Analysis of the(3+1)-Dimensional Fractional Kadomtsev–Petviashvili–Boussinesq Equation: Solitary, Bright, Singular, and Dark Solitons

Abstract

We looked at the (3+1)-dimensional fractional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which comes up in fluid dynamics, plasma physics, physics, and superfluids, as well as when connecting the optical model and hydrodynamic domains. Furthermore, unlike the Kadomtsev–Petviashvili equation (KPE), which permits the modeling of waves traveling in both directions, the zero-mass assumption, which is required for many scientific applications, is not required by the KP-B equation. In several applications in engineering and physics, taking these features into account allows researchers to acquire more precise conclusions, particularly in studies pertaining to the dynamics of water waves. The foremost purpose of this manuscript is to establish diverse solutions in the form of exponential, trigonometric, hyperbolic, and rational functions of the (3+1)-dimensional fractional (KP-B) via the application of four analytical methods. This KP-B model has fruitful applications in fluid dynamics and plasma physics. Additionally, in order to better explain the potential and physical behavior of the equation, the relevant models of the findings are visually indicated, and 2-dimensional (2D) and 3-dimensional (3D) graphics are drawn.

1. Introduction

Not long after classical calculus was expanded, in 1695, fractional calculus was developed. Since fractional calculus plays a crucial role in illustrating the internal aspect of the idea of this current authenticity, many experts have been interested in fractional-order nonlinear partial differential equations (FNLPDEs). The fractional calculus theorem itself has advanced, and these developments have been applied in many domains. Fractional calculus has close connections to the dynamics of complex real-world issues. Over the past 60 years, fractional calculus has become a valuable and proficient tool in mathematics, mostly because of its proven uses in a wide range of seemingly disparate and widespread scientific domains. Naturally occurring phenomena can be expressed in the form of FNLPDEs.

A traveling wave moves in a certain direction but keeps a constant shape. Furthermore, the propagation velocity of a traveling wave is known to be constant. These waves have been seen in a number of scientific domains, including combustion, which can result from a chemical process. In mathematical biology, the apparent impulses in nerve fibers are treated as moving waves. In conservation laws pertaining to fluid dynamics problems, shock profiles are referred to as traveling waves. Moreover, standing waves are widely employed to imitate solid mechanical structures. A traveling wave solution is found by solving a model that depicts the system. These models usually take the form of partial differential equations (PDEs), the solution to which provides insight into the dynamics of the systems. Many authors have recently focused on time-dependent variable coefficient NLPDEs in order to investigate analytical soliton solutions. To assist us understand the physical, dynamic, and future behaviors of these NLPDEs, these answers take on the appearance of traveling waves, solitary waves, bright-soliton, dark solitons, multisolitons, periodic solitons, etc.

There has been a growing focus on the study of solutions of nonlinear evolution equations (NLEEs) [1,2,3,4,5]. Numerous scientific and engineering fields, such as biology, optic fibers, plasmas, fluid mechanics, acoustics, and many more, use NLEEs to investigate phenomena [6,7,8,9,10]. The beginning of the selection of mathematical tools allowed for traveling wave solutions of NLEEs [11,12,13,14,15]. Currently, a multiscale analysis and an influential reductive perturbation technique have been used to solve several types of nonlinear evolution equations [16,17,18,19]. The W shape and M shape, two new types of brilliant and dark solitons, have recently been observed. A comparable quantity of works has been used to demonstrate the importance of these findings [20,21,22,23,24].

2. Applications of Fractional Calculus

A close connection exists between fractional calculus and the dynamics of challenging real-world issues. Fractional-order differential equations accurately rule many mathematical models. Liouville, Riemann, Leibniz, and others were credited with conducting the first systematic investigations [25,26]. Fractional calculus has long been thought of as a field of pure mathematics with no practical uses. However, things have altered in the last few decades. It has been discovered that fractional calculus can be effective and even powerful. Machado et al. [27] provided a brief history of fractional calculus, with an emphasis on its applications. These days, fractional calculus and its uses are developing quickly, with ever-more-persuasive real-world applications [28,29,30,31,32]. Intrinsically multidisciplinary, fractional differentiation and integration research finds applications in many domains, such as complex systems, medical imaging, and turbulence.

By applying traveling wave transformations to many fractional differential equations that include the applications mentioned above, the equations are converted into ordinary differential equations. In making this transformation, the fractional derivative used is applied by paying attention to the properties of that derivative. With the help of these transformations, the relationship between many fractional equations and partial differential equations can be demonstrated.

In this context, the fractional derivative version of the model we discuss below was transformed into the following equation with the help of similar transformations. Here, we discovered solitary wave solutions via four analytical approaches to the fractional (3+1)-dimensional KP-B model, which has the following form:

$$\begin{array}{c}\hfill U_{xxxy}+3{\left(U_x{U}_y\right)}_x+D_t^{\alpha }\left(U_x+U_y\right)+D_t^{2\alpha }U-U_{zz}=0,\end{array}$$

(1)

where $\alpha$ is the sense of the conformable derivative, and $0<\alpha \le 1$.

This KP-B model has many applications in science. The derived results via application of the extended simple equation scheme [33], $(G^{\prime }/G)$-expansion method [34], $Exp(-\mathsf{\Psi }(\varphi \left)\right)$-expansion scheme [35] and modified F-expansion scheme [33] have fruitful applications in different fields such as plasma physics.

