A Soliton Solution for the Kadomtsev–Petviashvili Model Using Two Novel Schemes

Abstract

Symmetries are crucial to the investigation of nonlinear physical processes, particularly the evaluation of a differential problem in the real world. This study focuses on the investigation of the Kadomtsev–Petviashvili $\left(\mathrm{KP}\right)$ model within a (3+1)-dimensional domain, governing the behavior of wave propagation in a medium characterized by both nonlinearity and dispersion. The inquiry employs two distinct analytical techniques to derive multiple soliton solutions and multiple solitary wave solutions. These methods include the modified Sardar sub-equation technique and the Darboux transformation $\left(\mathrm{DT}\right)$. The modified Sardar sub-equation technique is used to obtain multiple soliton solutions, while the $\mathrm{DT}$ is introduced to develop two bright and two dark soliton solutions. These solutions are presented alongside their corresponding constraint conditions and illustrated through 3-D, 2-D, and contour plots to physically portray the derived solutions. The results demonstrate that the employed analytical techniques are useful and have not yet been explored in the context of the analyzed models. The proposed methodologies are valuable and can be applied to additional nonlinear evolutionary models employed to describe nonlinear physical models within the domain of nonlinear science.

1. Introduction

Recently, a higher-dimensional integrable system’s physical and mathematical aspects have been addressed. A tremendous amount of research has gone into developing and understanding integrable models because of their major effects in scientific fields. Many investigations have been conducted on higher-dimensional integrable equations in many different sectors, including solitary wave theory and plasma acoustic waves, as well as many more scientific fields. Many soliton solutions can be found in [1,2,3,4,5,6,7,8,9,10,11,12,13,14].

A nonlinear partial differential equation, the (3+1)-dimensional Kadomtsev–Petviashvili (KP) model describes wave events in a four-dimensional space–time context. It is an extension of the (2+1)-dimensional KP model and is applied to the study of many physical systems, such as fluid dynamics, nonlinear optics, and plasma physics. The soliton solutions, which are localized wave packets that can travel through the medium without changing their shape or amplitude, are of great importance in the (3+1)-dimensional KP model. Numerous scientific phenomena, including wave propagation, energy transport, and information encoding, depend heavily on soliton behavior. A (3+1)-dimensional KP model was presented in [1] as

$$\alpha {\mathcal{M}}_{xt}-\frac{{\alpha }^4+{\beta }^4-6{\alpha }^2{\beta }^2}{16}{\mathcal{M}}_{xxxx}-\frac{3({\beta }^2-{\alpha }^2)}{4}{\left({\mathcal{M}}^2\right)}_{xx}-\frac{3}{4}{\mathcal{M}}_{yy}+\frac{3}{4}{\mathcal{M}}_{zz}=0,$$

(1)

which allows for a weak dispersion factor ${\mathcal{M}}_{xxxx}$. Its integrability was examined by using the Bäcklund transformation in Equation (1). The suggested version of Equation (1) is not in the form of Painlevé integrability. To expand and generalize integrable systems to larger-dimensional models, researchers have conducted a lot of work. The development of numerous higher-dimensionally integrable systems is the result of these efforts. Through the application of a number of potent approaches, the higher-order integrable model enables the determination of the dynamic behavior of the solution. Recently, there have been two thorough reviews of the subject of nonlinear wave structures in various physical contexts, including consideration of nonlinear optics, photonics, and matter waves in Einstein Bose and the condensation products [15,16]. Addressing the context of liquid crystals, molecular Einstein’s condensates, photonic systems, and other fundamental contexts, researchers have examined the two- and three-dimensional resonances and associated states, such as quantum droplets [17]. In addition, new information on light bullets, the generation of few-cycle (ultra-narrow) optical pulses and their many uses, and the appearance of rogue waves in a variety of media was formally provided in the overview published in [18]. The extended (3+1)-dimensional Painlevé integrable equation is now examined as:

$$\alpha {\mathcal{M}}_{xt}-\frac{{\alpha }^4+{\beta }^4-6{\alpha }^2{\beta }^2}{16}{\mathcal{M}}_{xxxx}-\frac{3({\beta }^2-{\alpha }^2)}{4}{\left({\mathcal{M}}^2\right)}_{xx}+a{\mathcal{M}}_{xx}+b{\mathcal{M}}_{xy}+c{\mathcal{M}}_{xz}+d{\mathcal{M}}_{zz}+e{\mathcal{M}}_{yz}+\frac{e^2}{4d}{\mathcal{M}}_{yy}=0,$$

(2)

$$\alpha {\mathcal{M}}_{xt}-\frac{{\alpha }^4+{\beta }^4-6{\alpha }^2{\beta }^2}{16}{\mathcal{M}}_{xxxx}-\frac{3({\beta }^2-{\alpha }^2)}{4}{\left({\mathcal{M}}^2\right)}_{xx}+a{\mathcal{M}}_{xx}+b{\mathcal{M}}_{xy}+c{\mathcal{M}}_{yy}=0.$$

(3)

The majority of phenomena can be represented by nonlinear differential equations (NDEs). Determining an exact solution for nonlinear differential equations is significantly more difficult in this circumstance. Numerous analytical techniques have been employed to solve nonlinear issues, including the Adomian decomposition technique and the variational homotopy perturbation technique, the variational iteration technique, the homotopy perturbation technique, the differential transform technique, and the variational iteration technique.

Additionally, several intriguing techniques have been researched to find precise solutions for nonlinear PDEs, including the prolonged sin-Gordon expansion technique, the modified Tanh-function technique, and the modified Kudryashov technique. For Schamel’s equation, accurate nondifferentiable solutions have been found. For nonlinear PDEs defined in the sense of conformable and local fractional derivatives, several of the aforementioned techniques have been attained. The Hirota bilinear technique is perhaps one of the most fascinating exact solution techniques in solution theory for solving nonlinear integrable equations.

Lie group theory plays a significant part in symmetry reduction, whenever the traditional methods for minimizing the number of independent components in a problem are the solution under some subgroup of the symmetry group of that system. Numerous scientists have claimed that finding the approximate solution to such problems is very important either by using numerical or analytical techniques. Therefore, symmetry analysis is a very useful system for identifying PDEs, especially when the equations are derived from the ideas of mathematical accounting. Although symmetry is an essential component of nature, the majority of the findings are not symmetrical. An effective approach for hiding symmetry is to offer unexpected symmetry-breaking phenomena. Discrete and continuous finite symmetries come in two different varieties. Parity and temporal inversion are two examples of natural symmetries that are discrete, whereas space is a continuous transformation.

Each of these techniques has pros and cons. The optimum auxiliary function technique, a novel methodology, has been similarly expanded to partial differential equations (PDEs). Recent years have seen a boom in research on multiple soliton solutions and lump solutions because of their significance in nonlinear scientific domains. High-amplitude localization is a feature of lumps. The Hirota bilinear form [19,20,21], can be used to produce lumps and solitons. A lump may physically separate (or emit) from a line soliton, survive for a momentary interval, and then meld with the following soliton. Recently, Wazwaz used this model to explore nonlinear integrable models and a number of potent approaches, including the Hirota method and the Darboux transformation method [22,23,24]. Darboux transformation is a powerful mathematical technique used in the study of integrable systems and nonlinear partial differential equations. It was first introduced by the French mathematician Gaston Darboux in the late 19th century as a way of constructing new solutions to differential equations from known ones. The Darboux transformation works by converting a given differential equation solution into a new solution that uses the same differential equation but has different boundary conditions. The Darboux matrix, a new function that is used to methodically alter the initial answer, is introduced in order to make this transformation. One of the key benefits of the Darboux transformation is that it enables the building of an endless number of new solutions to a given differential equation, which can provide important information about how the system under study behaves. Numerous physical systems, such as quantum mechanics, fluid dynamics, and nonlinear optics have all been studied using it.

The novelty of this work is that the Darboux transformation and the modified Sardar sub-equation technique are applied to the KP model in this study to evaluate and analyze their impacts. New soliton solutions, new aspects of the model’s behavior, or other characteristics that have not yet been investigated in the literature may be found through this approach [25,26,27]. This paper has the potential to increase knowledge and comprehension in the area of the KP equation by utilizing these methodologies and examining their consequences. It might offer fresh perspectives, illuminate the integrability characteristics, unearth more symmetries, or make known KP model-related events that were previously unknown. The study of systems that can be precisely solved using analytical techniques is the focus of the field of integrable systems, which includes the Darboux transformation as a key tool. It has played a central role in the development of many important mathematical concepts, such as soliton theory and the inverse scattering method [28,29,30].

In this paper, we use different techniques, the modified Sardar sub-equation technique $\mathrm{MSST}$ and $\mathrm{DT}$, to find soliton solutions.

