$\begin{eqnarray} \mathcal{E}_{\mathfrak{T}}(\mathfrak{Z}) = \frac{1}{6\,\mathfrak{H}^{3}}\left\{ \begin{array}{cc} (\mathfrak{Z}-\mathfrak{Z}_{\mathfrak{T}-2})^{3},& \mathfrak{Z}\in [\mathfrak{Z}_{\mathfrak{T}-2},\mathfrak{Z}_{\mathfrak{T}-1}],\\ -3\,(\mathfrak{Z}-\mathfrak{Z}_{\mathfrak{T}-1})^{3}+3\,\mathfrak{H}\, (\mathfrak{Z}-\mathfrak{Z}_{\mathfrak{T}-1})^{2}+3\,\mathfrak{H}^{2}\,(\mathfrak{Z}-\mathfrak{Z}_{\mathfrak{T}-1})+\mathfrak{H}^{3},&\mathfrak{Z}\in [\mathfrak{Z}_{\mathfrak{T}-1},\mathfrak{Z}_{i}],\\ -3\,(\mathfrak{Z}_{\mathfrak{T}+1}-\mathfrak{Z})^{3}+3\,\mathfrak{H}\, (\mathfrak{Z}_{\mathfrak{T}+1}-\mathfrak{Z})^{2}+3\,\mathfrak{H}^{2}\, (\mathfrak{Z}_{\mathfrak{T}+1}-\mathfrak{Z})+\mathfrak{H}^{3},&\mathfrak{Z}\in [\mathfrak{Z}_{\mathfrak{T}},\mathfrak{Z}_{\mathfrak{T}+1}],\\ (\mathfrak{Z}_{\mathfrak{T}+2}-\mathfrak{Z})^{3},& \mathfrak{Z}\in [\mathfrak{Z}_{\mathfrak{T}+1},\mathfrak{Z}_{\mathfrak{T}+2}],\\ 0,&\text{otherwise}. \end{array}\right. \end{eqnarray}$

(2.17)

For $\mathfrak{T} \in [-2, \mathfrak{M}+2]$ , we obtain

$\begin{eqnarray} \mathfrak{B}_{\mathfrak{T}}(\mathfrak{Z}) = \mathfrak{C}_{\mathfrak{T}-1}+4\,\mathfrak{C}_{\mathfrak{T}}+\mathfrak{C}_{\mathfrak{T}+1}. \end{eqnarray}$

(2.18)

2.2. Method's results

Here, we apply the TQBS and ECBS schemes to Eq (1.1) with the evaluated initial and boundary conditions (1.3) as following.

2.2.1. Numerical solution via TQBS scheme

Applying the TQBS scheme to Eq (1.2) with above-conditions (1.3), gets the following numerical values in Tables 2, 3, and Figure 1.

Table 2. Analytical, numerical, and absolute error of Eq (1.2) through the TQBS scheme under the shown values of boundary and initial conditions (1.3).

