# Study of Ion-Acoustic Solitary Waves in a Magnetized Plasma Using the Three-Dimensional Time-Space Fractional Schamel-KdV Equation

Authors: Min Guo; Chen Fu; Yong Zhang; Jianxin Liu; Hongwei Yang
Journal: Complexity
 (2018)

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## Abstract

The study of ion-acoustic solitary waves in a magnetized plasma has long been considered to be an important research subject and plays an increasingly important role in scientific research. Previous studies have focused on the integer-order models of ion-acoustic solitary waves. With the development of theory and advancement of scientific research, fractional calculus has begun to be considered as a method for the study of physical systems. The study of fractional calculus has opened a new window for understanding the features of ion-acoustic solitary waves and can be a potentially valuable approach for investigations of magnetized plasma. In this paper, based on the basic system of equations for ion-acoustic solitary waves and using multi-scale analysis and the perturbation method, we have obtained a new model called the three-dimensional(3D) Schamel-KdV equation. Then, the integer-order 3D Schamel-KdV equation is transformed into the time-space fractional Schamel-KdV (TSF-Schamel-KdV) equation by using the semi-inverse method and the fractional variational principle. To study the properties of ion-acoustic solitary waves, we discuss the conservation laws of the new time-space fractional equation by applying Lie symmetry analysis and the Riemann-Liouville fractional derivative. Furthermore, the multi-soliton solutions of the 3D TSF-Schamel-KdV equation are derived using the Hirota bilinear method. Finally, with the help of the multi-soliton solutions, we explore the characteristics of motion of ion-acoustic solitary waves.


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## Body

## 1. Introduction

Ion-acoustic solitary waves are well-known to be an important example of nonlinear phenomena in modern plasma research [1–3]. Many researchers have studied ion-acoustic solitary waves in different plasma systems such as thermal, magnetized, and unmagnetized plasmas. Among the different plasma systems, magnetized plasma systems have attracted intense interest. Many authors have studied ion-acoustic solitary waves in magnetized plasma based on the quantum hydrodynamic (QHD) model [4, 5]. The QHD model is derived from the basic system of equations of ion-acoustic solitary waves and is one of the macroscopic mathematical models used to describe the hydrodynamic behavior of quantum plasmas.For simplicity, 1D and 2D nonlinear partial differential equations have been used to describe the evolution of nonlinear ion-acoustic solitary waves. For the simplest 1D geometry where the ion-acoustic solitary waves become solitons, Washimi and Taniuti [6] derived the KdV equation by using the reductive perturbation method. Kako and Rowlands [7] derived the 2D KP equation based on the results of Washimi and Taniuti. However, in the real magnetized plasma environment, 1D and 2D models cannot solve some of the problems encountered in the motion of ion-acoustic solitary waves. Thus, it is necessary to introduce higher-dimensional theories for the nonlinear ion-acoustic solitary waves. Therefore, in this paper, we discuss a new 3D model for nonlinear ion-acoustic solitary waves.Most of the QHD models, such as the KdV model, mKdV model, and KP model, are integer-order models. Fractional order models have rarely been considered. Fractional calculus is a generalization of integer calculus. Many of the physical processes that have been explored to date are nonconservative. It is important to be able to apply the power of fractional differentiation [8–10]. However, because of its nonlocal character, fractional calculus has not been used in physics and engineering. With the development of nonlinear science, fractional calculus theory has been continuously developed to date. Researchers have discovered that the derivatives and integrals of fractional order models are suitable for describing various physical phenomena. In recent years, the application of fractional differential equations has attracted increasing attention in plasma physics [11]. Thus, research on fractional order models is necessary.The solution of the integer equation is a research hot spot in the field of research and development of various models [12–14], and similarly, the solution of fractional models has been a focus of our research [15, 16]. Thus, many solution methods have been found and used to solve the fractional order equation. For instance, the iterative method [17–19], Hirota bilinear method [20, 21], trial function method [22], Homotopy perturbation [23], and other methods have all been developed in the recent decades. In the past, researchers solved integer-order models by using the Hirota bilinear method. Recently, the Hirota bilinear method has been used to solve fractional models. In this paper, using the Hirota bilinear method, we obtain soliton solutions for the new model. Various phenomena can be explained via the application of the solutions given by the above methods [24–26]. Additionally, the use of these methods enables a better understanding of various magnetized plasma phenomena. Therefore, based on the solutions derived by the abovementioned methods, we seek to determine the properties of ion-acoustic solitary waves. The properties of the model include conservation laws [27, 28], boundary value problems [29, 30], and integrable systems [31, 32].The research on conservation laws plays an important role in the study of the physical phenomena in nonlinear magnetized plasma. Conservation laws are a mathematical formulation, and they indicate that the total amount of a certain physical quantity remains the same during the evolution of a physical system [33, 34]. In 1918, Noether [35] proved that each conservation law is associated with an appropriate symmetry and can be derived from the Lagrangian function and the invariance principle. In 1996, Riewe [36] introduced the Lagrangian function for the fractional derivative. In the past two decades, many different types of fractional Euler-Lagrangian equations have been generalized. Based on these conclusion, some fractional generalizations of Noether’s theorem were proved [37], and many fractional conservation laws were obtained [38]. To study the conservation laws of the fractional differential equations, we use Lie symmetry analysis to construct the conserved vectors [39, 40]In this paper, applying the basic system of equations of ion-acoustic solitary waves [41], we develop a new 3D model. Using the new model, we study the conservation laws and the solution of ion-acoustic solitary waves. The rest of the paper is structured as follows: In Section 2, based on the basic system of equations of ion-acoustic solitary waves, we obtain a new 3D Schamel-KdV equation by using multi-scale analysis and the perturbation method [42]. A new 3D TSF-Schamel-KdV equation is obtained in Section 3 according to the new integer-order model and by using the semi-inverse method and the fractional variational principle [43, 44]. In Section 4, applying the Riemann-Liouville fractional derivative [39, 40], we discuss the conservation laws of the new fractional model. In Section 5, according to the Hirota bilinear method, we obtain the soliton solutions of the 3D TSF-Schamel-KdV equation. The propagation of solitary waves is important because it describes the characteristic nature of the interaction of the waves and the plasmas. Therefore, using soliton solutions [17, 18], we study the characteristics of motion of ion-acoustic solitary waves.