Additionally, we improve the discussion on the theoretical implications of our research. We illustrate our ability to achieve significant theoretical advancement by describing the exact analytical framework used and highlighting the innovative methods used, such as the $\mathrm{MSST}$ and $\mathrm{DT}$. With the help of these innovations, we identify new solitons, which are essential for comprehending the behavior of the (3+1)-dimensional KP model. The discussion of the potential applications resulting from the derived soliton solutions is expanded. Numerous disciplines are included in these applications, including wave phenomena, nonlinear optics, fluid dynamics, and plasma physics.

The layout of the paper is as follows: Section 2 explains the general description of the used techniques and the application of the governing model. In Section 3, we discuss the $\mathrm{DT}$ of the given model. The results and discussion are provided in Section 4. Section 5 provides the conclusion.

2. General Description and Application of the Proposed Methods

This section provides a detailed explanation of the methodology used for Equation (PDE) to produce soliton solutions. The following form is the generic NLPDE:

$$\mathcal{S}(w,w_x,w_t,w_{xx},w_{xt},...)=0,$$

(4)

where $w=w(x,t)$ is an unexplained function. Applying the transformation

$$w(x,t)=\mathcal{W}\left(\tau \right),\tau =x-vt,$$

(5)

Equation (1) can be changed into an ODE, as

$$\mathcal{R}(\mathcal{W},{\mathcal{W}}^{\prime },{\mathcal{W}}^{″},...)=0,$$

(6)

where v and ${\mathcal{W}}^{\prime }$ represent the constant velocity and $\frac{dw}{d\tau }$, respectively.

The methodology is described in the following subsection.

2.1. The Modified Sardar Sub-Equation Technique

The given form describes the general solution of Equation (6), as per the method.

$$\mathcal{W}\left(\tau \right)=R_0+\sum _{i=1}^N{R}_i{\mathcal{P}}^i\left(\tau \right),R_i\ne 0,$$

(7)

where $W=W\left(\tau \right)$, and the following equation is the first-order differential equation that satisfies the general solution to Equation (7).

$${\mathcal{P}}^{\prime }{\left(\tau \right)}^2={ϵ}_2\mathcal{P}{\left(\tau \right)}^4+{ϵ}_1\mathcal{P}{\left(\tau \right)}^2+{ϵ}_0,$$

(8)

where ${ϵ}_0\ne 1$ and ${ϵ}_1$ and ${ϵ}_2\ne 0$ are integers. We calculate the constants $R_0,R_1$, and $R_2$, and additionally, it is possible for $R_i$ to be zero. We determine the value of N using the balance principle. Following are the solutions to Equation (8).

• Case 1:

• If ${ϵ}_0=0, {ϵ}_1>0$, and ${ϵ}_2\ne 0$, then

  $${\mathcal{P}}_1\left(\tau \right)=\sqrt{-\frac{{ϵ}_1}{{ϵ}_2}}\mathrm{sech}\left(\sqrt{{ϵ}_1}(\eta +\tau )\right),$$

  (9)
• If ${ϵ}_0=0, {ϵ}_1>0$, and ${ϵ}_2\ne 0$, then

  $${\mathcal{P}}_2\left(\tau \right)=\sqrt{-\frac{{ϵ}_1}{{ϵ}_2}}\mathrm{csch}\left(\sqrt{{ϵ}_1}(\eta +\tau )\right).$$

  (10)

• Case 2:

• For constants $f_1$ and $f_2$, let ${ϵ}_0=0, {ϵ}_1>0$, and ${ϵ}_2=+4f_1{f}_2$; then,

  $${\mathcal{P}}_3\left(\zeta \right)=\frac{4f_1\sqrt{{ϵ}_1}}{\left(4f_1^2-{ϵ}_2\right)sinh\left(\sqrt{{ϵ}_1}(\eta +\tau )\right)+\left(4f_1^2-{ϵ}_2\right)cosh\left(\sqrt{{ϵ}_1}(\eta +\tau )\right)}.$$

  (11)

• Case 3:

• For constants $R_1$ and $R_2$, let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1<0,$ and ${ϵ}_2>0$; then,

  $${\mathcal{P}}_4\left(\tau \right)=\sqrt{-\frac{{ϵ}_1}{2{ϵ}_2}}tanh\left(\sqrt{-\frac{{ϵ}_1}{2}}(\eta +\tau )\right).$$

  (12)
• For constants $R_1$ and $R_2$, let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1<0$, and ${ϵ}_2>0$; then,

  $${\mathcal{P}}_5\left(\tau \right)=\sqrt{-\frac{{ϵ}_1}{2{ϵ}_2}}coth\left(\sqrt{-\frac{{ϵ}_1}{2}}(\eta +\tau )\right).$$

  (13)
• For constants $R_1$ and $R_2$, let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1<0$, and ${ϵ}_2>0$; then,

  $${\mathcal{P}}_6\left(\tau \right)=\sqrt{-\frac{{ϵ}_1}{2{ϵ}_2}}\left(tanh\left(\sqrt{-\frac{{ϵ}_1}{2}}(\eta +\tau )\right)+i\mathrm{sech}\left(\sqrt{-2{ϵ}_1}(\eta +\tau )\right)\right).$$

  (14)
• For constants $R_1$ and $R_2$, let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1<0$, and ${ϵ}_2>0$; then,

  $${\mathcal{P}}_7\left(\tau \right)=\sqrt{-\frac{{ϵ}_1}{8{ϵ}_2}}\left(tanh\left(\sqrt{-\frac{{ϵ}_1}{8}}(\eta +\tau )\right)+coth\left(\sqrt{-\frac{{ϵ}_1}{8}}(\eta +\tau )\right)\right).$$

  (15)
• For constants $R_1$ and $R_2$, let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1<0$, and ${ϵ}_2>0$; then,

  $${\mathcal{P}}_8\left(\tau \right)=\frac{\sqrt{-\frac{{ϵ}_1}{2{ϵ}_2}}cosh\left(\sqrt{-2{ϵ}_1}(\eta +\tau )\right)}{sinh\left(\sqrt{-2{ϵ}_1}(\eta +\tau )\right)+i}.$$

  (16)

• Case 4:

• Let ${ϵ}_0=0,{ϵ}_1<0$, and ${ϵ}_2\ne 0$; then,

  $${\mathcal{P}}_9\left(\tau \right)=\sqrt{-\frac{{ϵ}_1}{{ϵ}_2}}sec\left(\sqrt{-{ϵ}_1}(\eta +\tau )\right).$$

  (17)
• Let ${ϵ}_0=0,{ϵ}_1<0$, and ${ϵ}_2\ne 0$; then,

  $${\mathcal{P}}_{10}\left(\tau \right)=\sqrt{-\frac{{ϵ}_1}{{ϵ}_2}}csc\left(\sqrt{-{ϵ}_1}(\eta +\tau )\right).$$

  (18)

• Case 5:

• Let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1>0$, ${ϵ}_2>0$, and $R_1^2-R_2^2>0$; then,

  $${\mathcal{P}}_{11}\left(\tau \right)=\sqrt{-\frac{{ϵ}_1}{2{ϵ}_2}}tan\left(\sqrt{\frac{{ϵ}_1}{2}}(\eta +\tau )\right).$$

  (19)
• Let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1>0$, ${ϵ}_2>0$, and $R_1^2-R_2^2>0$; then,

  $${\mathcal{P}}_{12}\left(\tau \right)=-\sqrt{-\frac{{ϵ}_1}{2{ϵ}_2}}cot\left(\sqrt{\frac{{ϵ}_1}{2}}(\eta +\tau )\right).$$

  (20)
• Let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1>0$, ${ϵ}_2>0$, and $R_1^2-R_2^2>0$; then,

  $${\mathcal{P}}_{13}\left(\tau \right)=-\sqrt{-\frac{{ϵ}_1}{2{ϵ}_2}}\left(tan\left(\sqrt{2{ϵ}_1}(\eta +\tau )\right)-sec\left(\sqrt{2{ϵ}_1}(\eta +\tau )\right)\right).$$

  (21)
• Let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1>0$, ${ϵ}_2>0$, and $R_1^2-R_2^2>0$; then,

  $${\mathcal{P}}_{14}\left(\tau \right)=\sqrt{-\frac{{ϵ}_1}{8{ϵ}_2}}\left(tan\left(\sqrt{\frac{{ϵ}_1}{8}}(\eta +\tau )\right)-cot\left(\sqrt{\frac{{ϵ}_1}{8}}(\eta +\tau )\right)\right).$$

  (22)
• Let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1>0$, ${ϵ}_2>0$, and $R_1^2-R_2^2>0$; then,

  $${\mathcal{P}}_{15}\left(\tau \right)=\frac{\sqrt{-\frac{{ϵ}_1}{2{ϵ}_2}}\left(\sqrt{R_1^2-R_2^2}-W_{1}cos\left(\sqrt{2{ϵ}_1}(\eta +\tau )\right)\right)}{R_2+W_{1}sin\left(\sqrt{2{ϵ}_1}(\eta +\tau )\right)},$$