• Definition of Conformable Derivative:

A conformable derivative [36] is defined as

$\left.\mathrm{Let}\mathrm{CD}\mathrm{of}\phi :(0,\infty )\to \mathbb{R}\mathrm{of}\mathrm{order}\alpha ϵ(0,1\right]\mathrm{as};$

$D_x^{\alpha }\left(x\right)=\underset{\sigma \to 0}{\mathrm{Limit}}\left({\displaystyle \frac{\phi \left(\sigma {\mathrm{x}}^{1-\alpha }+x\right)-\phi \left(x\right)}{\sigma }}\right)$

• Some Properties of Conformable Derivative:

Let G and H be differentiable functions at $x>0$,

$$\begin{array}{c}\hfill D_{\alpha }(\mathrm{aG}+\mathrm{bH})=a{\mathrm{D}}_{\alpha }\left(G\right)+b{\mathrm{D}}_{\alpha }\left(H\right).\end{array}$$

(2)

$$\begin{array}{c}\hfill D_{\delta }\left(x^p\right)={\mathrm{px}}^{p-1}.\end{array}$$

(3)

$$\begin{array}{c}\hfill D_{\alpha }(\mathrm{G}.\mathrm{H})=G{\mathrm{D}}_{\alpha }\left(H\right)+H{\mathrm{D}}_{\alpha }\left(G\right)\end{array}$$

(4)

$$\begin{array}{c}\hfill D_{\alpha }\left({\displaystyle \frac{G}{H}}\right)={\displaystyle \frac{H{\mathrm{D}}_{\alpha }\left(G\right)-H{\mathrm{D}}_{\alpha }\left(G\right)}{H^2}}\end{array}$$

(5)

$$\begin{array}{c}\hfill D_{{\alpha }_1}\left(K\right)=0,forallconstantfunctionG\left(x\right)=K\end{array}$$

(6)

$$\begin{array}{c}\hfill D_{\alpha }\left(G\right)\left(x\right)=x^{\alpha -1}\left({\displaystyle \frac{dG\left(x\right)}{dx}}\right),ForGisdifferentiable.\end{array}$$

(7)

3. Formation of Proposed Methods

Let NFDEs

$$L_1\left(U,U_x,U_t,D_t^{\alpha },\dots \right)=0,0<\alpha \le 1$$

(8)

Suppose

$$U=U\left(\xi \right),\xi =x+\gamma y+\delta z-\omega \left({\displaystyle \frac{t^{\alpha }}{\alpha }}\right)$$

(9)

Put Equation (9) into Equation (8),

$$L_2\left(U,U^{\prime },U^{″},U^{‴},\dots \right)=0,$$

(10)

3.1. Extended Simple Equation Method

Assume that Equation (10) has the following solution:

$$U=\sum _{i=-N}^N{A}_i{\mathsf{\Psi }}^i\left(\xi \right)$$

(11)

Let ${\mathsf{\Psi }}^{\prime }$ satisfy,

$${\mathsf{\Psi }}^{\prime }=c_0+c_1\mathsf{\Psi }+c_2{\mathsf{\Psi }}^2+c_3{\mathsf{\Psi }}^3$$

(12)

If $c_3=0$, then the solutions of Equation (12) are

$$\mathsf{\Psi }\left(\xi \right)=-\left({\displaystyle \frac{c_1-\sqrt{4c_0{c}_2-c_1^2}tan\left(\frac{1}{2}\sqrt{4c_0{c}_2-c_1^2}\left(\xi +{\xi }_0\right)\right)}{2c_2}}\right),4c_0{c}_2>c_1^2,$$

(13)

If $c_0=0,c_3=0$, the simple Ansatz Equation (12) reduces to the Bernoulli equation, which has the following solutions:

$$\mathsf{\Psi }\left(\xi \right)={\displaystyle \frac{c_{1}exp\left(c_1\left(\xi +{\xi }_0\right)\right)}{1-c_{2}exp\left(c_1\left(\xi +{\xi }_0\right)\right)}},c_1>0$$

(14)

$$\mathsf{\Psi }\left(\xi \right)=-{\displaystyle \frac{c_{1}exp\left(c_1\left(\xi +{\xi }_0\right)\right)}{c_{2}exp\left(c_1\left(\xi +{\xi }_0\right)\right)+1}},c_1<0$$

(15)

If $c_1=0,c_3=0$, the Ansatz Equation (12) has the following solutions:

$$\mathsf{\Psi }\left(\xi \right)={\displaystyle \frac{\sqrt{c_0{c}_2}tan\left(\sqrt{c_0{c}_2}\left(\xi +{\xi }_0\right)\right)}{c_2}},c_0{c}_2>0$$

(16)

$$\mathsf{\Psi }\left(\xi \right)={\displaystyle \frac{\sqrt{-c_0{c}_2}tanh\left(\sqrt{-c_0{c}_2}\left(\xi +{\xi }_0\right)\right)}{c_2}},c_0{c}_2<0$$

(17)

Equation (10) can be solved to obtain the necessary solution of Equation (8) with the use of Equations (13)–(17). This can be achieved by substituting Equations (11) and (12) into (10).

3.2. $(G^{\prime }/G)$-Expansion Method

Let (10) have the following solution:

$$U=A_0+\sum _{i=1}^N{A}_i\left({\displaystyle \frac{G^{\prime }}{G}}\right)$$

(18)

Let

$$G^{″}=-\lambda G^{\prime }-\mu G$$

(19)

Combining Equations (18) and (19) with Equation (10), we have a system of equations with the following examples of solutions.