## 2. Derivation of the 3D Schamel-KdV Equation

We use the basic system of equations of ion-acoustic solitary waves given by(1)∂n∂t+∂nu∂x+∂nv∂y+∂nw∂z=0,∂u∂t+u∂u∂x+v∂u∂y+w∂u∂z=-∂ϕ∂x+uΛΩex,∂v∂t+u∂v∂x+v∂v∂y+w∂v∂z=-∂ϕ∂y+vΛΩex,∂w∂t+u∂w∂x+v∂w∂y+w∂w∂z=-∂ϕ∂z+wΛΩex,∂2ϕ∂x2+∂2ϕ∂y2+∂2ϕ∂z2=ne-n,where n is the ion number density, and u, v, w are the ion fluid velocities in the x-, y-, and z-directions, respectively. ϕ is the electric field potential, ne is the electron number density, and Ω is the uniform external magnetic field. Ion-acoustic solitary waves are assumed to propagate in the x-direction, and the direction is specified by the unit vector ex.We consider the propagation of ion-acoustic solitary waves in 3D space(x,y,z) and introduce the following independent stretched variables:(2)T=ϵ3/4t,X=ϵ1/4x-t,Y=ϵ1/4y,Z=ϵ1/4z,where ϵ is a small parameter characterizing the strength of the nonlinearity. Thus, we can obtain(3)∂∂t=ϵ3/4∂∂T-ϵ1/4∂∂X,∂∂x=ϵ1/4∂∂X,∂∂y=ϵ1/4∂∂Y,∂∂z=ϵ1/4∂∂Z.The dependent variables are expanded in the following form:(4)nX,Y,Z,T=1+ϵn1X,Y,Z,T+ϵ3/2n2X,Y,Z,T+⋯,ϕX,Y,Z,T=ϵϕ1X,Y,Z,T+ϵ3/2ϕ2X,Y,Z,T+⋯,uX,Y,Z,T=ϵu1X,Y,Z,T+ϵ3/2u2X,Y,Z,T+⋯,vX,Y,Z,T=ϵ5/4v1X,Y,Z,T+ϵ3/2v2X,Y,Z,T+⋯,wX,Y,Z,T=ϵ5/4w1X,Y,Z,T+ϵ3/2w2X,Y,Z,T+⋯,neX,Y,Z,T=1+ϕX,Y,Z,T-43bϕ3/2X,Y,Z,T+12ϕ2X,Y,Z,T,and the boundary conditions are given by(5)n=ne=1,u=v=w=ϕ=0,asX⟶∞.Substituting (3) and (4) into (1), we can obtain the approximate equations for ϵ in the following form:(6)ϵ:n1=ϕ1,(7)ϵ5/4:∂ϕ1∂X=∂u1∂X=∂n1∂X,∂ϕ1∂Y=v1ΛΩex=θw1,∂ϕ1∂Z=w1ΛΩex=-θv1,(8)ϵ3/2:∂v1∂Y=-∂w1∂Z,∂v1∂Y=-v2ΛΩex=-θw2,∂w1∂X=-w2ΛΩex=θv2,(9)ϵ7/4:∂n1∂T-∂n2∂X+∂u2∂X+∂v2∂Y+∂w2∂Z=0,∂u1∂T-∂u2∂X=∂ϕ2∂X,∂v2∂X=∂ϕ2∂Y,∂w2∂X=∂ϕ2∂Z.According to (6) and (7), we can obtain(10)ϕ1=u1=n1,w1=1θ∂ϕ1∂Y,v1=-1θ∂ϕ1∂Z.Substituting (10) into (8) and (9) and eliminating ϕ2, u2, v2, w2 and n2, we can obtain(11)∂3ϕ1∂X3+∂3ϕ1∂X∂Y2+∂3ϕ1∂X∂Z2+2bϕ11/2∂ϕ1∂X+2∂ϕ1∂T=0.Letting ϕ1(X,Y,Z,T)=A(X,Y,Z,T), (11) can be rewritten as(12)AT+a1AAX+a2AXXX+a3AXYY+a4AXZZ=0,where a1=b, a2=1/2 and a3=a4=(1/2)(1+1/θ2).Remark 1.
Because of the nonlinear termAAX, when a1≠0 and a3=a4=0, (12) can be reduced to the 1D Schamel-KdV equation. When a3=a4≠0, (12) is a 3D equation. Therefore, (12) is called the 3D Schamel-KdV equation. Compared to the KdV and mKdV models [6], the nonlinearity of the 3D Schamel-KdV equation is relatively weak. Therefore, the 3D Schamel-KdV equation presents a new research direction for the study of ion-acoustic solitary waves.