  (23)

  $${\mathcal{P}}_{16}\left(\tau \right)=\frac{\sqrt{-\frac{{ϵ}_1}{2{ϵ}_2}}cos\left(\sqrt{2{ϵ}_1}(\eta +\tau )\right)}{sin\left(\sqrt{2{ϵ}_1}(\eta +\tau )\right)-1}.$$

  (24)

• Case 6:

• Let ${ϵ}_0=0$ and ${ϵ}_1>0;$ then,

  $${\mathcal{P}}_{17}\left(\tau \right)=\frac{4{ϵ}_1{e}^{\sqrt{{ϵ}_1}(\eta +\tau )}}{e^{2\sqrt{{ϵ}_1}(\eta +\tau )}-4{ϵ}_1{ϵ}_2}.$$

  (25)
• Let ${ϵ}_0=0$ and ${ϵ}_1>0;$ then,

  $${\mathcal{P}}_{18}\left(\tau \right)=\frac{4{ϵ}_1{e}^{\sqrt{{ϵ}_1}(\eta +\tau )}}{1-4{ϵ}_1{ϵ}_2{e}^{2\sqrt{{ϵ}_1}(\eta +\tau )}}.$$

  (26)

• Case 7:

• Let ${ϵ}_0=0, {ϵ}_1=0$, and ${ϵ}_2>0$; then,

  $${\mathcal{H}}_{19}\left(\tau \right)=\frac{1}{\sqrt{{ϵ}_2}(\eta +\tau )}.$$

  (27)
• Let ${ϵ}_0=0, {ϵ}_1=0$, and ${ϵ}_2>0$; then,

  $${\mathcal{P}}_{20}\left(\tau \right)=\frac{i}{\sqrt{{ϵ}_2}(\eta +\tau )}.$$

  (28)

2.2. Application of the Modified Sardar Sub-Equation Technique

The extended form of the (3+1)-dimensional Kadomtsev–Petviashvili model reads as

$$\begin{array}{c}\hfill \alpha {\mathcal{M}}_{xt}-\frac{{\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4}{16}{\mathcal{M}}_{xxxx}-\frac{3({\beta }^2-{\alpha }^2)}{4}{\left({\mathcal{M}}^2\right)}_{xx}+a{\mathcal{M}}_{xx}+b{\mathcal{M}}_{xy}+c{\mathcal{M}}_{xz}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+d{\mathcal{M}}_{zz}+e{\mathcal{M}}_{yz}+\frac{e^2}{4d}{\mathcal{M}}_{yy}=0,\end{array}$$

(29)

where $\beta ,a,b,c,d,$ and e are real coefficients, but $\alpha ,d\ne 0$, and $\mathcal{M}=s(x,y,z,t)$ is a function that, with relation to the spatial and temporal variables, is sufficiently differentiable. The modification described by

$$\mathcal{M}(x,y,z,t)=s\left(\tau \right),\tau =x+y+z-vt,$$

(30)

is used to reduce Equation (1) into the following ordinary differential equation

$$-\frac{{\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4}{16}{s}^{″}-\frac{3({\beta }^2-{\alpha }^2)}{4}{s}^2+(a+b+c+d+e+\frac{e^2}{4d}-\alpha v)s=0,$$

(31)

which is second-order nonlinear. Applying the homogeneous balancing principle, we obtain $N=2$. By utilizing the modified Sardar sub-equation technique, we consider the following form of analytical solution

$$\mathcal{M}\left(\tau \right)=R_2\mathcal{P}{\left(\tau \right)}^2+R_1\mathcal{P}\left(\tau \right)+R_0.$$

(32)

Putting Equation (27) into Equation (26) and utilizing Equation (8), we obtain an algebraic system by equating all the coefficients of the various powers of $\left(\mathcal{P}\right(\ll \left)\right)$ to zero. The following is the solution to the system of algebraic equations.

Family 1:

$$\begin{array}{c}\hfill \{R_0\to \frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2},\\ \hfill R_1\to 0,R_2\to \frac{{ϵ}_2\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)}{2\left({\alpha }^2-{\beta }^2\right)},a\to \frac{1}{4}(\frac{\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{{\alpha }^2-{\beta }^2}\\ \hfill -4b-4c-\frac{e^2}{d}-4d-4e+4\alpha v\left)\right\}.\end{array}$$

(33)

The following solutions have been acknowledged as satisfying Family 1:

• Case 1:

• If ${ϵ}_0=0, {ϵ}_1>0$, and ${ϵ}_2\ne 0$, then

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,1}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right){\mathrm{sech}}^2\left(\sqrt{{ϵ}_1}(\eta -tv+x+y+z)\right)}{2\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (34)
• If ${ϵ}_0=0, {ϵ}_1>0$, and ${ϵ}_2\ne 0$, then

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,2}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right){\mathrm{csch}}^2\left(\sqrt{{ϵ}_1}(\eta -tv+x+y+z)\right)}{2\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (35)

• Case 2:

• For constants $f_1$ and $f_2$, let ${ϵ}_0=0, {ϵ}_1>0$, and ${ϵ}_2=+4f_1{f}_2$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,3}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}+\\ \hfill \frac{8f_1^2{ϵ}_1{ϵ}_2\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)}{\left({\alpha }^2-{\beta }^2\right)\left(\left(4f_1^2-{ϵ}_2\right)sinh\left(\sqrt{{ϵ}_1}(\eta -tv+x+y+z)\right)+\left(4f_1^2-{ϵ}_2\right)cosh\left(\sqrt{{ϵ}_1}(\eta -tv+x+y+z)\right)\right){}^2}.\end{array}$$

  (36)

• Case 3:

• For constants $R_1$ and $R_2$, let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1<0,$ and ${ϵ}_2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,4}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right){tanh}^2\left(\frac{\sqrt{-{ϵ}_1}(\eta -tv+x+y+z)}{\sqrt{2}}\right)}{4\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (37)
• For constants $R_1$ and $R_2$, let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1<0$, and ${ϵ}_2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,5}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right){coth}^2\left(\frac{\sqrt{-{ϵ}_1}(\eta -tv+x+y+z)}{\sqrt{2}}\right)}{4\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (38)
• For constants $R_1$ and $R_2$, let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2},{ϵ}_1<0,$ and ${ϵ}_2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,6}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)\left(tanh\left(\frac{\sqrt{-{ϵ}_1}(\eta -tv+x+y+z)}{\sqrt{2}}\right)+i\mathrm{sech}\left(\sqrt{2}\sqrt{-{ϵ}_1}(\eta -tv+x+y+z)\right)\right){}^2}{4\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (39)
• For constants $R_1$ and $R_2$, let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1<0,$ and ${ϵ}_2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,7}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)\left(tanh\left(\frac{\sqrt{-{ϵ}_1}(\eta -tv+x+y+z)}{2\sqrt{2}}\right)+coth\left(\frac{\sqrt{-{ϵ}_1}(\eta -tv+x+y+z)}{2\sqrt{2}}\right)\right){}^2}{16\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (40)
• For constants $R_1$ and $R_2$, let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1<0,$ and ${ϵ}_2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,8}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right){cosh}^2\left(\sqrt{2}\sqrt{-{ϵ}_1}(\eta -tv+x+y+z)\right)}{4\left({\alpha }^2-{\beta }^2\right)\left(sinh\left(\sqrt{2}\sqrt{-{ϵ}_1}(\eta -tv+x+y+z)\right)+i\right){}^2}.\end{array}$$

  (41)

• Case 4:

• Let ${ϵ}_0=0,{ϵ}_1<0$, and ${ϵ}_2\ne 0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,9}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right){sec}^2\left(\sqrt{-{ϵ}_1}(\eta -tv+x+y+z)\right)}{2\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (42)
• Let ${ϵ}_0=0,{ϵ}_1<0$, and ${ϵ}_2\ne 0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,10}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right){csc}^2\left(\sqrt{-{ϵ}_1}(\eta -tv+x+y+z)\right)}{2\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (43)

• Case 5:

• Let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1>0$, ${ϵ}_2>0$, and $R_1^2-R_2^2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,11}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right){tan}^2\left(\frac{\sqrt{{ϵ}_1}(\eta -tv+x+y+z)}{\sqrt{2}}\right)}{4\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (44)
• Let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1>0, {ϵ}_2>0$, and $R_1^2-R_2^2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,12}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right){cot}^2\left(\frac{\sqrt{{ϵ}_1}(\eta -tv+x+y+z)}{\sqrt{2}}\right)}{4\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (45)
• Let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1>0, {ϵ}_2>0$, and $R_1^2-R_2^2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,13}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)\left(tan\left(\sqrt{2}\sqrt{{ϵ}_1}(\eta -tv+x+y+z)\right)-sec\left(\sqrt{2}\sqrt{{ϵ}_1}(\eta -tv+x+y+z)\right)\right){}^2}{4\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (46)
• Let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1>0, {ϵ}_2>0,$ and $R_1^2-R_2^2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,14}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)\left(tan\left(\frac{\sqrt{{ϵ}_1}(\eta -tv+x+y+z)}{2\sqrt{2}}\right)-cot\left(\frac{\sqrt{{ϵ}_1}(\eta -tv+x+y+z)}{2\sqrt{2}}\right)\right){}^2}{16\left({\alpha }^2-{\beta }^2\right)}.\end{array}$$

  (47)
• Let ${ϵ}_0=\frac{{ϵ}_1^2}{4{ϵ}_2}, {ϵ}_1>0, {ϵ}_2>0$, and $R_1^2-R_2^2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,15}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)\left(\frac{1}{2}\sqrt{-\frac{{ϵ}_2^2{\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)}^2}{{\left({\alpha }^2-{\beta }^2\right)}^2}}-W_{1}cos\left(\sqrt{2}\sqrt{{ϵ}_1}(\eta -tv+x+y+z)\right)\right){}^2}{4\left({\alpha }^2-{\beta }^2\right)\left(\frac{{ϵ}_2\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)}{2\left({\alpha }^2-{\beta }^2\right)}+W_{1}sin\left(\sqrt{2}\sqrt{{ϵ}_1}(\eta -tv+x+y+z)\right)\right){}^2}.\end{array}$$

  (48)

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,16}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}-\\ \hfill \frac{{ϵ}_1\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right){cos}^2\left(\sqrt{2}\sqrt{{ϵ}_1}(\eta -tv+x+y+z)\right)}{4\left({\alpha }^2-{\beta }^2\right)\left(sin\left(\sqrt{2}\sqrt{{ϵ}_1}(\eta -tv+x+y+z)\right)-1\right){}^2}.\end{array}$$

  (49)

• Case 6:

• Let ${ϵ}_0=0$ and ${ϵ}_1>0;$ then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,17}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}+\\ \hfill \frac{8{ϵ}_2{ϵ}_1^2\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)e^{2\sqrt{{ϵ}_1}(\eta -tv+x+y+z)}}{\left({\alpha }^2-{\beta }^2\right)\left(e^{2\sqrt{{ϵ}_1}(\eta -tv+x+y+z)}-4{ϵ}_1{ϵ}_2\right){}^2}.\end{array}$$

  (50)
• Let ${ϵ}_0=0$ and ${ϵ}_1>0;$ then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,18}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}+\\ \hfill \frac{8{ϵ}_2{ϵ}_1^2\left({\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4\right)e^{2\sqrt{{ϵ}_1}(\eta -tv+x+y+z)}}{\left({\alpha }^2-{\beta }^2\right)\left(1-4{ϵ}_1{ϵ}_2{e}^{2\sqrt{{ϵ}_1}(\eta -tv+x+y+z)}\right){}^2}.\end{array}$$

  (51)

• Case 7:

• Let ${ϵ}_0=0, {ϵ}_1=0,$ and ${ϵ}_2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,19}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}+\\ \hfill \frac{{\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4}{2\left({\alpha }^2-{\beta }^2\right){(\eta -tv+x+y+z)}^2}.\end{array}$$

  (52)
• Let ${ϵ}_0=0, {ϵ}_1=0,$ and ${ϵ}_2>0$; then,

  $$\begin{array}{c}\hfill {\mathcal{M}}_{1,20}=\frac{{ϵ}_1\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)-\sqrt{\left({ϵ}_1^2-3{ϵ}_0{ϵ}_2\right){\left({\alpha }^6-7{\alpha }^4{\beta }^2+7{\alpha }^2{\beta }^4-{\beta }^6\right)}^2}}{6{\left({\alpha }^2-{\beta }^2\right)}^2}\\ \hfill -\frac{{\alpha }^4-6{\alpha }^2{\beta }^2+{\beta }^4}{2\left({\alpha }^2-{\beta }^2\right){(\eta -tv+x+y+z)}^2}.\end{array}$$

  (53)

3. Darboux Transformation

The corresponding Lax representation of Equation (1) takes the form:

$${\mathsf{\Psi }}_x=V\mathsf{\Psi },{\mathsf{\Psi }}_t=(2\alpha \mu +4\beta {\mu }^2){\mathsf{\Psi }}_y+(+6\alpha {\mu }^3){\mathsf{\Psi }}_z+U\mathsf{\Psi },$$

(54)

in which

$$\begin{array}{c}\hfill V=-i\mu {\omega }_1+V_0,U=\mu U_1+U_0+\frac{i}{\mu +\gamma }{U}_{-1},\\ \\ \hfill {\omega }_1=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),V_0=\left(\begin{array}{cc}0& m\\ -n& 0\end{array}\right),U_{-1}=\left(\begin{array}{cc}\xi & -l\\ -q& \xi \end{array}\right),\\ \\ \hfill U_1=\gamma {\omega }_1+2i\beta {\omega }_1{V}_{0y},U_0=-\frac{i}{2}u{\omega }_1+\left(\begin{array}{cc}0& -\alpha m_y-\beta m_{xy}+i\gamma m\\ i\alpha n_y+\beta n_{xy}-i\gamma n& 0\end{array}\right).\end{array}$$

If $\lambda$ is a complex spectral parameter that is independent of time $\left(t\right)$, then $\mathsf{\Psi }=\mathsf{\Psi }(x,t)={(\psi (x,t),\varphi (x,t))}^{\mathcal{T}}$ ($\mathcal{T}$) denote the transpose of the complex eigenfunctions. The zero curvatures equation $V_t-U_x+\left[V,U\right]=0,$ where V and U are $2\times 2$ matrices that meet the linear isospectral problem, is identical to Equation (1). According to Equation (54), the Lax pair of Equation (1) is:

$$\mathsf{\Psi }=V\mathsf{\Psi },V=\left(\begin{array}{cc}-i\mu & m\\ -\sigma m^{\ast }& i\mu \end{array}\right),$$

(55)

$${\mathsf{\Psi }}_t=U\mathsf{\Psi },U=\left(\begin{array}{cc}\frac{i\xi }{\mu +\gamma }& -\frac{i}{2(\xi +\gamma )}{m}_t\\ \\ -\frac{i\sigma }{2(\xi +\gamma )}{m}_t^{\ast }& -\frac{i\xi }{\mu +\gamma }\end{array}\right).$$

(56)

3.1. The N-Fold $\mathrm{DT}$

The N-fold $\mathrm{DT}$ of (1) is developed in this section. First, we apply the gauge modification, shown as follows.

$$\tilde{\mathsf{\Psi }}=\mathcal{T}\mathsf{\Psi },$$

(57)

whereas $\mathsf{\Psi }$ satisfies the Lax pairs Equations (55) and (56), $\mathcal{T}$ is a $2\times 2$ Darboux matrix, which is provided here, and $\tilde{\mathsf{\Psi }}=\tilde{\mathsf{\Psi }}(x,t)={(\tilde{\varphi }(x,t),\tilde{\phi }(x,t))}^{\mathcal{T}}$ should maintain the same shape as the Lax pairing. Equations (4) and (5), excluding the replacement of $VandU$ with $\tilde{\mathcal{V}}and\tilde{\mathcal{U}}$, are

$${\tilde{\mathsf{\Psi }}}_x=\tilde{\mathcal{V}}\tilde{\mathsf{\Psi }},\tilde{\mathcal{V}}=\left(\begin{array}{cc}-i\mu & \tilde{m}\\ -\sigma {\tilde{m}}^{\ast }& i\mu \end{array}\right),$$

(58)

$${\tilde{\mathsf{\Psi }}}_t=\tilde{\mathcal{U}}\tilde{\mathsf{\Psi }},\tilde{\mathcal{U}}=\left(\begin{array}{cc}\frac{i\tilde{\xi }}{\mu +\gamma }& -\frac{i}{2(\mu +\gamma )}\frac{\partial \tilde{m}}{\partial t}\\ \\ -\frac{i\sigma }{2(\mu +\gamma )}\frac{\partial {\tilde{m}}^{\ast }}{\partial t}& -\frac{i\tilde{\xi }}{\mu +\gamma }\end{array}\right),$$

(59)

where $\tilde{m}=\tilde{m}(x,t),$ and $\tilde{\xi }=\tilde{\xi }(x,t)$ are novel prospective capabilities that can also address Equation (1). Using Equations (55)–(59), we have

$$\tilde{\mathcal{V}}=({\mathcal{T}}_x+\mathcal{T}V){\mathcal{T}}^{-1},\tilde{\mathcal{U}}=({\mathcal{T}}_t+\mathcal{T}U){\mathcal{T}}^{-1},$$