Value of $\mathfrak{Z}$	Analytica	Numerical	Absolute Error	Value of $\mathfrak{Z}$	Analytical	Numerical	Absolute Error
0	0	-2.71051E-20	2.71051E-20	0.2578125	-0.034185856	-0.034185856	4.37705E-14
0.0078125	-0.001041661	-0.001041661	1.00159E-15	0.265625	-0.035209885	-0.035209885	4.49224E-14
0.015625	-0.002083291	-0.002083291	2.52879E-15	0.2734375	-0.036232859	-0.036232859	4.60743E-14
0.0234375	-0.003124857	-0.003124857	3.91484E-15	0.28125	-0.037254747	-0.037254747	4.72261E-14
0.03125	-0.004166328	-0.004166328	5.33774E-15	0.2890625	-0.038275521	-0.038275521	4.83363E-14
0.0390625	-0.005207671	-0.005207671	6.74634E-15	0.296875	-0.039295151	-0.039295151	4.94466E-14
0.046875	-0.006248856	-0.006248856	8.1584E-15	0.3046875	-0.040313607	-0.040313607	5.05221E-14
0.0546875	-0.00728985	-0.00728985	9.56613E-15	0.3125	-0.041330861	-0.041330861	5.15907E-14
0.0625	-0.008330622	-0.008330622	1.09721E-14	0.3203125	-0.042346885	-0.042346885	5.26246E-14
0.0703125	-0.00937114	-0.00937114	1.23738E-14	0.328125	-0.043361648	-0.043361648	5.36515E-14
0.078125	-0.010411372	-0.010411372	1.37685E-14	0.3359375	-0.044375124	-0.044375124	5.46438E-14
0.0859375	-0.011451287	-0.011451287	1.51632E-14	0.34375	-0.045387282	-0.045387282	5.56291E-14
0.09375	-0.012490853	-0.012490853	1.65527E-14	0.3515625	-0.046398096	-0.046398096	5.65936E-14
0.1015625	-0.013530039	-0.013530039	1.7937E-14	0.359375	-0.047407536	-0.047407536	5.75304E-14
0.109375	-0.014568812	-0.014568812	1.93196E-14	0.3671875	-0.048415576	-0.048415576	5.84324E-14
0.1171875	-0.015607143	-0.015607143	2.06935E-14	0.375	-0.049422187	-0.049422187	5.93275E-14
0.125	-0.016644999	-0.016644999	2.20587E-14	0.3828125	-0.050427341	-0.050427341	6.02018E-14
0.1328125	-0.017682349	-0.017682349	2.34222E-14	0.390625	-0.051431011	-0.051431011	6.10553E-14
0.140625	-0.018719162	-0.018719162	2.47719E-14	0.3984375	-0.052433171	-0.052433171	6.1888E-14
0.1484375	-0.019755406	-0.019755406	2.6118E-14	0.40625	-0.053433792	-0.053433792	6.2686E-14
0.15625	-0.020791051	-0.020791051	2.74503E-14	0.4140625	-0.054432847	-0.054432847	6.34492E-14
0.1640625	-0.021826065	-0.021826065	2.87825E-14	0.421875	-0.055430311	-0.055430311	6.41917E-14
0.171875	-0.022860418	-0.022860418	3.01044E-14	0.4296875	-0.056426157	-0.056426157	6.48925E-14
0.1796875	-0.023894078	-0.023894078	3.14124E-14	0.4375	-0.057420357	-0.057420357	6.56003E-14
0.1875	-0.024927014	-0.024927014	3.26926E-14	0.4453125	-0.058412887	-0.058412887	6.62456E-14
0.1953125	-0.025959197	-0.025959197	3.39763E-14	0.453125	-0.059403719	-0.059403719	6.68632E-14
0.203125	-0.026990595	-0.026990595	3.52322E-14	0.4609375	-0.060392829	-0.060392829	6.74669E-14
0.2109375	-0.028021178	-0.028021178	3.64916E-14	0.46875	-0.06138019	-0.06138019	6.80359E-14
0.21875	-0.029050915	-0.029050915	3.77406E-14	0.4765625	-0.062365777	-0.062365777	6.8591E-14
0.2265625	-0.030079776	-0.030079776	3.89688E-14	0.484375	-0.063349565	-0.063349565	6.91253E-14
0.234375	-0.03110773	-0.03110773	4.01866E-14	0.4921875	-0.064331528	-0.064331528	6.95971E-14
0.2421875	-0.032134749	-0.032134749	4.13905E-14	0.5	-0.065311643	-0.065311643	7.00412E-14
0.25	-0.0331608	-0.0331608	4.25909E-14	0.5078125	-0.066289885	-0.066289885	7.04575E-14

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Table 3. Analytical, numerical, and absolute error of Eq (1.2) through the TQBS scheme under the shown values of boundary and initial conditions (1.3).