## 3. Derivation of the 3D TSF-Schamel-KdV Equation

In Section2, we have obtained a new 3D integer-order Schamel-KdV equation. To learn more about ion-acoustic solitary waves, we seek to obtain the 3D TSF-Schamel-KdV equation by using the semi-inverse method and the fractional variational principle. First, we introduce some definitions as follows.Definition 2 (see [44]).
The left Riemann-Liouville fractional derivation of a functionA(X,Y,Z,T) is defined as(13)D0TωA=1Γn-ωdndTn∫0TT-tn-ω-1Adt,n-1≤ω<n.Definition 3 (see [45]).
The Riemann-Liouville fractional derivation of a functionA(X,Y,Z,T) is defined as(14)
D
T
ω
A
=
∂
ω
A
∂
T
ω
=
1
Γ
n
-
ω
∂
n
∂
T
n
∫
0
T
0
T
-
s
n
-
ω
-
1
A
d
s
,
n
-
1
≤
ω
<
n
,
∂
n
A
∂
T
n
,
ω
=
n
.According to the integer-order 3D Schamel-KdV equation,(15)AT+a1AAX+a2AXXX+a3AXYY+a4AXZZ=0,assuming A(X,Y,Z,T)=BX(X,Y,Z,T), where B(X,Y,Z,T) is a potential function, and therefore, the potential equation of the 3D Schamel-KdV equation can be written in the following form:(16)BXT+a1BXBXX+a2BXXXX+a3BXXYY+a4BXXZZ=0.Then, the function of the potential equation (16) can be described as(17)JB=∭VdXdYdZ∫T∗dTBX,Y,Z,Tb1BXT+b2a1BXBXX+b3a2BXXXX+b4a3BXXYY+b5a4BXXZZ,where bi, i=1,2,3,4,5, are Lagrangian multipliers which can be obtained later.Using integration by parts for (17) and taking BXR=BYR=BZR=BTT∗=BXXR=BXYR=BXZR=0, we obtain(18)JB=∭VdXdYdZ∫T∗dT-b1BTBX-23b2a1BX5/2+b3a2BXX2+b4a3BXY2+b5a4BXZ2.Using the variation of the above function, integrating each term by parts and applying the variation optimum condition, we obtain(19)FX,Y,Z,T,B,BT,BX,BXX,BXY,BXZ=∂F∂B-∂∂T∂F∂BT-∂∂X∂F∂BX+∂2∂X2∂F∂BXX+∂∂X∂Y∂F∂BXY+∂∂X∂Z∂F∂BXZ=2c1BXT+52b2a1BX1/2BXX+2b3a2BXXXX+2b4a3BXXYY+2b5a4BXXZZ=0.Comparing (19) with (16), we obtain the following Lagrangian multipliers:(20)b1=12,b2=15,b3=12,b4=12,b5=12.Therefore, the Lagrangian form of the integer-order 3D Schamel-KdV equation is given by(21)LBT,BX,BXX,BXY,BXZ=-12BTBX-415a1BX5/2+12a2BXX2+12a3BXY2+12a4BXZ2.Similarly, the Lagrangian form of the 3D TSF-Schamel-KdV equation is given by(22)LDTωB,DXαB,DXααB,DXαDYβB,DXαDZγB=-12DTωBDXαB-415a1DXαB5/2+12a2DXααB2+12a3DXαDYβB2+12a4DXαDZγB2,where DXααB=DXα(DXαB). Thus, the function of the 3D TSF-Schamel-KdV equation can be obtained as(23)JFB=∫RdXα∫RdYα∫RdZγ∫T∗dTωFDTωB,DXαB,DXααB,DXαDYβB,DXαDZγB.According to the Agrawal’s method [46, 47], the variation of functional Eq. (23) can be written as(24)δJFB=∫RdXα∫RdYα∫RdZγ∫T∗dTω∂F∂DTωBδDTωB+∂F∂DXαBδDXαB+∂F∂DXααBδDXααB+∂F∂DXαDYβBδDXαDYβB+∂F∂DXαDZγBδDXαDZγB,where(25)∫aTdτjfτ=j∫aTdxT-τjfτ.Using the fractional integration by parts,(26)∫abdτjfzDzjgz=Γ1+jgzfzab-∫abdzjgzDzjfz,fz,gz∈a,b,we can obtain(27)δJFB=∫RdXα∫RdYα∫RdZγ∫T∗dTω-DTω∂F∂DTωB-DXα∂F∂DXαB+DXαα∂F∂DXααB+DXαDYβ∂F∂DXαDYβB+DXαDZγ∂F∂DXαDZγB.Optimizing the variation Eq. (24), δJF(B)=0, we can obtain the Euler-Lagrange equation of the 3D TSF-Schamel-KdV equation as(28)-DTω∂F∂DTωB-DXα∂F∂DXαB+DXαα∂F∂DXααB+DXαDYβ∂F∂DXαDYβB+DXαDZγ∂F∂DXαDZγB=0.Substituting (22) into (28), we obtain(29)DTωDXαB+a1DXαB1/2DXααB+a2DXααααB+a3DXααDYββB+a4DXααDZγγB=0.Letting DXαB(X,Y,Z,T)=A(X,Y,Z,T) and substituting DXαB(X,Y,Z,T) into (29), we can obtain(30)DTωA+a1ADXαA+a2DXαααA+a3DXαDYββA+a4DXαDZγγA=0.Eq. (30) is the 3D TSF-Schamel-KdV equation.