(60)

wherein we deduce

$${\tilde{\mathcal{V}}}_t-{\tilde{\mathcal{U}}}_x+\tilde{\mathcal{V}}\tilde{\mathcal{U}}-\tilde{\mathcal{U}}\tilde{\mathcal{V}}=\mathcal{T}(V_t-U_x+VU-UV){\mathcal{T}}^{-1},$$

(61)

considering that the equation for zero curvature $V_t-U_x+VU-UV$ in the Darboux matrices is zero $\mathcal{T}$, that is, nonsingular. So, ${\tilde{\mathcal{V}}}_t-{\tilde{\mathcal{U}}}_x+\tilde{\mathcal{V}}\tilde{\mathcal{U}}-\tilde{\mathcal{U}}\tilde{\mathcal{V}}=0$, and it can also provide the same Equation (1) in m, $\xi \to \tilde{m},\tilde{\xi }$, i.e., $\tilde{m},\tilde{\xi }$ are still a solution for Equation (1) determined by an innovative spectral issue of Equations (58) and (59). In other words, for solitons, we can generate novel solutions to the existing ones via the gauge transformation. It is crucial to choose a suitable Darboux matrix ($\mathcal{T}$). By doing this, we develop a certain $\mathcal{T}$ in the following form:

$$\mathcal{T}=\mathcal{T}\left(\mu \right)=\left(\begin{array}{cc}{\mathcal{T}}_{11}\left(\mu \right)& {\mathcal{T}}_{12}\left(\mu \right)\\ \\ {\mathcal{T}}_{21}\left(\mu \right)& {\mathcal{T}}_{22}\left(\mu \right)\end{array}\right)=\left(\begin{array}{cc}{\gamma }^M+\sum _{i=0}^{M-1}{\mathcal{A}}^i{\gamma }^i& \sum _{i=0}^{M-1}{\mathcal{B}}^i{\gamma }^i\\ \\ -\sigma \sum _{i=0}^{M-1}{\mathcal{B}}^{i^{\ast }}{\gamma }^i& {\gamma }^M+\sum _{i=0}^{M-1}{\mathcal{A}}^{i^{\ast }}{\gamma }^i\end{array}\right),$$

(62)

whereas the complex variables ${\mathcal{A}}^i$ and ${\mathcal{B}}^i(i=0,1,...,M-1)$ are $2M$. When the linear arithmetic problem is solved, it is possible to provide indeterminate functions $\mathcal{T}{\left(\mu \right)}_q{\mathsf{\Psi }}_q{\left(\mu \right)}_q=0(q=1,2,...,M)$ with $2M$ equations, i.e.,

$$\begin{array}{c}\hfill \left[{\mu }_q^M+\sum _{i=0}^{M-1}{\mathcal{A}}^i\left({\mu }_q\right){\mu }_q^i\right]{\varphi }_q\left({\mu }_q\right)+\left[\sum _{i=0}^{M-1}{\mathcal{B}}^i\left({\mu }_q\right){\mu }_q^i\right]{\phi }_q\left({\mu }_q\right)=0,\\ \\ \hfill \left[-\sigma +\sum _{i=0}^{M-1}{\mathcal{B}}^{i^{\ast }}\left({\mu }_q\right){\mu }_q^i\right]{\varphi }_q\left({\mu }_q\right)+\left[{\mu }_q^M+\sum _{i=0}^{M-1}{\mathcal{B}}^i\left({\mu }_q\right){\mu }_q^i\right]{\phi }_q\left({\mu }_q\right)=0,\end{array}$$

(63)

as ${\mathsf{\Psi }}_q\left({\mu }_q\right)=\left({\varphi }_q\right({\mu }_q,{\phi }_q{\left({\mu }_q\right)}^{\mathcal{T}}={({\varphi }_q,{\phi }_q)}^{\mathcal{T}}(q=1,2,...,M)$, which are M solutions for the Lax pairing of Equations (55) and (56) regarding the particular spectrum characteristics ${\mu }_q$ and the original solutions $m_0=m_0(x,t),{\xi }_0={\xi }_0(x,t),{\mu }_q({\mu }_q\ne {\mu }_i,q\ne i,q,i=1,2,...,M)$, according to the determinant of the equation’s coefficients, have certain diverse parameters that have been chosen appropriately (63).

$$det\mathcal{T}\left(\mu \right)=\prod _{q=1}^M\left(\mu -{\mu }_q\right)\left(\mu -{\mu }_q^{\ast }\right).$$

(64)

We substitute Equation (62) into Equation (59) with Equation (63).

Theorem 1.

Assume ${\mathsf{\Psi }}_1\left({\mu }_1\right),{\mathsf{\Psi }}_2\left({\mu }_2\right),\dots ,{\mathsf{\Psi }}_M\left({\mu }_M\right)$ are the responses to M, a varied unit vector addressing the spectral issues for Equations (55) and (56) of the spectrum parameter ${\mu }_1,{\mu }_2,\dots ,{\mu }_M$ and the original solution $m_0,{\xi }_0$ to Equation (1); then, using the following formulas,

$${\tilde{m}}_M=m_0+2i{\mathcal{B}}^{(M-1)},{\tilde{\xi }}_M={\xi }_0-i{\mathcal{A}}_t^{(M-1)},$$

(65)

where ${\mathcal{B}}^{(M-1)}=\frac{\Delta {\mathcal{B}}^{(M-1)}}{{\Delta }_M},{\mathcal{A}}^{(M-1)}=\frac{\Delta {\mathcal{A}}^{(M-1)}}{{\Delta }_M}$ with

$${\Delta }_M=\left|\begin{array}{cccccccc}{\mu }_1^{M-1}{\varphi }_1& {\mu }_1^{M-2}{\varphi }_1& \dots & {\varphi }_1& {\mu }_1^{M-1}{\phi }_1& {\mu }_1^{M-2}{\phi }_1& \dots & {\phi }_1\\ {\mu }_2^{M-1}{\varphi }_2& {\mu }_2^{M-2}{\varphi }_2& \dots & {\varphi }_2& {\mu }_2^{M-1}{\varphi }_2& {\mu }_2^{M-2}{\phi }_2& \dots & {\phi }_2\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ {\mu }_M^{M-1}{\varphi }_M& {\mu }_M^{M-2}{\varphi }_M& \dots & {\varphi }_M& {\mu }_M^{M-1}{\phi }_M& {\mu }_M^{M-2}{\phi }_M& \dots & {\phi }_M\\ {\left({\mu }_1^{M-1}\right)}^{\ast }{\phi }_1^{\ast }& {\left({\mu }_1^{M-2}\right)}^{\ast }{\phi }_1^{\ast }& \dots & {\phi }_1^{\ast }& -{\sigma }^{\ast }{\left({\mu }_1^{M-1}\right)}^{\ast }{\varphi }_1^{\ast }& -{\sigma }^{\ast }{\left({\mu }_1^{M-2}\right)}^{\ast }{\varphi }_1^{\ast }& \dots & -{\sigma }^{\ast }{\varphi }_1^{\ast }\\ {\left({\mu }_2^{M-1}\right)}^{\ast }{\phi }_2^{\ast }& {\left({\mu }_2^{M-2}\right)}^{\ast }{\phi }_2^{\ast }& \dots & {\phi }_2^{\ast }& -{\sigma }^{\ast }{\left({\mu }_2^{M-1}\right)}^{\ast }{\varphi }_2^{\ast }& -{\sigma }^{\ast }{\left({\mu }_2^{M-2}\right)}^{\ast }{\varphi }_2^{\ast }& \dots & -{\sigma }^{\ast }{\varphi }_2^{\ast }\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ {\left({\mu }_M^{M-1}\right)}^{\ast }{\phi }_M^{\ast }& {\left({\mu }_M^{M-2}\right)}^{\ast }{\phi }_M^{\ast }& \dots & {\phi }_M^{\ast }& -{\sigma }^{\ast }{\left({\mu }_M^{M-1}\right)}^{\ast }{\varphi }_M^{\ast }& -{\sigma }^{\ast }{\left({\mu }_M^{M-2}\right)}^{\ast }{\varphi }_M^{\ast }& \dots & -{\sigma }^{\ast }{\varphi }_M^{\ast }\end{array}\right|,$$

while $\Delta {\mathcal{A}}^{(M-1)}$ and $\Delta {\mathcal{B}}^{(M-1)}$ are expressed through the determinant ${\Delta }_M$ by replacing its first and $(M+1)$-th columns with the column vector ${(-{\mu }_1^M{\varphi }_1,\cdots ,-{\mu }_M^M{\varphi }_M,-{\left({\mu }_1^M\right)}^{\ast }{\phi }_1^{\ast },-{\left({\mu }_2^M\right)}^{\ast }{\phi }_2^{\ast },\cdots ,-{\left({\phi }_M^M\right)}^{\ast }{\phi }_M^{\ast })}^{\mathcal{T}}$, respectively.