• CASE I: When ${\lambda }^2-4\mu >0$

$$\left(G^{\prime }/G\right)={\displaystyle \frac{\sqrt{{\lambda }^2-4\mu }\left(\xi P_{1}sinh\left(\frac{1}{2}\sqrt{{\lambda }_1^2-4\mu }\right)+\xi P_{2}cosh\left(\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\right)\right)}{2\left(\xi P_{2}sinh\left(\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\right)+\xi P_{1}cosh\left(\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\right)\right)}}-{\displaystyle \frac{\lambda }{2}}$$

(20)

• CASE II: When ${\lambda }^2-4\mu <0$

$$\left(G^{\prime }/G\right)={\displaystyle \frac{\sqrt{4\mu -{\lambda }^2}\left(\xi P_{2}cos\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)-\xi P_{1}sin\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)\right)}{2\left(\xi P_{2}sin\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)+\xi P_{1}cos\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)\right)}}-{\displaystyle \frac{\lambda }{2}}$$

(21)

• CASE III: When ${\lambda }^2-4\mu =0$

$$\left(G^{\prime }/G\right)=\left({\displaystyle \frac{P_2}{\xi {\mathrm{P}}_2+P_1}}-{\displaystyle \frac{\lambda }{2}}\right)$$

(22)

To solve Equation (8) and obtain the necessary solution, substitute Equations (18) and (19) into Equation (10), which may be solved with the aid of Equations (20)–(22).

3.3. The $Exp(-\mathsf{\Psi }(\varphi \left)\right)$-Expansion Method

Let (10) have solution

$$U=A_N{\left(Exp(-\mathsf{\Psi }(-\varphi )\right)}^N+\dots ,A_N\ne 0$$

(23)

Let

$${\mathsf{\Psi }}^{\prime }=Exp(-\mathsf{\Psi }\left(\varphi \right))+\mu Exp(\mathsf{\Psi }\left(\varphi \right))+\lambda$$

(24)

When ${\lambda }^2-4\mu >0$, $\mu \ne 0$, Equation (24) has the following solution:

$$\mathsf{\Psi }=ln\left({\displaystyle \frac{-\sqrt{{\lambda }^2-4\mu }tanh\left(\frac{1}{2}\sqrt{{\lambda }^2-4\mu }(\xi +{\xi }_0)\right)-\lambda }{2\mu }}\right)$$

(25)

When ${\lambda }^2-4\mu >0$, $\mu =0$, Equation (24) has the following solution:

$$\mathsf{\Psi }=-ln\left({\displaystyle \frac{\lambda }{e^{\lambda \left(\xi +{\xi }_0\right)}-1}}\right)$$

(26)

When ${\lambda }^2-4\mu =0$, $\lambda \ne 0$, Equation (24) has the following solution:

$$\mathsf{\Psi }=ln\left({\displaystyle \frac{2-2\lambda \left(\xi +{\xi }_0\right)}{{\lambda }^2\left(\xi +{\xi }_0\right)}}\right)$$

(27)

When ${\lambda }^2-4\mu =0$, $\mu ,\lambda =0$, Equation (24) has the following solution:

$$\mathsf{\Psi }=ln(\xi +{\xi }_0)$$

(28)

When ${\lambda }^2-4\mu <0$, Equation (24) has the following solution

$$\mathsf{\Psi }=\left(ln\left(-{\displaystyle \frac{\sqrt{4\mu -{\lambda }^2}tan\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}(\xi +{\xi }_0)\right)+\lambda }{2\mu }}\right)\right)$$

(29)

Enter Equations (23) and (24) into Equation (10) and solve to obtain the necessary solution of Equation (8) using Equations (25)–(29).

3.4. Modified F-Expansion Method

Let (10) have solution

$$U=a_0+\sum _{i=1}^N{a}_i{F}^i\left(\xi \right)+\sum _{i=1}^N{b}_i{F}^{-i}\left(\xi \right)$$

(30)

Let $F^{\prime }$ fulfill

$$F^{\prime }=A+BF+C{F}^2$$

(31)

The relationship between equivalent values of $F\left(\xi \right)$ for A, B, C in Equation (34) is as follows:

Values of A, B, C	$F\left(\xi \right)$
$A=0$, $B=1$, $C=-1$	${\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}tanh\left({\displaystyle \frac{1}{2}}\xi \right)$
$A=0$, $B=-1$, $C=1$	${\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}coth\left({\displaystyle \frac{1}{2}}\xi \right)$
$A={\displaystyle \frac{1}{2}}$, $B=0$, $C=-{\displaystyle \frac{1}{2}}$	$coth\left(\xi \right)\pm csch\left(\xi \right)$
$A=1$, $B=0$, $C=-1$	$tanh\left(\xi \right),coth\left(\xi \right)$
$A={\displaystyle \frac{1}{2}}$, $B=0$, $C={\displaystyle \frac{1}{2}}$	$sec\left(\xi \right)+tan\left(\xi \right)$
$A=-{\displaystyle \frac{1}{2}}$, $B=0$, $C=-{\displaystyle \frac{1}{2}}$	$sec\left(\xi \right)-tan\left(\xi \right)$
$A=1(-1)$, $B=0$, $C=1(-1)$	$tan\left(\xi \right)\left(\cot \right(\xi \left)\right)$
$A=0$, $B=0$, $C\ne 0$	$-{\displaystyle \frac{1}{C\xi +ϵ}}(ϵ$ is arbitary constant)
$A\ne 0$, $B=0$, $C=0$	$A\xi$
$A\ne 0$, $B\ne 0$, $C=0$	${\displaystyle \frac{exp\left(B\xi \right)-A}{B}}$

To obtain the necessary answer to (8) with the aid of Equation (31)’s solutions, substitute Equations (30) and (31) into Equation (10).