Value of $\mathfrak{Z}$	Analytical	Numerical	Absolute error	Value of $\mathfrak{Z}$	Analytical	Numerical	Absolute error
0.515625	-0.067266228	-0.067266228	7.08461E-14	0.7578125	-0.096468595	-0.096468595	6.22696E-14
0.5234375	-0.068240649	-0.068240649	7.11931E-14	0.765625	-0.097372658	-0.097372658	6.1201E-14
e 0.53125	-0.069213124	-0.069213124	7.14984E-14	0.7734375	-0.098274146	-0.098274146	6.00908E-14
0.5390625	-0.070183629	-0.070183629	7.18037E-14	0.78125	-0.099173043	-0.099173043	5.88973E-14
0.546875	-0.071152141	-0.071152141	7.20674E-14	0.7890625	-0.100069331	-0.100069331	5.76483E-14
0.5546875	-0.072118635	-0.072118635	7.22755E-14	0.796875	-0.100962996	-0.100962996	5.63855E-14
0.5625	-0.07308309	-0.07308309	7.24559E-14	0.8046875	-0.101854021	-0.101854021	5.50393E-14
0.5703125	-0.074045483	-0.074045483	7.25808E-14	0.8125	-0.102742391	-0.102742391	5.36515E-14
0.578125	-0.075005789	-0.075005789	7.2678E-14	0.8203125	-0.103628091	-0.103628091	5.21666E-14
0.5859375	-0.075963988	-0.075963988	7.27196E-14	0.828125	-0.104511107	-0.104511107	5.06262E-14
0.59375	-0.076920057	-0.076920057	7.27474E-14	0.8359375	-0.105391423	-0.105391423	4.89747E-14
0.6015625	-0.077873973	-0.077873973	7.27057E-14	0.84375	-0.106269025	-0.106269025	4.72677E-14
0.609375	-0.078825716	-0.078825716	7.26502E-14	0.8515625	-0.107143899	-0.107143899	4.55053E-14
0.6171875	-0.079775264	-0.079775264	7.25253E-14	0.859375	-0.108016031	-0.108016031	4.36595E-14
0.625	-0.080722594	-0.080722594	7.23588E-14	0.8671875	-0.108885407	-0.108885407	4.17999E-14
0.6328125	-0.081667688	-0.081667688	7.21367E-14	0.875	-0.109752015	-0.109752015	3.98431E-14
0.640625	-0.082610522	-0.082610522	7.18869E-14	0.8828125	-0.110615841	-0.110615841	3.78308E-14
0.6484375	-0.083551078	-0.083551078	7.15816E-14	0.890625	-0.111476871	-0.111476871	3.57908E-14
0.65625	-0.084489334	-0.084489334	7.12486E-14	0.8984375	-0.112335095	-0.112335095	3.36675E-14
0.6640625	-0.08542527	-0.08542527	7.086E-14	0.90625	-0.113190498	-0.113190498	3.14748E-14
0.671875	-0.086358867	-0.086358867	7.04575E-14	0.9140625	-0.11404307	-0.11404307	2.92405E-14
0.6796875	-0.087290105	-0.087290105	6.99718E-14	0.921875	-0.114892798	-0.114892798	2.68674E-14
0.6875	-0.088218965	-0.088218965	6.94583E-14	0.9296875	-0.11573967	-0.11573967	2.44249E-14
0.6953125	-0.089145427	-0.089145427	6.88755E-14	0.9375	-0.116583676	-0.116583676	2.18991E-14
0.703125	-0.090069472	-0.090069472	6.82371E-14	0.9453125	-0.117424804	-0.117424804	1.92901E-14
0.7109375	-0.090991083	-0.090991083	6.75571E-14	0.953125	-0.118263043	-0.118263043	1.66117E-14
0.71875	-0.09191024	-0.09191024	6.68215E-14	0.9609375	-0.119098383	-0.119098383	1.38639E-14
0.7265625	-0.092826925	-0.092826925	6.60166E-14	0.96875	-0.119930814	-0.119930814	1.10745E-14
0.734375	-0.093741121	-0.093741121	6.51562E-14	0.9765625	-0.120760325	-0.120760325	8.18789E-15
0.7421875	-0.094652809	-0.094652809	6.42403E-14	0.984375	-0.121586906	-0.121586906	5.31519E-15
0.75	-0.095561973	-0.095561973	6.32688E-14	0.9921875	-0.122410548	-0.122410548	2.13718E-15

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Figure 1. analytical, numerical, and absolute error for the nonlinear KGZ equations through the TQBS scheme.