The $\mathit{N}-\mathit{fold}$ $\mathrm{DT}$ of Equation (1) is transformed to Equations (57) and (65) using the spectrum variables M and ${\mu }_q$. Theorem 1’s proof is provided by simple calculation; the readers are left with the task of determining the specifics of the proof’s procedure in light of the fact that they can consult the pertinent literature [27] for further information.

3.2. Asymptotic State Analysis and Solutions to Bright–Dark Multi-Soliton Systems

By utilizing the $\mathrm{DT}$ obtained in the preceding part, we can generate solitons on a constant background of Equation (1). In order to produce a single fundamental solution with $\mu ={\mu }_q$ as shown below, we first take the initial solutions $m_0=0$ and ${\xi }_0=1$ as the constant solution.

$$\begin{array}{c}\hfill {\mathsf{\Psi }}_q=\left(\begin{array}{c}\hfill {\varphi }_q\hfill \\ \hfill {\phi }_q\hfill \end{array}\right)=\left(\begin{array}{c}{e}^{\frac{i\left(-{\mu }_q^{2}x-{\mu }_q\gamma x+t\right)}{{\mu }_q+\gamma }}\\ e^{\frac{-i\left(-{\mu }_q^{2}x-{\mu }_q\gamma x+t\right)}{{\mu }_q+\gamma }}\end{array}\right).\end{array}$$

(66)

The following always uses the parameters $\gamma =\sigma =1$ to conveniently gain the dynamic properties of soliton solutions. The M-soliton solutions of Equation (1) can therefore be obtained from Equation (65). We depict their structures as seen in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 to better comprehend their physical characteristics and interactions in Figure 14, Figure 15 and Figure 16 when $M=1,2,3,4$.

3.2.1. Dynamic Analysis and One-Soliton Solutions

For $M=1,{\mu }_1=c+di$, where c and d are constant, from Theorem 1, we obtain the solution

$${\tilde{m}}_1=2i\frac{\Delta {\mathcal{B}}^{\left(0\right)}}{{\Delta }_1},{\tilde{\xi }}_1=1-i{\left(\frac{\Delta {\mathcal{A}}^{\left(0\right)}}{{\Delta }_1}\right)}_t$$

(67)

where

$${\Delta }_1=\left|\begin{array}{cc}{\varphi }_1& {\phi }_1\\ {\phi }_1^{\ast }& -{\sigma }^{\ast }{\varphi }_1^{\ast }\end{array}\right|,\Delta {\mathcal{A}}^{\left(0\right)}=\left|\begin{array}{cc}-{\mu }_1{\varphi }_1& {\phi }_1\\ -{\mu }_1^{\ast }{\phi }_1^{\ast }& -{\sigma }^{\ast }{\varphi }_1^{\ast }\end{array}\right|,\Delta {\mathcal{B}}^{\left(0\right)}=\left|\begin{array}{cc}{\varphi }_1& -{\mu }_1{\varphi }_1\\ {\phi }_1^{\ast }& -{\mu }_1^{\ast }{\phi }_1^{\ast }\end{array}\right|.$$

We convert the responses to their equations to examine their physical characteristics in Equation (67) as

$${\tilde{m}}_1=2d{e}^{2i{\zeta }_{Im}}sech\left(2{\zeta }_{Re}\right),{\tilde{\xi }}_1=1-\frac{2d^2}{q}{sech}^2\left(2{\zeta }_{Re}\right),$$

(68)

as ${\zeta }_{Re}=\left(dx+\frac{d}{q}t\right),{\zeta }_{Im}=\left[-cx+(c+1)\frac{t}{q}\right]$, with $q=c^2+d^2+2c+1$.

It is simple to observe that ${\tilde{m}}_1$ is a bright soliton framework, and ${\tilde{\xi }}_1$ is a dark soliton, from the solutions to Equation (68), and the physical characteristics, such as the intensity, dimension, speed, wave numbers, main phase, and energy, which are listed in

Table 1. The one-soliton solutions’ physical characteristics.

Solitons	Intensity	Dimensions	Speed	Wave Numbers	Main Phases	Energies
${\tilde{m}}_1$	$2c$	$\frac{1}{2c}$	$-\frac{1}{q}$	$2c$	0	$4c$
${\tilde{\xi }}_1-1$	$\frac{2c^2}{q}$	$\frac{1}{2c}$	$-\frac{1}{q}$	$2c$	0	$\frac{8c^3}{3q^2}$

, can also be easily probed. For $m_1$ and ${\xi }_1$, the energies are defined as $F_{m_1}={\int }_{-\infty }^{\infty }{\left|m_1\right|}^{2}dx,F_{{\xi }_1}={\int }_{-\infty }^{\infty }{\left({\xi }_1-1\right)}^{2}dx$. We can easily observe the amplitudes and velocities of one-soliton solutions depending on the spectral parameter ${\mu }_1$. Figure 15 and Figure 16 show the bell-shaped bright one-soliton structure of ${\tilde{m}}_1$ and the anti-bell-shaped dark one-soliton structure of ${\tilde{\xi }}_1$, when ${\mu }_1=1+2i$ (i.e., $c=1,d=2$).

3.2.2. Analysis of Two-Soliton Solutions Asymptotically

When $M=2,{\mu }_1=c_1+d_{1}i,{\mu }_2=c_2+d_{2}i$, where $c_1,c_2,d_1,d_2$, according to Theorem 1, are positive independent variables, and twofold $\mathrm{DT}$ can provide the following:

$${\tilde{m}}_2=2i\frac{\mathcal{A}{\mathcal{B}}^{\left(i\right)}}{{\mathcal{A}}_2},{\tilde{\xi }}_2=1-i{\left(\frac{\mathcal{A}{\mathcal{A}}^{\left(1\right)}}{{\mathcal{A}}_2}\right)}_l,$$

(69)

where

$$\widetilde{m_2}=\frac{n_1{e}^{{\zeta }_2^{\ast }}+n_2{e}^{{\zeta }_1^{\ast }+{\zeta }_2+{\zeta }_2^{\ast }}+n_2^{\ast }{e}^{{\zeta }_1^{\ast }}+n_1^{\ast }{e}^{{\zeta }_1+{\zeta }_1^{\ast }+{\zeta }_2^{\ast }}}{n_3\left(e^{{\zeta }_1^{\ast }+{\zeta }_2}+e^{{\zeta }_1+{\zeta }_2^{\ast }}\right)+n_4\left(1+e^{{\zeta }_1+{\zeta }_1^{\ast }+{\zeta }_2+{\zeta }_2^{\ast }}\right)+n_5\left(e^{{\zeta }_1+{\zeta }_1^{\ast }}+e^{{\zeta }_2+{\zeta }_2^{\ast }}\right)},{\tilde{\zeta }}_2=\frac{G_1}{G_2},$$

(70)

where

$$\begin{array}{cc}{G}_1=\hfill & W_{1}cosh\left({\zeta }_1^{\ast }-{\zeta }_2^{\ast }\right)+W_{2}cosh\left({\zeta }_1+{\zeta }_2^{\ast }\right)+W_2^{\ast }cosh\left({\zeta }_1^{\ast }+{\zeta }_2\right)+W_1^{\ast }cosh\left({\zeta }_1-{\zeta }_2\right)+W_{3}cosh\left[\left({\zeta }_2-{\zeta }_2^{\ast }\right)-\left({\zeta }_1-{\zeta }_1^{\ast }\right)\right]\hfill \\ & +W_{4}cosh\left({\zeta }_1+{\zeta }_1^{\ast }+{\zeta }_2+{\zeta }_2^{\ast }\right)+W_{5}cosh\left({\zeta }_2+{\zeta }_2^{\ast }\right)+W_{6}cosh\left[{\zeta }_1+{\zeta }_1^{\ast }-\left({\zeta }_2+{\zeta }_2^{\ast }\right)\right]+W_7+W_{8}cosh\left({\zeta }_1+{\zeta }_1^{\ast }\right),\hfill \\ G_2=\hfill & 2q_1{q}_2{\left[W_{9}cosh\left(\frac{{\zeta }_1-{\zeta }_1^{\ast }-\left({\zeta }_2-{\zeta }_2^{\ast }\right)}{2}\right)+W_{10}cosh\left(\frac{{\zeta }_1+{\zeta }_1^{\ast }+{\zeta }_2+{\zeta }_2^{\ast }}{2}\right)+W_{11}cosh\left(\frac{{\zeta }_1+{\zeta }_1^{\ast }-\left({\zeta }_2+{\zeta }_2^{\ast }\right)}{2}\right)\right]}^2,\hfill \end{array}$$