4. Applications

Substitute Equation (9) into Equation (1).

$$\gamma U^{″2}+U\left(-(\gamma +1)\omega -{\delta }^2+{\omega }^2\right)=0$$

(32)

4.1. Application of Extended Simple Equation Method

Since homogeneous balance N = 2, Equation (32) has the following solution form:

$$U=A_1\mathsf{\Psi }+{\displaystyle \frac{A_{-1}}{\mathsf{\Psi }}}+A_0+A_2{\mathsf{\Psi }}^2+{\displaystyle \frac{A_{-2}}{{\mathsf{\Psi }}^2}}$$

(33)

Put (33) with (12) in (32),

• CASE 1: $c_3=0$,
• FAMILY-I

$$\begin{array}{cc}\hfill & A_0={\displaystyle \frac{1}{3}}\left(-c_1^2-2c_0{c}_2\right),A_{-2}=0,A_{-1}=0,A_2=-2c_2^2,A_1=-2c_1{c}_2,\hfill \\ \hfill & \delta =-\sqrt{-\gamma c_1^2+4\gamma c_0{c}_2-\gamma \omega +{\omega }^2-\omega }\hfill \end{array}$$

(34)

Put (34) in (33),

$$\begin{array}{cc}\hfill U_1=& {\displaystyle \frac{1}{6}}\left(12\sqrt{4c_0{c}_2-c_1^2}{c}_{1}tan\left({\displaystyle \frac{1}{2}}\sqrt{4c_0{c}_2-c_1^2}\left(\xi +{\xi }_0\right)\right)-11c_1^2-4c_0{c}_2\right.\hfill \\ \hfill & \left.+3\left(c_1^2-4c_0{c}_2\right){tan}^2\left({\displaystyle \frac{1}{2}}\sqrt{4c_0{c}_2-c_1^2}\left(\xi +{\xi }_0\right)\right)\right),4c_0{c}_2>c_1^2.\hfill \end{array}$$

(35)

• FAMILY-II

$$\begin{array}{cc}\hfill & A_0={\displaystyle \frac{1}{3}}\left(-c_1^2-2c_0{c}_2\right),A_{-2}=-2c_0^2,A_{-1}=-2c_0{c}_1,A_2=0,A_1=0,\hfill \\ \hfill & \delta =\sqrt{-\gamma c_1^2+4\gamma c_0{c}_2-\gamma \omega +{\omega }^2-\omega }\hfill \end{array}$$

(36)

Substitute (36) in (33),

$$\begin{array}{cc}\hfill U_2=& {\displaystyle \frac{2}{3}}{c}_0{c}_2\left(-{\displaystyle \frac{12c_0{c}_2}{{\left(c_1-\sqrt{4c_0{c}_2-c_1^2}tan\left(\frac{1}{2}\sqrt{4c_0{c}_2-c_1^2}\left(\xi +{\xi }_0\right)\right)\right)}^2}}-1\right)\hfill \\ \hfill & -{\displaystyle \frac{4c_0{c}_2{c}_1}{c_1-\sqrt{4c_0{c}_2-c_1^2}tan\left(\frac{1}{2}\sqrt{4c_0{c}_2-c_1^2}\left(\xi +{\xi }_0\right)\right)}}-\left({\displaystyle \frac{c_1^2}{3}}\right),4c_0{c}_2>c_1^2\hfill \end{array}$$

(37)

• CASE 2: $c_0=c_3=0$,

$$\begin{array}{c}\hfill A_0=-{\displaystyle \frac{c_1^2}{3}},A_{-2}=0,A_{-1}=0,A_2=-2c_2^2,A_1=-2c_1{c}_2,\delta =\sqrt{-\gamma c_1^2-\gamma \omega +{\omega }^2-\omega }\end{array}$$

(38)

Put (38) in (33),

$$U_3=\left(-{\displaystyle \frac{c_1^2\left(c_2{e}^{c_1\left(\xi +{\xi }_0\right)}\left(c_2{e}^{c_1\left(\xi +{\xi }_0\right)}+4\right)+1\right)}{3{\left(c_2{e}^{c_1\left(\xi +{\xi }_0\right)}-1\right)}^2}}\right),c_1>0.$$

(39)

$$U_4=\left(-{\displaystyle \frac{c_1^2\left(c_2{e}^{c_1\left(\xi +{\xi }_0\right)}\left(c_2{e}^{c_1\left(\xi +{\xi }_0\right)}-4\right)+1\right)}{3{\left(c_2{e}^{c_1\left(\xi +{\xi }_0\right)}+1\right)}^2}}\right),c_1<0.$$

(40)

• CASE 3: $c_1=0,c_3=0$,
• FAMILY-I

$$\begin{array}{c}\hfill A_0={\displaystyle \frac{1}{3}}(-2)c_0{c}_2,A_{-2}=A_{-1}=A_1=0,A_2=-2c_2^2,\delta =-\sqrt{4\gamma c_0{c}_2-\gamma \omega +{\omega }^2-\omega }\end{array}$$

(41)

Put (41) in (33),

$$\begin{array}{c}\hfill U_5=\left(-2c_2^2{\left({\displaystyle \frac{\sqrt{c_0{c}_2}tan\left(\sqrt{c_0{c}_2}\left(\xi +{\xi }_0\right)\right)}{c_2}}\right)}^2-{\displaystyle \frac{2c_0{c}_2}{3}}\right),c_0{c}_2>0,\end{array}$$