## 4. Conservation Laws of the 3D TSF-Schamel-KdV Equation

### 4.1. Lie Symmetry Analysis

In the previous section, we have obtained the 3D TSF-Schamel-KdV equation. To learn about the properties of the new model, we study the conservation laws [48, 49]. First, we convert (30) to the following fractional partial differential equation form:(31)DTγA=QX,Y,Z,T,Am,DTωA,DXαA,DXαααA,DXαDYββA,DXαDZγγA,…,ω,α,β,γ>0.We assume that (31) is invariant under a one parameter Lie group of point transformations in the following form:(32)X′=x+ϵξX,Y,Z,T,A+Oϵ2,Y′=Y+ϵζX,Y,Z,T,A+Oϵ2,Z′=Z+ϵψX,Y,Z,T,A+Oϵ2,T′=T+ϵτX,Y,Z,T,A+Oϵ2,A′=x+ϵηX,Y,Z,T,A+Oϵ2,DTωA′⟶DTωA+ϵηωT+Oϵ2,DXαA′⟶DXαA+ϵηαX+Oϵ2,DXαααA′⟶DXαααA+ϵηαXXX+Oϵ2,DXαDYββA′⟶DXαDYββA+ϵηα,βXYY+Oϵ2,DXαDZγγA′⟶DXαDZγγA+ϵηα,γXZZ+Oϵ2,where ξ, ζ, ψ, τ, and η are infinitesimal functions, and ηωT, ηαX, ηαXXX, ηα,γXYY, and ηα,γXZZ are the prolongations of infinitesimal functions defined as(33)ηγT=DTωη+ξDTωAX-DTωξAX+ζDTωAY-DTωζAY+ψDTωAZ-DTωψAZ+DTωDTτA-DTγ+1τA+τDTγ+1A,ηαX=DXαη+DXαADXξ-DYβADXζ-DZγADZψ,ηαXXX=DXαηαXX-AXXXDXαξ-AXXYDXαζ-AXXZDZψ-AXXTDXατ,ηα,βXYY=DXαηβYY-AXXYDXαξ-AXYYDXαζ-AXYZDXαψ-AXYTDXατ,ηα,βXZZ=DXαηγZZ-AXXZDXαξ-AXYZDXαζ-AXZZDXαψ-AXZTDXατ,where DT and DX are the total derivative operators given by(34)DT=∂∂T+AT∂∂T+ATT∂∂AT+AXT∂∂AX+AYT∂∂AY+AZT∂∂AZ+⋯,DX=∂∂X+AX∂∂A+AXX∂∂AX+ATX∂∂AT+AYX∂∂AY+AZX∂∂AZ+⋯.Applying the generalized Leibnitz rule as given by(35)DTωftgt=∑n=0∞ωnDtω-nftDtngt,ω>0,where(36)ωn=-1n-1ωΓn-ωΓ1-ωΓn+1,and the chain rule for a compound function defined as(37)dmfgtdtm=∑k=0m∑r=0kkr1k!-gtk-rdkfgtdtk,we can obtain the following equation:(38)ηωT=DTωη-ωDTωτ∂ωA∂Tω-∑n=1∞ωnDTnξDTω-nAX-∑n=1∞ωnDTnζ·DTω-nAY-∑n=1∞ωnDTnψDTω-nAZ-∑n=1∞ωn+1DTn+1τDTω-nA.For the chain rule given by (37), when f(t)=1, we obtain(39)DTω=∂ωη∂Tω+ηA∂ωA∂Tω-A∂ωηA∂Tω+∑n=1∞ωn∂nηA∂TnDTω-nA+Ra,where(40)Ra=∑n=2∞∑m=2n∑k=2m∑r=0k-1ωnnmkr1k!Tn-ωΓn+1-ω-Ar∂A∂TAAk-r∂n-m+kη∂Tn-m∂Ak.Therefore, (38) can be rewritten as(41)ηωT=∂ωη∂Tω+ηA-ωDTτ∂ωA∂Tω-A∂ωηA∂Tω+∑n=1∞ωn∂ωηA∂Tω-ωn+1DTn+1τDTω-n-∑n=1∞ωnDTnξDTω-nAX+DTnζDTω-nAY+DTnψDTω-nAZ+Ra.Similarly, using the generalized Leibnitz rule and the chain rule for a compound function, we also obtain the following equation:(42)ηαX=∂αη∂Xα+ηA-αDXξ∂αA∂Xα-A∂αηA∂Xα+∑n=1∞αn∂αηA∂Xα-αn+1DTn+1ξDXα-n-∑n=1∞αnDTnξDTα-nAY+DXnψDXα-nAZ+DTnτDTα-nAT+Rb,where(43)Rb=∑n=2∞∑m=2n∑k=2m∑r=0k-1αnnmkr1k!Xn-αΓn+1-α-Ar∂m∂XmAk-r∂n-m+kη∂Xn-m∂Ak.The infinitesimal generatorM can be defined as follows:(44)M=ξ∂∂X+ζ∂∂Y+ψ∂∂Z+τ∂∂T+η∂∂A.Under the infinitesimal transformations, the invariance of the system (31) leads to the following invariance condition:(45)PrnMΔΔ=0=0,n=1,2,3,…,Δ=DTωA+a1ADXαA+a2DXαααA+a3DXαDYββA+a4DXαDZγγA.According to (42) and (43), we can obtain(46)Prω,α,β,γ,4MΔ=τ∂ω∂Tω+ξ∂α∂Xα+ζ∂β∂Yβ+ψ∂γ∂Zγ+η∂∂A+ηγT∂∂DTγA+ηαX∂∂DXαA+ηαXXX∂∂DXαααA+ηα,βXYY∂∂DXαDYββA+ηα,γXZZ∂∂DXαDZγγA.Then, we can obtain the following invariance criterion:(47)ηωT+12a11AηDXαA+a1AηαX+a2ηαXXX+a3ηα,βXYY+a4ηα,γXZZ=0.Substituting (33), (34), (41), and (42) into (47) and equating the coefficients of alike partial derivatives, fractional derivatives and powers of A, the set of determining equations can be obtained as(48)ωn∂γηA∂Tγ-ωn+1DTn+1τ=0,αn∂αηA∂Xα-αn+1DXn+1ξ=0,ξA=ξT=0,ζA=ζT=0,τX=τY=0,η=A3ξX-ωτT,ηAA=0,αξX-ωτT=0,∂ωη∂Tω-A∂ωηA∂Tω+a1A∂αη∂Xα-A∂αηA∂Xα+a2DXαααη+a3DXαDYββη+a4DXαDZγγη=0.By solving the above equations, we can obtain a series of Lie algebra of point symmetries as(49)τ=c2αT+c1,ξ=c2ωX+c3,ζ=c2γY+c4Z+c5,ψ=c2βZ-c4Y+c6,η=-c2α3-ωA.Hence, a series of Lie algebra of point symmetries can be written as(50)M1=∂∂T,M2=∂∂X,M3=∂∂Y,M4=∂∂Z,M5=ωX∂∂X+γY∂∂Y+βZ∂∂Z+αT∂∂T-α3-ωA∂∂A.


### 4.2. Conservation Laws

We have obtained the Lie symmetry generator in Section4.2. In this section, we will discuss conservation laws of the 3D TSF-Schamel-KdV equation based on the obtained Lie symmetry generator. We know that the conservation laws of (30) satisfy the following equation:(51)DTCT+DXCX+DYCY+DZCZ=0,where CT, CX, CY and CZ are the conserved vectors.A formal Lagrangian for the 3D TSF-Schamel-KdV equation can be presented as(52)L=sX,Y,Z,TDTωA+a1ADXαA+a2DXαααA+a3DXαDYββA+a4DXαDZγγA,where s(X,Y,T) is a new dependent variable. According to the formal Lagrangian, an action integral is defined as(53)∫R∫R∫R∫T∗LX,Y,Z,T,A,s,DTωA,DXαA,DXαααA,DXαDYββA,DXαDZγγAdXdYdZdT.Therefore, we can obtain the adjoint equation of (30) as the Euler-Lagrange equation(54)F∗=δLδA=0,where δ/δm is the Euler-Lagrange operator defined as(55)δδA=∂∂A+DTω∗∂∂DTω+DXα∗∂∂DXαA-DXααα∗∂∂DXαααA-DXαDYββ∗∂∂DXαDYββA-DXαDZγγ∗∂∂DXαDZγγA,where (DTω)∗, (DXα)∗, (DXααα)∗, (DXαDYββ)∗, and (DXαDZγγ)∗ are the adjoint operators of the Riemann-Liouville fractional differential operators DTω, DXα, DXααα, DXαDYββ, and DXαDZγγ, respectively. These are given by(56)DTω∗=-1nIpn-ωDTn=DTCpω,DXα∗=-1mIqm-αDXm=DXCqα,where Ipn-ω and Iqm-α are the right-sided fractional integral operators of orders n-ω and m-β, respectively. TCDpω and XCDqα are the right-sided Caputo fractional differential operators of orders ω and α, respectively. Therefore, the adjoint equation (54) can be rewritten as(57)F∗=DTγ∗s+a1ADXα∗s-a2DXααα∗s-a3DXαDYββ∗s-a4DXαDZγγ∗s=0.Based on Section4.1, we obtain infinitesimal symmetry of (30). We assume that the Lie characteristic function W is given by(58)W=η-τAT-ξAX-ζAY-ψAZ.Applying this on the M5 of the symmetry (50), we obtain(59)W1=AX,W2=AY,W3=AZ,W4=AT,W5=-α3-ωA-αTAT-ωXAX-γYAY-βZAZ.Using the Riemann-Liouville fractional derivative, the components of the conserved vectors of (30) are defined as(60)CT=τI+∑k=0n-1-10kDTγ-1-kWDTk∂L∂D0TγA--1nJW,DTn∂L∂D0TγA,CX=ξI+W∂L∂DXαA+DXαα∂L∂DXαααA+DYββ∂L∂DXαDYββA+DZγγ∂L∂DXαDZγγA-DXαWDXα∂L∂DXαααA-DYβWDYβ∂L∂DXαDYββA-DZγWDZγ∂L∂DXαDZγγA+DXααW∂L∂DXαααA+DYββW∂L∂DXαDYββA+DZγγW∂L∂DXαDZγγA,CY=ζI+WDXαDYβ∂L∂DXαDYββA-DYβWDXα∂L∂DXαDYββA-DXαWDYβ∂L∂DXαDYββA+DXαDYβW∂L∂DXαDYββA,CZ=ζI+WDXαDZγ∂L∂DXαDZγγA-DZγWDXα∂L∂DXαDZγγm-DXαWDZγ∂L∂DXαDZγγA+DXαDZγW∂L∂DXαDZγγA,where n=[ω]+1, and J is the integral given by(61)Ja,b=1Γn-ω∫0T∫TT0fτ,X,Ygμ,X,Yμ-τω+1-ndμdτ,with the property that(62)DTJf,g=fTIT′n-γg-g0ITn-γf.The conservation laws of the 3D TSF-Schamel-KdV equation are explained in detail below (see the appendix).