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2.2.2. Numerical solution via ECBS scheme

Applying the ECBS scheme to Eq (1.2) with above-conditions (1.3), gets the following numerical values in Table 4, and Figure 2.

Table 4. Analytical, numerical, and absolute error of Eq (1.2) through the ECBS scheme under the shown values of boundary and initial conditions (1.3).

Value of $\mathfrak{Z}$	Analytical	Numerical	Absolute error
0	0	0	0
0.001	-0.000133333	-0.000133333	2.75062E-16
0.002	-0.000266667	-0.000266667	5.33482E-16
0.003	-0.0004	-0.0004	7.58508E-16
0.004	-0.000533333	-0.000533333	9.33498E-16
0.005	-0.000666665	-0.000666665	1.04181E-15
0.006	-0.000799998	-0.000799998	1.06685E-15
0.007	-0.00093333	-0.00093333	9.9172E-16
0.008	-0.001066661	-0.001066661	7.99924E-16
0.009	-0.001199992	-0.001199992	4.75097E-16
0.01	-0.001333322	-0.001333322	0

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Figure 2. Exact, approximate, and absolute error of the nonlinear KGZ equations through the ECBS scheme.

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3. Results and discussion

Here, we explain the accuracy and novelty of the obtained numerical result in this research paper by comparing them with the previously calculated in ^[42] through four-different schemes (Adomian decomposition (AD), El-kalla (EK), cubic B-spline (CB), and extended cubic B-spline (ECB) schemes)) and one common scheme (exponential cubic B-spline (ExCB) scheme). Although, comparing our obtained result with each other, shows the accuracy of the ECBS scheme over the TQBS scheme where the absolute error is smaller than that have been obtained by the TQBS scheme which have been shown in Figures 1, 2 and Tables 2–4. Now, comparing the accuracy between our solutions and that have been evaluated in ^[42], shows our solution is more accurate than their solutions that have been explained in Table 5, and Figure 3.

Table 5. Numerical methods' absolute error.

Value of $\mathfrak{Z}$	AD	EK	CBS	EtCBS	ECBS	ECBS	TQBS
0	0	0	0	0	5.45828E-18	0	2.71E-20
0.001	2.71051E-20	2.71051E-20	1.83257E-16	3.29993E-09	1.10002E-09	2.75E-16
0.002	5.42101E-20	5.42101E-20	3.55401E-16	6.39986E-09	2.13337E-09	5.33E-16
0.003	0	0	5.05455E-16	9.0998E-09	3.03339E-09	7.59E-16
0.004	0	0	6.22007E-16	1.11997E-08	3.7334E-09	9.33E-16
0.005	0	0	6.94215E-16	1.24997E-08	4.16674E-09	1.04E-15
0.006	1.0842E-19	1.0842E-19	7.10695E-16	1.27997E-08	4.26674E-09	1.07E-15
0.007	1.0842E-19	1.0842E-19	6.60821E-16	1.18997E-08	3.96674E-09	9.92E-16
0.008	0	0	5.32994E-16	9.59976E-09	3.20006E-09	8E-16
0.009	2.1684E-19	0	3.1637E-16	5.69985E-09	1.90003E-09	4.75E-16
0.01	0	2.1684E-19	0	0	2.1684E-19	0	2.53E-15

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Figure 3. Absolute value of error for AD, EK, CBS, ECBS, ExCBS, ECBS, and TQBS schemes.

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4. Conclusions

This manuscript has employed the TQBS and ECBS numerical schemes for evaluating the numerical solutions of the nonlinear KGZ model. The matching between analytical and numerical solutions has been explained through the shown tables and figures. The accuracy of the modified Khater method has been proved through six numerical schemes. The novelty and originality of our obtained solutions have been explained. the powerful and effectiveness of the used techniques are also explained and verified.