(42)

$$\begin{array}{c}\hfill U_6=\left(-2c_2^2{\left({\displaystyle \frac{\sqrt{-c_0{c}_2}tanh\left(\sqrt{-c_0{c}_2}\left(\xi +{\xi }_0\right)\right)}{c_2}}\right)}^2-{\displaystyle \frac{2c_0{c}_2}{3}}\right),c_0{c}_2<0.\end{array}$$

(43)

• FAMILY-II

$$\begin{array}{c}\hfill A_0=-2c_0{c}_2,A_{-2}=-2c_0^2,A_{-1}=0,A_2=0,A_1=0,\delta =\sqrt{-4\gamma c_0{c}_2-\gamma \omega +{\omega }^2-\omega }\end{array}$$

(44)

Put (44) in (33),

$$\begin{array}{c}\hfill U_7=\left(-{\displaystyle \frac{2c_0^2}{{\left(\frac{\sqrt{c_0{c}_2}tan\left(\sqrt{c_0{c}_2}\left(\xi +{\xi }_0\right)\right)}{c_2}\right)}^2}}-2c_2{c}_0\right),c_0{c}_2>0,\end{array}$$

(45)

$$\begin{array}{c}\hfill U_8=\left(-{\displaystyle \frac{2c_0^2}{{\left(\frac{\sqrt{-c_0{c}_2}tanh\left(\sqrt{-c_0{c}_2}\left(\xi +{\xi }_0\right)\right)}{c_2}\right)}^2}}-2c_2{c}_0\right),c_0{c}_2<0.\end{array}$$

(46)

• FAMILY-III

$$\begin{array}{cc}\hfill & A_0={\displaystyle \frac{4c_0{c}_2}{3}},A_{-2}=-2c_0^2,A_{-1}=0,A_2=-2c_2^2,A_1=0,\hfill \\ \hfill & \delta =\sqrt{16\gamma c_0{c}_2-\gamma \omega +{\omega }^2-\omega }\hfill \end{array}$$

(47)

Put (47) in (33),

$$\begin{array}{c}\hfill U_9=\left({\displaystyle \frac{2}{3}}{c}_0{c}_2\left(-3{tan}^2\left(\sqrt{c_0{c}_2}\left(\xi +{\xi }_0\right)\right)-3{cot}^2\left(\sqrt{c_0{c}_2}\left(\xi +{\xi }_0\right)\right)+2\right)\right),c_0{c}_2>0,\end{array}$$

(48)

$$\begin{array}{c}\hfill U_{10}=\left({\displaystyle \frac{2}{3}}{c}_0{c}_2\left(3{tanh}^2\left(\sqrt{-c_0{c}_2}\left(\xi +{\xi }_0\right)\right)+3{coth}^2\left(\sqrt{-c_0{c}_2}\left(\xi +{\xi }_0\right)\right)+2\right)\right),c_0{c}_2<0.\end{array}$$

(49)

4.2. The $(G^{\prime }/G)$-Expansion Method

Let (19) have solution

$$\begin{array}{c}\hfill U=A_0+A_1\left({\displaystyle \frac{G^{\prime }}{G}}\right)+A_2{\left({\displaystyle \frac{G^{\prime }}{G}}\right)}^2\end{array}$$

(50)

Put (50) with (14) in (19):

$$\begin{array}{c}\hfill A_0=-2\mu ,A_1=-2\lambda ,A_2=-2,\delta =\sqrt{\gamma {\lambda }^2-4\gamma \mu -\gamma \omega +{\omega }^2-\omega }\end{array}$$

(51)

Put (51) in (50),

• CASE I: ${\lambda }^2-4\mu >0$

$$\begin{array}{c}\hfill U_{11}=\left({\displaystyle \frac{\left(P_1^2-P_2^2\right)\left({\lambda }^2-4\mu \right)}{2{\left(P_{2}sinh\left(\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\right)+P_{1}cosh\left(\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\right)\right)}^2}}\right)\end{array}$$

(52)

• CASE II: ${\lambda }^2-4\mu <0$

$$\begin{array}{cc}\hfill U_{12}=& \left({\lambda }^2-2\mu +{\displaystyle \frac{\lambda \sqrt{4\mu -{\lambda }^2}\left(P_{1}sin\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)-P_{2}cos\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)\right)}{P_{2}sin\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)+P_{1}cos\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)}}\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\lambda {\left(\lambda +{\displaystyle \frac{\sqrt{4\mu -{\lambda }^2}\left(P_{1}sin\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)-P_{2}cos\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)\right)}{P_{2}sin\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)+P_{1}cos\left(\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\right)}}\right)}^2\hfill \end{array}$$

(53)

• CASE III: ${\lambda }^2-4\mu =0$

$$\begin{array}{c}\hfill U_{13}=\left({\displaystyle \frac{{\lambda }^2}{2}}-2\mu -{\displaystyle \frac{2P_2^2}{{\left(\xi {\mathrm{P}}_2+P_1\right)}^2}}\right)\end{array}$$

(54)

4.3. The $Exp(-\mathsf{\Psi }(\varphi \left)\right)$-Expansion Method

Let (19) have solution

$$\begin{array}{c}\hfill U=A_0+A_{1}exp(-\mathsf{\Psi }\left(\varphi \right))+A_{2}exp{(-\mathsf{\Psi }\left(\varphi \right))}^2\end{array}$$

(55)

Put (55) with (16) in (19):

$$\begin{array}{c}\hfill A_0={\displaystyle \frac{1}{3}}\left(-{\lambda }^2-2\mu \right),A_1=-2\lambda ,A_2=-2,\delta =\sqrt{-\gamma {\lambda }^2+4\gamma \mu -\gamma \omega +{\omega }^2-\omega }\end{array}$$