## 4.1. Lie Symmetry Analysis

In the previous section, we have obtained the 3D TSF-Schamel-KdV equation. To learn about the properties of the new model, we study the conservation laws [48, 49]. First, we convert (30) to the following fractional partial differential equation form:(31)DTγA=QX,Y,Z,T,Am,DTωA,DXαA,DXαααA,DXαDYββA,DXαDZγγA,…,ω,α,β,γ>0.We assume that (31) is invariant under a one parameter Lie group of point transformations in the following form:(32)X′=x+ϵξX,Y,Z,T,A+Oϵ2,Y′=Y+ϵζX,Y,Z,T,A+Oϵ2,Z′=Z+ϵψX,Y,Z,T,A+Oϵ2,T′=T+ϵτX,Y,Z,T,A+Oϵ2,A′=x+ϵηX,Y,Z,T,A+Oϵ2,DTωA′⟶DTωA+ϵηωT+Oϵ2,DXαA′⟶DXαA+ϵηαX+Oϵ2,DXαααA′⟶DXαααA+ϵηαXXX+Oϵ2,DXαDYββA′⟶DXαDYββA+ϵηα,βXYY+Oϵ2,DXαDZγγA′⟶DXαDZγγA+ϵηα,γXZZ+Oϵ2,where ξ, ζ, ψ, τ, and η are infinitesimal functions, and ηωT, ηαX, ηαXXX, ηα,γXYY, and ηα,γXZZ are the prolongations of infinitesimal functions defined as(33)ηγT=DTωη+ξDTωAX-DTωξAX+ζDTωAY-DTωζAY+ψDTωAZ-DTωψAZ+DTωDTτA-DTγ+1τA+τDTγ+1A,ηαX=DXαη+DXαADXξ-DYβADXζ-DZγADZψ,ηαXXX=DXαηαXX-AXXXDXαξ-AXXYDXαζ-AXXZDZψ-AXXTDXατ,ηα,βXYY=DXαηβYY-AXXYDXαξ-AXYYDXαζ-AXYZDXαψ-AXYTDXατ,ηα,βXZZ=DXαηγZZ-AXXZDXαξ-AXYZDXαζ-AXZZDXαψ-AXZTDXατ,where DT and DX are the total derivative operators given by(34)DT=∂∂T+AT∂∂T+ATT∂∂AT+AXT∂∂AX+AYT∂∂AY+AZT∂∂AZ+⋯,DX=∂∂X+AX∂∂A+AXX∂∂AX+ATX∂∂AT+AYX∂∂AY+AZX∂∂AZ+⋯.Applying the generalized Leibnitz rule as given by(35)DTωftgt=∑n=0∞ωnDtω-nftDtngt,ω>0,where(36)ωn=-1n-1ωΓn-ωΓ1-ωΓn+1,and the chain rule for a compound function defined as(37)dmfgtdtm=∑k=0m∑r=0kkr1k!-gtk-rdkfgtdtk,we can obtain the following equation:(38)ηωT=DTωη-ωDTωτ∂ωA∂Tω-∑n=1∞ωnDTnξDTω-nAX-∑n=1∞ωnDTnζ·DTω-nAY-∑n=1∞ωnDTnψDTω-nAZ-∑n=1∞ωn+1DTn+1τDTω-nA.For the chain rule given by (37), when f(t)=1, we obtain(39)DTω=∂ωη∂Tω+ηA∂ωA∂Tω-A∂ωηA∂Tω+∑n=1∞ωn∂nηA∂TnDTω-nA+Ra,where(40)Ra=∑n=2∞∑m=2n∑k=2m∑r=0k-1ωnnmkr1k!Tn-ωΓn+1-ω-Ar∂A∂TAAk-r∂n-m+kη∂Tn-m∂Ak.Therefore, (38) can be rewritten as(41)ηωT=∂ωη∂Tω+ηA-ωDTτ∂ωA∂Tω-A∂ωηA∂Tω+∑n=1∞ωn∂ωηA∂Tω-ωn+1DTn+1τDTω-n-∑n=1∞ωnDTnξDTω-nAX+DTnζDTω-nAY+DTnψDTω-nAZ+Ra.Similarly, using the generalized Leibnitz rule and the chain rule for a compound function, we also obtain the following equation:(42)ηαX=∂αη∂Xα+ηA-αDXξ∂αA∂Xα-A∂αηA∂Xα+∑n=1∞αn∂αηA∂Xα-αn+1DTn+1ξDXα-n-∑n=1∞αnDTnξDTα-nAY+DXnψDXα-nAZ+DTnτDTα-nAT+Rb,where(43)Rb=∑n=2∞∑m=2n∑k=2m∑r=0k-1αnnmkr1k!Xn-αΓn+1-α-Ar∂m∂XmAk-r∂n-m+kη∂Xn-m∂Ak.The infinitesimal generatorM can be defined as follows:(44)M=ξ∂∂X+ζ∂∂Y+ψ∂∂Z+τ∂∂T+η∂∂A.Under the infinitesimal transformations, the invariance of the system (31) leads to the following invariance condition:(45)PrnMΔΔ=0=0,n=1,2,3,…,Δ=DTωA+a1ADXαA+a2DXαααA+a3DXαDYββA+a4DXαDZγγA.According to (42) and (43), we can obtain(46)Prω,α,β,γ,4MΔ=τ∂ω∂Tω+ξ∂α∂Xα+ζ∂β∂Yβ+ψ∂γ∂Zγ+η∂∂A+ηγT∂∂DTγA+ηαX∂∂DXαA+ηαXXX∂∂DXαααA+ηα,βXYY∂∂DXαDYββA+ηα,γXZZ∂∂DXαDZγγA.Then, we can obtain the following invariance criterion:(47)ηωT+12a11AηDXαA+a1AηαX+a2ηαXXX+a3ηα,βXYY+a4ηα,γXZZ=0.Substituting (33), (34), (41), and (42) into (47) and equating the coefficients of alike partial derivatives, fractional derivatives and powers of A, the set of determining equations can be obtained as(48)ωn∂γηA∂Tγ-ωn+1DTn+1τ=0,αn∂αηA∂Xα-αn+1DXn+1ξ=0,ξA=ξT=0,ζA=ζT=0,τX=τY=0,η=A3ξX-ωτT,ηAA=0,αξX-ωτT=0,∂ωη∂Tω-A∂ωηA∂Tω+a1A∂αη∂Xα-A∂αηA∂Xα+a2DXαααη+a3DXαDYββη+a4DXαDZγγη=0.By solving the above equations, we can obtain a series of Lie algebra of point symmetries as(49)τ=c2αT+c1,ξ=c2ωX+c3,ζ=c2γY+c4Z+c5,ψ=c2βZ-c4Y+c6,η=-c2α3-ωA.Hence, a series of Lie algebra of point symmetries can be written as(50)M1=∂∂T,M2=∂∂X,M3=∂∂Y,M4=∂∂Z,M5=ωX∂∂X+γY∂∂Y+βZ∂∂Z+αT∂∂T-α3-ωA∂∂A.