Acknowledgments

The authors would like to thank Taif University Researchers supporting Project number (TURSP-2020/159), Taif University-Saudi Arabia.

Conflict of interest

There is no conflict of interest.

References

[1]	H. Zhang, X. Yang, Q. Tang, D. Xu, A robust error analysis of the OSC method for a multi-term fourth-order sub-diffusion equation, Comput. Math. Appl., 109 (2022), 180–190. https://doi.org/10.1016/j.camwa.2022.01.007 doi: 10.1016/j.camwa.2022.01.007
[2]	X. Yang, H. Zhang, J. Tang, The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions, Comput. Math. Appl., 82 (2021), 1–12. https://doi.org/10.1016/j.camwa.2020.11.015 doi: 10.1016/j.camwa.2020.11.015
[3]	H. Zhang, X. Yang, D. Xu, Unconditional convergence of linearized orthogonal spline collocation algorithm for semilinear subdiffusion equation with nonsmooth solution, Numer. Meth. Part. Differ. Equ., 37 (2021), 1361–1373. https://doi.org/10.1002/num.22583 doi: 10.1002/num.22583
[4]	A. F. Daghistani, A. M. T. Abd El-Bar, A. M. Gemeay, M. A. E. Abdelrahman, S. Z. Hassan, A hyperbolic secant-squared distribution via the nonlinear evolution equation and its application, Mathematics, 11 (2023), 4270. https://doi.org/10.3390/math11204270 doi: 10.3390/math11204270
[5]	M. A. E. Abdelrahman, G. Alshreef, Closed-form solutions to the new coupled Konno–Oono equation and the Kaup-Newell model equation in magnetic field with novel statistic application, Eur. Phys. J. Plus, 136 (2021), 455. https://doi.org/10.1140/epjp/s13360-021-01472-2 doi: 10.1140/epjp/s13360-021-01472-2
[6]	Y. Cheng, A. Chertock, M. Herty, A. Kurganov, T. Wu, A new approach for designing moving-water equilibria preserving schemes for the shallow water equations, J. Sci. Comput., 80 (2019), 538–554. https://doi.org/10.1007/s10915-019-00947-w doi: 10.1007/s10915-019-00947-w
[7]	P. Ripa, Conservation laws for primitive equations models with inhomogeneous layers, Geophys. Astrophys. Fluid Dynam., 70 (1993), 85–111. https://doi.org/10.1080/03091929308203588 doi: 10.1080/03091929308203588
[8]	G. Laibe, D. J. Price, Dusty gas with one fluid, Mon. Not. R. Astron. Soc., 440 (2014), 2136–2146. https://doi.org/10.1093/mnras/stu355 doi: 10.1093/mnras/stu355
[9]	Y. Shi, X. Yang, A time two-grid difference method for nonlinear generalized viscous Burgers' equation, J. Math. Chem., 62 (2024), 1323–1356. https://doi.org/10.1007/s10910-024-01592-x doi: 10.1007/s10910-024-01592-x
[10]	C. Li, H. Zhang, X. Yang, A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel, J. Appl. Math. Comput., 70 (2024), 2045–2077. https://doi.org/10.1007/s12190-024-02039-x doi: 10.1007/s12190-024-02039-x
[11]	H. Zhang, X. Yang, Y. Liu, Y. Liu, An extrapolated CN-WSGD OSC method for a nonlinear time fractional reaction-diffusion equation, Appl. Numer. Math., 157 (2020), 619–633. https://doi.org/10.1016/j.apnum.2020.07.017 doi: 10.1016/j.apnum.2020.07.017
[12]	H. Zhang, X. Yang, D. Xu, An efficient spline collocation method for a nonlinear fourth-order reaction subdiffusion equation, J. Sci. Comput., 85 (2020), 7. https://doi.org/10.1007/s10915-020-01308-8 doi: 10.1007/s10915-020-01308-8
[13]	X. Yang, H. Zhang, Q. Tang, A spline collocation method for a fractional mobile–immobile equation with variable coefficients, Comput. Appl. Math., 39 (2020), 34. https://doi.org/10.1007/s40314-019-1013-3 doi: 10.1007/s40314-019-1013-3