(56)

Put (56) with (55),

CASE I: ${\lambda }^2-4\mu >0$, $\mu \ne 0$

$$\begin{array}{cc}\hfill U_{14}=& {\displaystyle \frac{1}{3}}\left(\left(-6ln\left(-{\displaystyle \frac{\sqrt{{\lambda }^2-4\mu }tanh\left(\frac{1}{2}\left(\xi +{\xi }_0\right)\sqrt{{\lambda }^2-4\mu }\right)+\lambda }{2\mu }}\right)-{\lambda }^2-2\mu \right.\right)\hfill \\ \hfill & \left(\left.ln\left(-{\displaystyle \frac{\sqrt{{\lambda }^2-4\mu }tanh\left(\frac{1}{2}\left(\xi +{\xi }_0\right)\sqrt{{\lambda }^2-4\mu }\right)+\lambda }{2\mu }}\right)+\lambda \right)\right)\hfill \end{array}$$

(57)

• CASE II: ${\lambda }^2-4\mu >0$, $\mu =0$

$$\begin{array}{c}\hfill U_{15}={\displaystyle \frac{1}{3}}\lambda \left(\lambda \left(-{\displaystyle \frac{6}{{\left(e^{\lambda \left(\xi +{\xi }_0\right)}-1\right)}^2}}-1\right)+6ln\left({\displaystyle \frac{\lambda }{e^{\lambda \left(\xi +{\xi }_0\right)}-1}}\right)\right)\end{array}$$

(58)

• CASE III: ${\lambda }^2-4\mu =0$, $\mu ,\lambda \ne 0$

$$\begin{array}{c}\hfill U_{16}={\displaystyle \frac{1}{3}}\left(-6ln\left({\displaystyle \frac{2\left(\frac{1}{\xi +{\xi }_0}-\lambda \right)}{{\lambda }^2}}\right)\left(ln\left({\displaystyle \frac{2\left(\frac{1}{\xi +{\xi }_0}-\lambda \right)}{{\lambda }^2}}\right)+\lambda \right)-{\lambda }^2-2\mu \right)\end{array}$$

(59)

• CASE IV: ${\lambda }^2-4\mu =0$, $\mu =\lambda =0$

$$\begin{array}{c}\hfill U_{17}=\left(-2ln{\left(\left[\xi +{\xi }_0\right]\right)}^2\right)\end{array}$$

(60)

• CASE V: ${\lambda }^2-4\mu <0$

$$\begin{array}{cc}\hfill U_{18}=& \left({\displaystyle \frac{1}{3}}\left(-6ln\left(-{\displaystyle \frac{\sqrt{4\mu -{\lambda }^2}tan\left(\frac{1}{2}\left(\xi +{\xi }_0\right)\sqrt{4\mu -{\lambda }^2}\right)+\lambda }{2\mu }}\right)-{\lambda }^2-2\mu \right.\right)\hfill \\ \hfill & \left.ln\left(-{\displaystyle \frac{\sqrt{4\mu -{\lambda }^2}tan\left(\frac{1}{2}\left(\xi +{\xi }_0\right)\sqrt{4\mu -{\lambda }^2}\right)+\lambda }{2\mu }}\right)+\lambda \right)\hfill \end{array}$$

(61)

4.4. Modified F-Expansion Method

Let (19) have solution

$$\begin{array}{c}\hfill U=a_{1}F+a_0+{\displaystyle \frac{b_1}{F}}+a_2{F}^2+{\displaystyle \frac{b_2}{F^2}}\end{array}$$

(62)

Put (62) with (18) in (19),

A = 0, B = 1, C = −1,

$$a_0=0,a_2=-2,a_1=2,b_1=0,b_2=0,\delta =\sqrt{-\gamma \omega +\gamma +{\omega }^2-\omega }$$

(63)

Put (63) in (62):

$$\begin{array}{c}\hfill U_{19}=\left({\displaystyle \frac{e^{\xi }+3}{2cosh\left(\xi \right)+2}}\right)\end{array}$$

(64)

A = 0, C = 1, B = −1,

$$a_0=0,a_2=-2,a_1=2,b_1=0,b_2=0,\delta =\sqrt{-\gamma \omega +\gamma +{\omega }^2-\omega }$$

(65)

Put (65) into (62):

$$\begin{array}{c}\hfill U_{20}=\left({\displaystyle \frac{1}{1-cosh\left(\xi \right)}}\right)\end{array}$$

(66)

A = 1/2, B = 0, C = −1/2

• FAMILY-I

$$a_0={\displaystyle \frac{1}{6}},a_2=-{\displaystyle \frac{1}{2}},a_1=0,b_1=0,b_2=0,\delta =-\sqrt{-\gamma \omega -\gamma +{\omega }^2-\omega }$$

(67)

Substitute (67) into (62):

$$\begin{array}{c}\hfill U_{21,1}=\left({\displaystyle \frac{1}{6}}-{\displaystyle \frac{1}{2}}{(coth\left(\xi \right)+\csc \mathrm{h}\left(\xi \right))}^2\right)\end{array}$$

(68)

• FAMILY-II

$$a_0={\displaystyle \frac{1}{2}},a_2=0,a_1=0,b_1=0,b_2=-{\displaystyle \frac{1}{2}},\delta =\sqrt{-\gamma \omega +\gamma +{\omega }^2-\omega }$$

(69)