## 4.2. Conservation Laws

We have obtained the Lie symmetry generator in Section4.2. In this section, we will discuss conservation laws of the 3D TSF-Schamel-KdV equation based on the obtained Lie symmetry generator. We know that the conservation laws of (30) satisfy the following equation:(51)DTCT+DXCX+DYCY+DZCZ=0,where CT, CX, CY and CZ are the conserved vectors.A formal Lagrangian for the 3D TSF-Schamel-KdV equation can be presented as(52)L=sX,Y,Z,TDTωA+a1ADXαA+a2DXαααA+a3DXαDYββA+a4DXαDZγγA,where s(X,Y,T) is a new dependent variable. According to the formal Lagrangian, an action integral is defined as(53)∫R∫R∫R∫T∗LX,Y,Z,T,A,s,DTωA,DXαA,DXαααA,DXαDYββA,DXαDZγγAdXdYdZdT.Therefore, we can obtain the adjoint equation of (30) as the Euler-Lagrange equation(54)F∗=δLδA=0,where δ/δm is the Euler-Lagrange operator defined as(55)δδA=∂∂A+DTω∗∂∂DTω+DXα∗∂∂DXαA-DXααα∗∂∂DXαααA-DXαDYββ∗∂∂DXαDYββA-DXαDZγγ∗∂∂DXαDZγγA,where (DTω)∗, (DXα)∗, (DXααα)∗, (DXαDYββ)∗, and (DXαDZγγ)∗ are the adjoint operators of the Riemann-Liouville fractional differential operators DTω, DXα, DXααα, DXαDYββ, and DXαDZγγ, respectively. These are given by(56)DTω∗=-1nIpn-ωDTn=DTCpω,DXα∗=-1mIqm-αDXm=DXCqα,where Ipn-ω and Iqm-α are the right-sided fractional integral operators of orders n-ω and m-β, respectively. TCDpω and XCDqα are the right-sided Caputo fractional differential operators of orders ω and α, respectively. Therefore, the adjoint equation (54) can be rewritten as(57)F∗=DTγ∗s+a1ADXα∗s-a2DXααα∗s-a3DXαDYββ∗s-a4DXαDZγγ∗s=0.Based on Section4.1, we obtain infinitesimal symmetry of (30). We assume that the Lie characteristic function W is given by(58)W=η-τAT-ξAX-ζAY-ψAZ.Applying this on the M5 of the symmetry (50), we obtain(59)W1=AX,W2=AY,W3=AZ,W4=AT,W5=-α3-ωA-αTAT-ωXAX-γYAY-βZAZ.Using the Riemann-Liouville fractional derivative, the components of the conserved vectors of (30) are defined as(60)CT=τI+∑k=0n-1-10kDTγ-1-kWDTk∂L∂D0TγA--1nJW,DTn∂L∂D0TγA,CX=ξI+W∂L∂DXαA+DXαα∂L∂DXαααA+DYββ∂L∂DXαDYββA+DZγγ∂L∂DXαDZγγA-DXαWDXα∂L∂DXαααA-DYβWDYβ∂L∂DXαDYββA-DZγWDZγ∂L∂DXαDZγγA+DXααW∂L∂DXαααA+DYββW∂L∂DXαDYββA+DZγγW∂L∂DXαDZγγA,CY=ζI+WDXαDYβ∂L∂DXαDYββA-DYβWDXα∂L∂DXαDYββA-DXαWDYβ∂L∂DXαDYββA+DXαDYβW∂L∂DXαDYββA,CZ=ζI+WDXαDZγ∂L∂DXαDZγγA-DZγWDXα∂L∂DXαDZγγm-DXαWDZγ∂L∂DXαDZγγA+DXαDZγW∂L∂DXαDZγγA,where n=[ω]+1, and J is the integral given by(61)Ja,b=1Γn-ω∫0T∫TT0fτ,X,Ygμ,X,Yμ-τω+1-ndμdτ,with the property that(62)DTJf,g=fTIT′n-γg-g0ITn-γf.The conservation laws of the 3D TSF-Schamel-KdV equation are explained in detail below (see the appendix).