with

$$\begin{array}{cc}& {\zeta }_1=2\left(d_1+c_{1}i\right)x+\frac{2\left(d_1-c_{1}i-i\right)}{q_1}t,\hfill \\ & {\zeta }_2=-2\left(d_2-c_{2}i\right)x-\frac{2\left(d_2-c_{2}i-i\right)}{q_2}t,\hfill \\ & q_1=c_1^2+d_1^2+2c_1+1,\hfill \\ & q_2=c_2^2+d_2^2+2c_2+1,\hfill \\ & n_1=\left[4d_2{\left(c_1-c_2\right)}^2+4d_2^3-4d_1^2{d}_2\right]-8d_1{d}_2\left(c_1-c_2\right)i,\hfill \\ & n_2=\left[4d_1{\left(c_1-c_2\right)}^2+4d_1^3-4d_1{d}_2^2\right]-8d_1{d}_2\left(c_2-c_1\right)i^2,\hfill \\ & n_3=W_9=-4d_1{d}_2,\hfill \\ & n_4=W_{10}={\left(c_1-c_2\right)}^2+{\left(d_1+d_2\right)}^2,\hfill \\ & n_5=W_{11}={\left(c_1-c_2\right)}^2+{\left(d_1-d_2\right)}^2,\hfill \\ & W_1=\left[-8d_1{d}_2{n}_5\left(c_1^3{c}_2-c_1^2{c}_2^2-c_1^2{d}_1{d}_2-c_1^2{d}_2^2+c_1{c}_2^3+c_1{c}_2{d}_1^2+c_1{c}_2{d}_2^2-c_2^2{d}_1^2-c_2^2{d}_1{d}_2-d_1^3{d}_2-d_1^2{d}_2^2\right.\right.\hfill \\ & -d_1{d}_2^3+c_1^3+c_1^2{c}_2+c_1{c}_2^2+c_1{d}_1^2-2c_1{d}_1{d}_2-c_1{d}_2^2+c_2^3-c_2{d}_1^2-2c_2{d}_1{d}_2+c_2{d}_2^2+2c_1^2+2c_1{c}_2\hfill \\ & \left.\left.+2c_2^2-2d_1{c}_2+2c_1+2c_2+1\right)\right]+8d_1{d}_2{n}_5\left(c_1{d}_2+c_2{d}_1+d_1+d_2\right)\left(c_1^2-c_2^2+d_1^2-d_2^2+2c_1-2c_2\right)i,\hfill \\ & W_2=\left[-8d_1{d}_2{n}_4\left(c_1^3{c}_2-c_1^2{c}_2^2+c_1^2{d}_1{d}_2-c_1^2{d}_2^2+c_1{c}_2^3+c_1{c}_2{d}_1^2+c_1{c}_2{d}_2^2-c_2^2{d}_1^2+c_2^2{d}_1{d}_2+d_1^3{d}_2-d_1^2{d}_2^2\right.\right.\hfill \\ & +d_1{d}_2^3+c_1^3+c_1^2{c}_2+c_1{c}_2^2+c_1{d}_1^2+2c_1{d}_1{d}_2-c_1{d}_2^2+c_2^3-c_2{d}_1^2+2c_2{d}_1{d}_2+c_2{d}_2^2+2c_1^2+2c_1{c}_2\hfill \\ & \left.\left.+2c_2^2+2d_1{d}_2+2c_1+2c_2+1\right)\right]+8d_1{d}_2{n}_4\left(c_1{d}_2-c_2{d}_1-d_1+d_2\right)\left(c_1^2-c_2^2+d_1^2-d_2^2+2c_1-2c_2\right)i,\hfill \end{array}$$

$$\begin{array}{cc}& W_3=d_1^2{d}_2^2\left[16{\left(c_1+1\right)}^2+16d_1^2\right]\left[{\left(c_2+1\right)}^2+d_2^2\right],\hfill \\ & W_4=n_4^2\left[{\left(c_1+1\right)}^2+d_1^2\right]\left[{\left(c_2+1\right)}^2+d_2^2\right],\hfill \\ & W_5=2n_4{n}_5\left[-3d_1^2+{\left(c_1+1\right)}^2\right]\left[{\left(c_2+1\right)}^2+d_2^2\right],\hfill \\ & W_6=n_5^2\left[{\left(c_1+1\right)}^2+d_1^2\right]\left[{\left(c_2+1\right)}^2+d_2^2\right],\hfill \\ & W_7=\left[36d_2^2-12{\left(c_2+1\right)}^2\right]d_1^6+\left\{-8d_2^4+\left[60c_1^2+\left(112c_2+232\right)c_1-96c_2^2-80c_2+76\right]d_2^2-20\left[c_1^2-\left(2.4c_2\right.\right.\right.\hfill \\ & \left.\left.+0.4)c_1+1.2c_2^2-0.2\right]{\left(c_2+1\right)}^2\right\}{d}_1^4+\left\{36d_2^6+\left[-96c_1^2+\left(112c_2-80\right)c_1+60c_2^2+232c_2+76\right]d_2^4\right.\hfill \\ & +\left[12c_1^4+\left(160c_2+208\right)c_1^3+\left(-288c_2^2-96c_2+264\right)c_1^2+\left(160c_2^3-96c_2^2-192c_2+112\right)c_1+12c_2^4\right.\hfill \\ & \left.\left.+208c_2^3+264c_2^2+112c_2+56\right]d_2^2-4{\left(c_1-c_2\right)}^2{\left(c_2+1\right)}^2\left[c_1^2+\left(-6c_2-4\right)c_1+3c_2^2-2\right]\right\}{d}_1^2+4\left[d_2^2\right.\hfill \\ & {\left.+{\left(c_1-c_2\right)}^2\right]}^2\left[-3d_2^2+{\left(c_2+1\right)}^2\right]{\left(c_1+1\right)}^2,\hfill \\ & W_8=2n_4{n}_5\left[{\left(c_1+1\right)}^2+d_1^2\right]\left[-3d_2^2+{\left(c_2+1\right)}^2\right].\hfill \end{array}$$

The expression of Equation (68) is a solution with two solitons. The distinctive feature of an elastic collision is that system’s total energy is unaffected by the collision either before or after it occurs. To determine that whenever two objects collide, the solitons are elastics, the asymptotic of the soliton collision is a very effective technique. These are the two states of the two solitons at zero and infinity. We correlate these to the two temporal patterns that we depict in the propagation process. Of course, we can also speculate about the particular set of parameters $c_1>0,c_2>0,d_1<0,d_2<0$. The examination of the asymptotic states for possibilities of Equation (68) obtains the following eight asymptotic expressions.

(i) With regard to this aspect ${\tilde{m}}_2$ in the addresses for Equation (68):

Before the collision $(t\to -\infty )$,

$$\begin{array}{cc}\hfill & \hfill {\tilde{m}}_1\to {ϵ}_1^{-}=\frac{n_2^{\ast }}{2\sqrt{n_4{n}_5}}{e}^{-{\zeta }_{1Im}i}sech\left({\zeta }_{1Re}-\frac{1}{2}ln\frac{n_4}{n_5}\right),\left({\zeta }_1+{\zeta }_1^{\ast }\sim 0,{\zeta }_2+{\zeta }_2^{\ast }\to -\infty \right),\hfill \\ \hfill & \hfill {\tilde{m}}_1\to {ϵ}_2^{-}=\frac{n_1^{\ast }}{2\sqrt{n_4{n}_5}}{e}^{-{\zeta }_{2Im}i}sech\left({\zeta }_{2Re}+\frac{1}{2}ln\frac{n_4}{n_5}\right),\left({\zeta }_2+{\zeta }_2^{\ast }\sim 0,{\zeta }_1+{\zeta }_1^{\ast }\to +\infty \right),\hfill \end{array}$$

(71)

where ${ϵ}_1^{-}$ and ${ϵ}_2^{-}$ are asymptotic illustrations of ${\tilde{m}}_2$ before they smash into one another.