[14]	H.S. Alayachi, The modulations of higher order solitonic pressure and energy of fluid filled elastic tubes, AIP Adv., 13 (2023), 115214. https://doi.org/10.1063/5.0179155 doi: 10.1063/5.0179155
[15]	X. Yang, W. Qiu, H. Zhang, L. Tang, An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation, Comput. Math. Appl., 102 (2021), 233–247. https://doi.org/10.1016/j.camwa.2021.10.021 doi: 10.1016/j.camwa.2021.10.021
[16]	H. Zhang, Y. Liu, X. Yang, An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space, J. Appl. Math. Comput., 69 (2023), 651–674. https://doi.org/10.1007/s12190-022-01760-9 doi: 10.1007/s12190-022-01760-9
[17]	X. Yang, W. Qiu, H. Chen, H. Zhang, Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space, Appl. Numer. Math., 172 (2022), 497–513. https://doi.org/10.1016/j.apnum.2021.11.004 doi: 10.1016/j.apnum.2021.11.004
[18]	H. Zhang, X. Jiang, F. Wang, X. Yang, The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation, J. Appl. Math. Comput., 70 (2024), 1127–1151. https://doi.org/10.1007/s12190-024-02000-y doi: 10.1007/s12190-024-02000-y
[19]	H. G. Abdelwahed, M. A. E. Abdelrahman, M. Inc, R. Sabry, New soliton applications in earth's magnetotail plasma at critical densities, Front. Phys., 8 (2020), 181. https://doi.org/10.3389/fphy.2020.00181 doi: 10.3389/fphy.2020.00181
[20]	S. Zhang, C. Tian, W. Y. Qian, Bilinearization and new multi-soliton solutions for the (4+1)-dimensional Fokas equation, Pramana-J. Phys., 86 (2016), 1259–1267. https://doi.org/10.1007/s12043-015-1173-7 doi: 10.1007/s12043-015-1173-7
[21]	L. Akinyemi, M. Şenol, U. Akpan, K. Oluwasegun, The optical soliton solutions of generalized coupled nonlinear Schrödinger-Korteweg-de Vries equations, Opt. Quant. Electron., 53 (2021), 394. https://doi.org/10.1007/s11082-021-03030-7 doi: 10.1007/s11082-021-03030-7
[22]	F. Mirzaee, S. Rezaei, N. Samadyar, Numerical solution of two-dimensional stochastic time-fractional sine-Gordon equation on non-rectangular domains using finite difference and meshfree methods, Eng. Anal. Bound. Elem., 127 (2021), 53–63. https://doi.org/10.1016/j.enganabound.2021.03.009 doi: 10.1016/j.enganabound.2021.03.009
[23]	M. A. E. Abdelrahman, H. AlKhidhr, A robust and accurate solver for some nonlinear partial differential equations and tow applications, Phys. Scr., 95 (2020), 065212. https://doi.org/10.1088/1402-4896/ab80e7 doi: 10.1088/1402-4896/ab80e7
[24]	Z. Zhou, H. Zhang, X. Yang, CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics, Appl. Math. Comput., 474 (2024), 128680. https://doi.org/10.1016/j.amc.2024.128680 doi: 10.1016/j.amc.2024.128680
[25]	X. F. Yang, Z. C. Deng, Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Differ. Equ., 2015 (2015), 117. https://doi.org/10.1186/s13662-015-0452-4 doi: 10.1186/s13662-015-0452-4
[26]	W. Wang, H. Zhang, Z. Zhou, X. Yang, A fast compact finite difference scheme for the fourth-order diffusion-wave equation, Int. J. Comput. Math., 101 (2024), 170–193. https://doi.org/10.1080/00207160.2024.2323985 doi: 10.1080/00207160.2024.2323985