## 5. Multi-Soliton Solutions for the 3D TSF-Schamel-KdV Equation

The solution of the model is a relatively broad research area in science [50, 51]. In this section, using the simplified Hirota bilinear method [24, 52], we seek multiple soliton solutions of the 3D TSF-Schamel-KdV equation.First, we introduce the following fractional transforms:(63)T′=p1TωΓ1+ω,X′=p2XαΓ1+α,Y′=p3YβΓ1+β,Z′=p4ZγΓ1+γ,where p1, p2, p3 and p4 are constants. Using the above transformations and omitting the apostrophe, we can convert the derivatives into classical derivatives,(64)∂ωA∂Tω=p1∂A∂T,∂αA∂Xα=p2∂A∂X,∂βA∂Yβ=p3∂A∂Y,∂γA∂Zγ=p4∂A∂Z.Then, (30) can be described as(65)AT+a1AAX+a2AXXX+a3AXYY+a4AXZZ=0.We assume that the solution of (65) has the form(66)AX,Y,Z,T=eθiX,Y,Z,T,where(67)θiX,Y,Z,T=kiX+riY+qiZ-uiT.Substituting (66) and (67) into the linear term of (65), we can obtain the following dispersion relation:(68)ui=a2ki3+a3kiri2+a4kiqi2.Hence, θi can be written as(69)θiX,Y,Z,T=kiX+riY+qiZ-a2ki3+a3kiri2+a4kiqi2T.


### 5.1. Single-Soliton Solution

We assume that the single-soliton solution of (65) has the following form:(70)AX,Y,Z,T=Rln⁡fXX,where f(X,Y,Z,T) is the auxiliary function defined as(71)fX,Y,Z,T=1+eθ1X,Y,Z,T=1+ek1X+r1Y+q1Z-a2k13+a3k1r12+a4k1q12T.Substituting (70) into (65), we obtain(72)R=18a2a12.Substituting (71) and (72) into (70), we obtain the following single-soliton solution:(73)AX,Y,Z,T=18a2a12k12eθ1X,Y,Z,T1+eθ1X,Y,Z,T2=18a2a12k12ek1X+r1Y+q1Z-a2k13+a3k1r12+a4k1q12T1+ek1X+r1Y+q1Z-a2k13+a3k1r12+a4k1q12T2.The above equation can be rewritten as(74)AX,Y,Z,T=18a2a12k12sech2⁡θ1X,Y,Z,T2,where(75)θ1=k1XαΓ1+α+r1YβΓ1+β+q1ZγΓ1+γ-a2k13+a3k1r12+a4k1q12TωΓ1+ω.


### 5.2. Two-Soliton Solution

We assume that the two-soliton solution has the following form:(76)fX,Y,Z,T=1+eθ1X,Y,Z,T+eθ2X,Y,Z,T+a12eθ1X,Y,Z,T+θ2X,Y,Z,T,where θ1 and θ2 are defined in (69). We know that(77)AX,Y,Z,T=18a2a12ln⁡fXX,and substituting this expression into (65), the coefficient a12 can be obtained as(78)a12=k12+k22-k1k2k12+k22+2k1k2.Therefore, the two-soliton solution for (65) has the following form:(79)AX,Y,Z,T=18a2a12k12eθ1+k22eθ2+a12k22eθ1+k12eθ2+a12k1+k22+k1-k22eθ1+θ21+eθ1+eθ2+a12eθ1+θ22,where(80)θ1=k1XαΓ1+α+r1YβΓ1+β+q1ZγΓ1+γ-a2k13+a3k1r12+a4k1q12TωΓ1+ω,θ2=k2XαΓ1+α+r2YβΓ1+β+q2ZγΓ1+γ-a2k23+a3k2r22+a4k2q22TωΓ1+ω.


### 5.3. Three-Soliton Solution

To investigate the three-soliton solution of (65), we assume that the auxiliary function f(X,Y,Z,T) has the following form:(81)fX,Y,Z,T=1+eθ1+eθ2+eθ3+a12eθ1+θ2a13eθ1+θ3+a23eθ2+θ3+a123eθ1+θ2+θ3,where(82)aij=ki2+kj2-kikjki2+kj2+2kikj.Substituting (77) and (81) into (65), we find the following pattern:(83)a123=a12a13a123.According to the pattern obtained in Section5.3, the N-soliton solutions for the 3D TSF-Schamel-KdV equation can be obtained, where N≥1. Based on the single-soliton solution and the two-soliton solution, we can study the characteristics of the motion of the ion-acoustic solitary waves.In this section, we describe the interaction of two small ion-acoustic solitary waves with finite amplitude in a weakly relativistic 3D magnetic plasma. Then, we can study the characteristics of motion of the solitary waves by changing the coefficients. Based on the single-soliton solution of ion-acoustic solitary waves, we obtain the evolution plots of the ion-acoustic solitary waves (see Figure1). Figure 1 shows that the solitonic amplitude increases with an increase in the a2/a1 ratio, and the initial superimposed solitons travel different distances over a period of time for the different choices of a1 and a2. Therefore, we conclude that the soliton moves along the positive x-axis with constant amplitude and velocity.Figure 1
Plots for the evolution of the one-soliton solution of ion-acoustic solitary waves with (a):t=1, a1=1, a3=0, a4=1, k1=1, r1=0.4, q1=0.4, α=0.8, β=1, γ=1, ω=2; (b): t=3, a1=1, a3=0, a4=1, k1=1, r1=0.4, q1=0.4, α=0.8, β=1, γ=1, ω=1.5; (c): t=5, a1=1, a3=0, a4=1, k1=1, r1=0.4, q1=0.4, α=0.8, β=1, γ=1, ω=1.5, and a2=0.1 (solid line), a2=0.09 (dotted line), a2=0.08 (dashed line).
(a) (b) (c)Examination of Figure2(a) shows that the propagation trajectory of the soliton exhibits a periodic oscillation. Figure 2(a) shows the curve propagation trajectory with constant amplitude and constantly changing velocity, where the velocity changes with time. Furthermore, Figure 2(b) shows the two-soliton interaction with constantly changing velocity. When T→0, the trajectory is sinusoidal with periodic oscillation. Otherwise, when T is far from the origin, the trajectory is parabolic-like. It can be seen from Figure 2(c) that the soliton generates a peak at the time of the interaction. Based on this, we conclude that, in addition to the periodic oscillation of the solitons in the local region, the large-scale propagation trajectories for such a structure show parabolic-type curves. Thus, if the variable coefficients are taken to have other forms, the corresponding characteristic curves will present different behaviors.Figure 2
Interaction of the two solitons of ion-acoustic solitary waves with (a):a1=1, a2=0.1, a3=1, a4=1, k1=1.6, r1=1, q1=1, k2=-1.9, r2=1, q2=1, α=0.9, β=0.8, γ=0.9, ω=1.9; (b): a1=1, a2=0.1, a3=1, a4=1, k1=1.6, r1=1, q1=1, k2=-1.9, r2=1, q2=1, α=0.7, β=1, γ=1, ω=2.8; (c): a1=1, a2=0.1, a3=1, a4=1, k1=1.6, r1=1, q1=1, k2=-1.9, r2=1, q2=1, α=0.8, β=1, γ=1, ω=2.8.
(a) (b) (c)Remark 4.
The present study describes the propagation and interaction of two small but finite amplitude ion-acoustic solitary waves in a weakly relativistic 3D magnetized plasma. Our conclusions can be considered as a generalization of the model suggested by Nejoh [53] and Pakira and Chowdhury [54] by including the effect of different coefficients. It has been found that the initial superposed solitons travel different distances over a period of time for the different choices of a1 and a2 and that the solitonic amplitude increases with an increase in the a2/a1 ratio. Moreover, the time fractional order ω and space fractional orders α, β, and γ play an important role in higher-order perturbation theory in the variation of the soliton amplitude. We believe that our research may be of basic interest for particle trapping experiments. Compared to other solitary waves, the unique features of the ion-acoustic solitary waves are the existence of an ultra-low frequency regime for wave propagation and the high charging of the grains, which can fluctuate because of the collection of plasma currents onto dust surfaces. It is difficult to carry out a reasonable comparison between previous studies and the present work. Nevertheless, due to the flexibility provided by the nonextensive method, we suggest that the quantitative discrepancies between the theory and experiment can be reduced. Thus, the application of our model may be particularly interesting in some plasma environments, such as space-plasmas, laser-plasma interactions, and the plasma sheet boundary layer of the earth’s magnetosphere.