After a collision $(t\to +\infty )$,

$$\begin{array}{cc}\hfill & \hfill {\tilde{m}}_2\to {ϵ}_1^{+}=\frac{n_2}{2\sqrt{n_4{n}_5}}{e}^{-{\zeta }_{1Im}i}sech\left({\zeta }_{1Re}+\frac{1}{2}ln\frac{n_4}{n_5}\right),\left({\zeta }_1+{\zeta }_1^{\ast }\sim 0,{\zeta }_2+{\zeta }_2^{\ast }\to +\infty \right),\hfill \\ \hfill & \hfill {\tilde{m}}_2\to {ϵ}_2^{+}=\frac{n_1}{2\sqrt{n_4{n}_5}}{e}^{-{\zeta }_{2Im}i}sech\left({\zeta }_{2Re}-\frac{1}{2}ln\frac{n_4}{n_5}\right),\left({\zeta }_2+{\zeta }_2^{\ast }\sim 0,{\zeta }_1+{\zeta }_1^{\ast }\to -\infty \right),\hfill \end{array}$$

(72)

where ${ϵ}_1^{+}$ and ${ϵ}_2^{+}$ are the formulas of $tilde{m}_2$ that asymptotically follow their collision. When assessed by Equations (71) and (72), through direct calculation, we can obtain

(ii) With regard to the element ${\tilde{\xi }}_2$ as solution to Equation (68):

Before the collision $(t\to -\infty )$,

$$\begin{array}{cc}\hfill & \hfill {\tilde{\xi }}_1-1\to {\tau }_1^{-}=-\frac{2d_1^2}{q_1}{sech}^2\left({\zeta }_{1Re}-\frac{1}{2}ln\frac{W_{10}}{W_{11}}\right),\left({\zeta }_1+{\zeta }_1^{\ast }\sim 0,{\zeta }_2+{\zeta }_2^{\ast }\to -\infty \right),\hfill \\ \hfill & \hfill {\tilde{\xi }}_1-1\to {\tau }_2^{-}=-\frac{2d_2^2}{q_2}{sech}^2\left({\zeta }_{2Re}+\frac{1}{2}ln\frac{W_{10}}{W_{11}}\right),\left({\zeta }_2+{\zeta }_2^{\ast }\sim 0,{\zeta }_1+{\zeta }_1^{\ast }\to +\infty \right),\hfill \end{array}$$

(73)

where ${\tau }_1^{-}$ and ${\tau }_2^{-}$, prior to their collision with one another, constitute the asymptotic interpretations of ${\tilde{\xi }}_2-1$.

After a collision $(t\to +\infty )$,

$$\begin{array}{cc}\hfill & \hfill {\tilde{\xi }}_2-1\to {\tau }_1^{+}=-\frac{2d_1^2}{q_1}{sech}^2\left({\zeta }_{1Re}+\frac{1}{2}ln\frac{W_{10}}{W_{11}}\right),\left({\zeta }_1+{\zeta }_1^{\ast }\sim 0,{\zeta }_2+{\zeta }_2^{\ast }\to +\infty \right),\hfill \\ \hfill & \hfill {\tilde{\xi }}_2-1\to {\tau }_2^{+}=-\frac{2d_1^2}{q_1}{sech}^2\left({\zeta }_{1Re}-\frac{1}{2}ln\frac{W_{10}}{W_{11}}\right),\left({\zeta }_2+{\zeta }_2^{\ast }\sim 0,{\zeta }_1+{\zeta }_1^{\ast }\to -\infty \right),\hfill \end{array}$$

(74)

where ${\tau }_1^{+},{\tau }_2^{+}$ represent the exponential formulations of ${\tilde{\xi }}_2-1$ following their collision. We deduce the respective physical characteristics of ${\tilde{\xi }}_2-1$, as shown in

Table 2. Specifications regarding the two-soliton solution’s physical state ${\tilde{m}}_2$.

Soliton	Amplitude	Width	Velocity	Wave Numbers	Primary Phase	Energy
${ϵ}_1^{-}$	$2d_1$	$\frac{1}{2d_1}$	$-\frac{1}{q_1}$	$2d_1$	$\frac{1}{2}ln\frac{n_4}{n_4}$	$4d_1$
${ϵ}_2^{-}$	$2d_2$	$\frac{1}{2d_2}$	$-\frac{1}{q_2}$	$2d_2$	$-\frac{1}{2}ln\frac{n_3}{n_4}$	$4d_2$
${ϵ}_1^{+}$	$2d_1$	$\frac{1}{2d_1}$	$-\frac{1}{q_1}$	$2d_1$	$-\frac{1}{2}ln\frac{n_s}{n_s}$	$4d_1$
${ϵ}_2^{+}$	$2d_2$	$\frac{1}{2d_2}$	$-\frac{1}{q_2}$	$2d_2$	$\frac{1}{2}ln\frac{n_4}{n_4}$	$4d_2$

, from the analysis above.

In view of the aforementioned discussion, we may conclude that the transmission and impact of two solitons possess the following attributes.

(i) The solitons’ amplitudes, lengths, velocity, pulse numbers, formats, and energy are unaffected by the collision in both the before and after states.

(ii) Only the stages of the two solitons that collided have changed, and they are now opposite. As a result, we conclude that the collisions of two solitons become elastic. For a solution to Equation (68), with the parameters ${\gamma }_1=2i$ and ${\gamma }_2=i$, Figure 16 shows the significant two-soliton configurations of $m_2$ and ${\xi }_2$. From this, we can see that part ${\tilde{m}}_2$ is an exceptional two-soliton framework, while part ${\tilde{\xi }}_2$ is a dark two-soliton framework.

4. Results and Discussion

This section covers the comparison of the new work to previous work. Wazwaz et al. [31], studied the (3+1)-dimensional Kadomtsev–Petviashvili equation to find multiple solutions. We compared the innovative soliton solution produced by employing the Sardar substitution and Darboux transformation methods with the preceding article’s analysis of the (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation determined using the streamlined Hirota’s direct method. The soliton solution that was found in the present research is different from the multiple-soliton solutions that Hirota’s direct technique yielded. The solutions of this study have potential applications in a variety of domains, including nonlinear optics, engineering, applied sciences, and communications. These solutions also expand our understanding of nonlinear phenomena. Furthermore, the KP model can be solved using a variety of solutions, and the suggested method’s robustness is demonstrated by the consistency and correctness of the results.

This study was used to investigate the (3+1)-dimensional KP model using two different useful techniques, including the $\mathrm{MSST}$ and the $\mathrm{DT}$. The aim was to obtain one and two bright and dark soliton solutions and multiple solitary wave solutions.

The studies show that any of the two approaches can be used to successfully arrive at the intended results. Solutions for hyperbolic, trigonometric, singular, dark, and bright solitons were found using the $\mathrm{MSST}$. Then, two bright and two dark solitons were created using the $\mathrm{DT}$. The solutions’ physical characteristics were reviewed, and the outcomes were shown using 3-D plots, contour plots, and 2-D curves. The study shows that the obtained solutions are reliable and may be applied to many physical phenomena in various scientific domains. In conclusion, the study illuminates the behavior of the KP model in (3+1)-dimensional space and emphasizes the efficiency of the $\mathrm{MSST}$ and $\mathrm{DT}$ in obtaining various soliton solutions. Future research in mathematical physics, engineering, and applied mathematics may be affected by the findings.

We were able to solve Equation (34)–(53) using the $\mathrm{MSST}$, and the results are provided in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, which show the dark, singular, bright, exponential, and periodic solutions as well as their 3-D, 2-D, and contour plot representations.

We were able to solve Equations (67)–(74) using the $\mathrm{DT}$, and the results are shown in Figure 14 and Figure 15, which represent the one-soliton dark and bright solutions, and Figure 16, which represent the two-soliton solutions provided and represented through 3D, 2D, and contour plots.

Finally, understanding the dynamics and characteristics of the (3+1)-dimensional KP equation improves our understanding of the behavior of nonlinear waves in a variety of physical conditions and advances our understanding of wave phenomena in multidimensional systems.

5. Conclusions

The $\mathrm{MSST}$ and $\mathrm{DT}$ were used in this study to analyze the KP model in (3+1)-dimensional space using two alternative methodologies. The major goal of the study was to find many solitary wave solutions, one and two bright and dark soliton solutions and so on. This paper exhibits the originality of the $\mathrm{MSST}$ and $\mathrm{DT}$ application to the (3+1)-dimensional KP model. The obtained innovative soliton solutions not only improve the knowledge of soliton dynamics in higher-dimensional spaces but also the viability of the suggested approach. The results show the potential of the $\mathrm{MSST}$ and $\mathrm{DT}$ for further research on nonlinear wave phenomena and broaden the KP equation’s applicability. The study’s findings show that both methods are successful in obtaining the desired solutions. While the $\mathrm{DT}$ was utilized to produce two bright and two dark soliton solutions, the $\mathrm{MSST}$ was employed to provide dark, single, bright, exponential, and periodic solutions. The physical properties of the solutions were discussed, and the results were depicted using various visualization techniques. The obtained solutions are stable and can be used to describe various physical phenomena in different fields of research, such as fluid dynamics, plasma physics, and nonlinear optics. Overall, this study contributes to the understanding of the behavior of the KP equation in (3+1)-dimensional space and highlights the effectiveness of the $\mathrm{MSST}$ and the $\mathrm{DT}$ in obtaining different types of soliton solutions. Exploring the (3+1)-dimensional KP model and its soliton solutions has a wide range of potential applications and future research avenues. Researchers can further expand their comprehension of nonlinear wave phenomena while also revealing useful applications in disciplines like optics, fluid dynamics, and plasma physics by exploring these topics.