[27]	B. Q. Li, Y. L. Ma, Interaction properties between rogue wave and breathers to the manakov system arising from stationary self-focusing electromagnetic systems, Chaos Soliton. Fract., 156 (2022), 111832. https://doi.org/10.1016/j.chaos.2022.111832 doi: 10.1016/j.chaos.2022.111832
[28]	X. Jin, J. Jiang, J. Chi, X. Wu, Adaptive finite-time pinned and regulation synchronization of disturbed complex networks, Commun. Nonlinear Sci., 124 (2023), 107319. https://doi.org/10.1016/j.cnsns.2023.107319 doi: 10.1016/j.cnsns.2023.107319
[29]	Z. J. Yang, S. M. Zhang, X. L. Li, Z. G. Pang, H. X. Bu, High-order revivable complex-valued hyperbolic-sine-Gaussian solitons and breathers in nonlinear media with a spatial nonlocality, Nonlinear Dyn., 94 (2018), 2563–2573. https://doi.org/10.1007/s11071-018-4510-9 doi: 10.1007/s11071-018-4510-9
[30]	Z. Sun, J. Li, R. Bian, D. Deng, Z. Yang, Transmission mode transformation of rotating controllable beams induced by the cross phase, Opt. Express, 32 (2024), 9201–9212. https://doi.org/10.1364/OE.520342 doi: 10.1364/OE.520342
[31]	M. A. E. Abdelrahman, N. F. Abdo, On the nonlinear new wave solutions in unstable dispersive environments, Phys. Scr., 95 (2020), 045220. https://doi.org/10.1088/1402-4896/ab62d7 doi: 10.1088/1402-4896/ab62d7
[32]	H. G. Abdelwahed, M. A. E. Abdelrahman, S. Alghanim, N. F. Abdo, Higher-order Kerr nonlinear and dispersion effects on fiber optics, Results Phys., 26 (2021), 104268. https://doi.org/10.1016/j.rinp.2021.104268 doi: 10.1016/j.rinp.2021.104268
[33]	J. L. Lebowitz, H. A. Rose, E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Stat. Phys., 50 (1988), 657–687. https://doi.org/10.1007/BF01026495 doi: 10.1007/BF01026495
[34]	G. D. McDonald, C. C. N. Kuhn, K. S. Hardman, S. Bennetts, P. J. Everitt, P. A. Altin, et al., Bright solitonic matter-wave interferometer, Phys. Rev. Lett., 113 (2014), 013002. https://doi.org/10.1103/PhysRevLett.113.013002 doi: 10.1103/PhysRevLett.113.013002
[35]	Y. L. Ma, $N$ th-order rogue wave solutions for a variable coefficient Schrödinger equation in inhomogeneous optical fibers, Optik, 251 (2022), 168103. https://doi.org/10.1016/j.ijleo.2021.168103 doi: 10.1016/j.ijleo.2021.168103
[36]	B. Q. Li, Y. L. Ma, Interaction properties between rogue wave and breathers to the manakov system arising from stationary self-focusing electromagnetic systems, Chaos Soliton. Fract., 156 (2022), 111832. https://doi.org/10.1016/j.chaos.2022.111832 doi: 10.1016/j.chaos.2022.111832
[37]	O. V. Marchukov, B. A. Malomed, V. A. Yurovsky, M. Olshanii, V. Dunjko, R. G. Hulet, Splitting of nonlinear-Schrödinger-equation breathers by linear and nonlinear localized potentials, Phys. Rev. A, 99 (2019), 063623. https://doi.org/10.1103/PhysRevA.99.063623 doi: 10.1103/PhysRevA.99.063623
[38]	S. Shen, Z. Yang, X. Li, S. Zhang, Periodic propagation of complex-valued hyperbolic-cosine-Gaussian solitons and breathers with complicated light field structure in strongly nonlocal nonlinear media, Commun. Nonlinear Sci., 103 (2021), 106005. https://doi.org/10.1016/j.cnsns.2021.106005 doi: 10.1016/j.cnsns.2021.106005
[39]	Z. Y. Sun, D. Deng, Z. G. Pang, Z. J. Yang, Nonlinear transmission dynamics of mutual transformation between array modes and hollow modes in elliptical sine-Gaussian cross-phase beams, Chaos Soliton. Fract., 178 (2024), 114398. https://doi.org/10.1016/j.chaos.2023.114398 doi: 10.1016/j.chaos.2023.114398