Put (69) in (62):

$$\begin{array}{c}\hfill U_{21,2}={\displaystyle \frac{1}{2}}-\left({\displaystyle \frac{1}{2{(coth\left(\xi \right)+\csc \mathrm{h}\left(\xi \right))}^2}}\right)\end{array}$$

(70)

• FAMILY-III

$$\begin{array}{c}\hfill a_0=1,a_2=-{\displaystyle \frac{1}{2}},a_1=0,b_1=0,b_2=-{\displaystyle \frac{1}{2}},\delta =\sqrt{-\gamma \omega +4\gamma +{\omega }^2-\omega }\end{array}$$

(71)

Put (71) in (62),:

$$\begin{array}{c}\hfill U_{21,3}=\left(-{\displaystyle \frac{{(coth\left(\xi \right)+\csc \mathrm{h}\left(\xi \right))}^4+1}{2{(coth\left(\xi \right)+\csc \mathrm{h}\left(\xi \right))}^2}}\right)\end{array}$$

(72)

A = 1, B = 0, C = −1

• FAMILY-I

$$a_0=2,a_2=-2,a_1=0,b_1=0,b_2=0,\delta =-\sqrt{-\gamma \omega +4\gamma +{\omega }^2-\omega }$$

(73)

Put (86) in (62):

$$\begin{array}{c}\hfill U_{22,1}=2-2{tanh}^2\left(\xi \right)\end{array}$$

(74)

• FAMILY-II

$$a_0=2,a_2=0,a_1=0,b_1=0,b_2=-2,\delta =\sqrt{-\gamma \omega +4\gamma +{\omega }^2-\omega }$$

(75)

Put (75) in (62):

$$\begin{array}{c}\hfill U_{21,2}=2-\left({\displaystyle \frac{2}{{tanh}^2\left(\xi \right)}}\right)\end{array}$$

(76)

• FAMILY-III

$$\begin{array}{c}\hfill a_0=4,a_2=-2,a_1=0,b_1=0,b_2=-2,\delta =\sqrt{-\gamma \omega +16\gamma +{\omega }^2-\omega }\end{array}$$

(77)

Put (77) in (62):

$$\begin{array}{c}\hfill U_{22,3}=-2{\csc \mathrm{h}}^2\left(\xi \right)+{\sec \mathrm{h}}^2\left(\xi \right)+1\end{array}$$

(78)

A = C = 1/2, B = 0,

• FAMILY-I

$$a_0=-{\displaystyle \frac{1}{2}},a_2=-{\displaystyle \frac{1}{2}},a_1=0,b_1=0,b_2=0,\delta =\sqrt{-\gamma \omega -\gamma +{\omega }^2-\omega }$$

(79)

Put (79) in (62):

$$\begin{array}{c}\hfill U_{23,1}=-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}{(tan\left(\xi \right)+sec\left(\xi \right))}^2\end{array}$$

(80)

• FAMILY-II

$$a_0=-{\displaystyle \frac{1}{6}},a_2=a_1=b_1=0,b_2=-{\displaystyle \frac{1}{2}},\delta =\sqrt{-\gamma \omega +\gamma +{\omega }^2-\omega }$$

(81)

Put (81) in (62):

$$\begin{array}{c}\hfill U_{23,2}=-{\displaystyle \frac{1}{6}}-\left({\displaystyle \frac{1}{2{(tan\left(\xi \right)+sec\left(\xi \right))}^2}}\right)\end{array}$$

(82)

• FAMILY-III

$$\begin{array}{c}\hfill a_0={\displaystyle \frac{1}{3}},a_2=-{\displaystyle \frac{1}{2}},a_1=0,b_1=0,b_2=-{\displaystyle \frac{1}{2}},\delta =\sqrt{-\gamma \omega +4\gamma +{\omega }^2-\omega }\end{array}$$

(83)

Put (83) in (62):

$$\begin{array}{c}\hfill U_{23,3}=-{(tan\left(\xi \right)+sec\left(\xi \right))}^2-\left({\displaystyle \frac{1}{2{(tan\left(\xi \right)+sec\left(\xi \right))}^2}}\right)+\left({\displaystyle \frac{1}{3}}\right)\end{array}$$

(84)

A = C = −1/2, B = 0,

• FAMILY-I

$$a_0=a_2=-{\displaystyle \frac{1}{2}},a_1=b_1=b_2=0,\delta =\sqrt{-\gamma \omega -\gamma +{\omega }^2-\omega }$$

(85)

Put (85) in (62):

$$\begin{array}{c}\hfill U_{24,1}=-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}{(sec\left(\xi \right)-tan\left(\xi \right))}^2\end{array}$$

(86)

• FAMILY-II

$$a_0=-{\displaystyle \frac{1}{6}},a_2=0,a_1=0,b_1=0,b_2=-{\displaystyle \frac{1}{2}},\delta =\sqrt{-\gamma \omega +\gamma +{\omega }^2-\omega }$$

(87)

Put (87) in (62):

$$\begin{array}{c}\hfill U_{24,2}=-{\displaystyle \frac{1}{6}}-\left({\displaystyle \frac{1}{2{(sec\left(\xi \right)-tan\left(\xi \right))}^2}}\right)\end{array}$$

(88)

• FAMILY-III

$$a_0={\displaystyle \frac{1}{3}},a_2=-{\displaystyle \frac{1}{2}},a_1=0,b_1=0,b_2=-{\displaystyle \frac{1}{2}},\delta =\sqrt{-\gamma \omega +4\gamma +{\omega }^2-\omega }$$

(89)