[40]	S. Shen, Z. J. Yang, Z. G. Pang, Y. R. Ge, The complex-valued astigmatic cosine-Gaussian soliton solution of the nonlocal nonlinear Schrödinger equation and its transmission characteristics, Appl. Math. Lett., 125 (2022), 107755. https://doi.org/10.1016/j.aml.2021.107755 doi: 10.1016/j.aml.2021.107755
[41]	L. M. Song, Z. J. Yang, X. L. Li, S. M. Zhang, Coherent superposition propagation of Laguerre-Gaussian and Hermite-Gaussian solitons, Appl. Math. Lett., 102 (2020), 106114. https://doi.org/10.1016/j.aml.2019.106114 doi: 10.1016/j.aml.2019.106114
[42]	M. Najafi, S. Arbabi, Traveling wave solutions for nonlinear Schrödinger equations, Optik, 126 (2015), 3992–3997. https://doi.org/10.1016/j.ijleo.2015.07.165 doi: 10.1016/j.ijleo.2015.07.165
[43]	M. Dehghan, A. Shokri, A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions, Comput. Math. Appl., 54 (2007), 136–146. https://doi.org/10.1016/j.camwa.2007.01.038 doi: 10.1016/j.camwa.2007.01.038
[44]	S. V. Mousavi, S. Miret-Artés, On non-linear Schrödinger equations for open quantum systems, Eur. Phys. J. Plus, 134 (2019), 431. https://doi.org/10.1140/epjp/i2019-12965-6 doi: 10.1140/epjp/i2019-12965-6
[45]	W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, The finite-difference vector beam propagation method: Analysis and assessment, J. Lightwave Technol., 10 (1992), 295–305. https://doi.org/10.1109/50.124490 doi: 10.1109/50.124490
[46]	A. I. Aliyu, M. Inc, A. Yusuf, D. Baleanu, Optical solitary waves and conservation laws to the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation, Mod. Phys. Lett. B, 32 (2018), 1850373. https://doi.org/10.1142/S0217984918503736 doi: 10.1142/S0217984918503736
[47]	H. Durur, E. Ilhan, H. Bulut, Novel complex wave solutions of the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation, Fractal Fract., 4 (2020), 41. https://doi.org/10.3390/fractalfract4030041 doi: 10.3390/fractalfract4030041
[48]	D. Baleanu, K. Hosseini, S. Salahshour, K. Sadri, M. Mirzazadeh, C. Park, A. Ahmadian, The (2+1)-dimensional hyperbolic nonlinear Schrödinger equation and its optical solitons, AIMS Mathematics, 6 (2021), 9568–9581. https://doi.org/10.3934/math.2021556 doi: 10.3934/math.2021556
[49]	G. Ai-Lin, L. Ji, Exact solutions of (2+1)-dimensional HNLS equation, Commun. Theor. Phys., 54 (2010), 401. https://doi.org/10.1088/0253-6102/54/3/04 doi: 10.1088/0253-6102/54/3/04
[50]	X. Yang, H. Zhang, The uniform $l^{1}$ long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data, Appl. Math. Lett., 124 (2022), 107644. https://doi.org/10.1016/j.aml.2021.107644 doi: 10.1016/j.aml.2021.107644
[51]	X. Yang, H. Zhang, Q. Zhang, G. Yuan, Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes, Nonlinear Dyn., 108 (2022), 3859–3886. https://doi.org/10.1007/s11071-022-07399-2 doi: 10.1007/s11071-022-07399-2
[52]	X. Yang, Z. Zhang, Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes, J. Sci. Comput., 99 (2024), 80. https://doi.org/10.1007/s10915-024-02511-7 doi: 10.1007/s10915-024-02511-7
[53]	X. Yang, Z. Zhang, On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations, Appl. Math. Lett., 150 (2024), 108972. https://doi.org/10.1016/j.aml.2023.108972 doi: 10.1016/j.aml.2023.108972

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