Put (89) in (62):

$$\begin{array}{c}\hfill U_{24,3}={\displaystyle \frac{1}{6}}\left(-3{(sec\left(\xi \right)-tan\left(\xi \right))}^2-{\displaystyle \frac{3}{{(sec\left(\xi \right)-tan\left(\xi \right))}^2}}+2\right)\end{array}$$

(90)

A = C = −1, B = 0,

• FAMILY-I

$$a_0=-{\displaystyle \frac{2}{3}},a_2=-2,a_1=b_1=b_2=0,\delta =\sqrt{-\gamma \omega +4\gamma +{\omega }^2-\omega }$$

(91)

Put (91) in (62):

$$\begin{array}{c}\hfill U_{25,1}=-{\displaystyle \frac{2}{3}}-2{tan}^2\left(\xi \right)\end{array}$$

(92)

• FAMILY-II

$$a_0=-{\displaystyle \frac{2}{3}},a_2=0,a_1=0,b_1=0,b_2=-2,\delta =\sqrt{-\gamma \omega +4\gamma +{\omega }^2-\omega }$$

(93)

Put (93) in (62):

$$\begin{array}{c}\hfill U_{25,2}=-{\displaystyle \frac{2}{3}}-{\displaystyle \frac{2}{{tan}^2\left(\xi \right)}}\end{array}$$

(94)

• FAMILY-III

$$\begin{array}{c}\hfill a_0={\displaystyle \frac{4}{3}},a_2=b_2=-2,a_1=b_1=0,\delta =\sqrt{-\gamma \omega +16\gamma +{\omega }^2-\omega }\end{array}$$

(95)

Put (95) in (62):

$$\begin{array}{c}\hfill U_{25,3}=-2{tan}^2\left(\xi \right)-2{cot}^2\left(\xi \right)+{\displaystyle \frac{4}{3}}\end{array}$$

(96)

• A = B = 0

$$\begin{array}{c}\hfill a_0=0,a_2=-2C^2,a_1=0,b_1=0,b_2=0,\delta =\sqrt{-\gamma \omega +{\omega }^2-\omega }\end{array}$$

(97)

Put (97) in (62):

$$\begin{array}{c}\hfill U_{26}=-2C^2{\left(-{\displaystyle \frac{1}{\mathrm{C}\xi +\eta }}\right)}^2\end{array}$$

(98)

• B = C = 0

$$\begin{array}{c}\hfill a_0=a_2=a_1=b_1=0,b_2=-2A^2,\delta =\sqrt{-\gamma \omega +{\omega }^2-\omega }\end{array}$$

(99)

Put (99) in (62):

$$\begin{array}{c}\hfill U_{27}=\left({\displaystyle \frac{-2}{{\xi }^2}}\right)\end{array}$$

(100)

• C = 0

$$\begin{array}{c}\hfill a_0=-{\displaystyle \frac{B^2}{3}},a_2=0,a_1=0,b_1=-2AB,b_2=-2A^2,\delta =\sqrt{-B^2\gamma -\gamma \omega +{\omega }^2-\omega }\end{array}$$

(101)

Put (101) in (62):

$$\begin{array}{c}\hfill U_{28}=\left(-{\displaystyle \frac{B^2\left(A^2+4A{e}^{B\xi }+e^{2B\xi }\right)}{3{\left(A-e^{B\xi }\right)}^2}}\right)\end{array}$$

(102)

5. Discussion of the Results

This section compares the solutions we looked into with those found in other papers that used various approaches to arrive at their conclusions. As a result of deriving distinct values for $A_N,(N=1,2)$ in Equations (50) and (55) and $Ai,(i=-2,-1,0,1,2)$ in Equation (33). Additionally, we obtained several findings in the form of rational, exponential, trigonometric, and rational functions for $a_i,b_i,(i=1,2)$ in Equation (62). However, because of the ensuing implications, several of our created solutions probably resemble each other.

• Our solutions $U_3$ and $U_4$ are likely similar to solutions $u_3(x,y,t)$ and $u_4(x,y,t)$ mentioned in Equations (3.12–3.13) respectively in [37].
• Our solutions $U_{11}$ and $U_{12}$ are likely similar to solutions $u_1(x,y,t)$ and $u_1(x,y,t)$ mentioned in Equations (4.4–4.5) respectively in [38].
• Our solution $U_{28}$ is likely similar to solution $u_{14}$ mentioned in [39].

Nothing in the literature is similar to our residual overall constructed findings, demonstrating that our suggested techniques offer a more potent and efficient mathematical tool for resolving nonlinear evolution problems in the physical sciences.Our method is trustworthy and supports a range of precise NFPDE solutions.

6. Conclusions

Using four mathematical approaches, we constructed numerous accurate solutions of the fractional Kadomtsev–Petviashvili–Boussinesq (3+1)-dimensional model. As a result, our proposed mathematical methods are stronger, and the findings of our investigation have numerous uses in plasma physics. This implies that these methods are more effective and robust in finding exact solutions to partial differential equations that are nonlinear. These methods can be applied to many creative solutions. To illustrate the dynamic nature of the solutions we discovered, we created three-dimensional charts using the extended simple equation approach, the (G′/G)-expansion method, the $exp(-\mathsf{\Psi }(\varphi \left)\right)$-expansion method, and the modified F-expansion method. These solutions demonstrated solitary, bright, singular, and dark solitons, which have numerous uses in physics, with various parameter values in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7 and Figure A8. Other comparable nonlinear equations and systems of equations can be resolved using these methods. The Mathematica package software was used to find the solutions that were found in this research. This equation has accurate and traveling wave solutions, which we successfully obtained in this study. Many of the outcomes are novel answers that had never been proposed before in the literature.