## 5.1. Single-Soliton Solution

We assume that the single-soliton solution of (65) has the following form:(70)AX,Y,Z,T=Rln⁡fXX,where f(X,Y,Z,T) is the auxiliary function defined as(71)fX,Y,Z,T=1+eθ1X,Y,Z,T=1+ek1X+r1Y+q1Z-a2k13+a3k1r12+a4k1q12T.Substituting (70) into (65), we obtain(72)R=18a2a12.Substituting (71) and (72) into (70), we obtain the following single-soliton solution:(73)AX,Y,Z,T=18a2a12k12eθ1X,Y,Z,T1+eθ1X,Y,Z,T2=18a2a12k12ek1X+r1Y+q1Z-a2k13+a3k1r12+a4k1q12T1+ek1X+r1Y+q1Z-a2k13+a3k1r12+a4k1q12T2.The above equation can be rewritten as(74)AX,Y,Z,T=18a2a12k12sech2⁡θ1X,Y,Z,T2,where(75)θ1=k1XαΓ1+α+r1YβΓ1+β+q1ZγΓ1+γ-a2k13+a3k1r12+a4k1q12TωΓ1+ω.


## 5.2. Two-Soliton Solution

We assume that the two-soliton solution has the following form:(76)fX,Y,Z,T=1+eθ1X,Y,Z,T+eθ2X,Y,Z,T+a12eθ1X,Y,Z,T+θ2X,Y,Z,T,where θ1 and θ2 are defined in (69). We know that(77)AX,Y,Z,T=18a2a12ln⁡fXX,and substituting this expression into (65), the coefficient a12 can be obtained as(78)a12=k12+k22-k1k2k12+k22+2k1k2.Therefore, the two-soliton solution for (65) has the following form:(79)AX,Y,Z,T=18a2a12k12eθ1+k22eθ2+a12k22eθ1+k12eθ2+a12k1+k22+k1-k22eθ1+θ21+eθ1+eθ2+a12eθ1+θ22,where(80)θ1=k1XαΓ1+α+r1YβΓ1+β+q1ZγΓ1+γ-a2k13+a3k1r12+a4k1q12TωΓ1+ω,θ2=k2XαΓ1+α+r2YβΓ1+β+q2ZγΓ1+γ-a2k23+a3k2r22+a4k2q22TωΓ1+ω.


## 5.3. Three-Soliton Solution

To investigate the three-soliton solution of (65), we assume that the auxiliary function f(X,Y,Z,T) has the following form:(81)fX,Y,Z,T=1+eθ1+eθ2+eθ3+a12eθ1+θ2a13eθ1+θ3+a23eθ2+θ3+a123eθ1+θ2+θ3,where(82)aij=ki2+kj2-kikjki2+kj2+2kikj.Substituting (77) and (81) into (65), we find the following pattern:(83)a123=a12a13a123.According to the pattern obtained in Section5.3, the N-soliton solutions for the 3D TSF-Schamel-KdV equation can be obtained, where N≥1. Based on the single-soliton solution and the two-soliton solution, we can study the characteristics of the motion of the ion-acoustic solitary waves.In this section, we describe the interaction of two small ion-acoustic solitary waves with finite amplitude in a weakly relativistic 3D magnetic plasma. Then, we can study the characteristics of motion of the solitary waves by changing the coefficients. Based on the single-soliton solution of ion-acoustic solitary waves, we obtain the evolution plots of the ion-acoustic solitary waves (see Figure1). Figure 1 shows that the solitonic amplitude increases with an increase in the a2/a1 ratio, and the initial superimposed solitons travel different distances over a period of time for the different choices of a1 and a2. Therefore, we conclude that the soliton moves along the positive x-axis with constant amplitude and velocity.Figure 1
Plots for the evolution of the one-soliton solution of ion-acoustic solitary waves with (a):t=1, a1=1, a3=0, a4=1, k1=1, r1=0.4, q1=0.4, α=0.8, β=1, γ=1, ω=2; (b): t=3, a1=1, a3=0, a4=1, k1=1, r1=0.4, q1=0.4, α=0.8, β=1, γ=1, ω=1.5; (c): t=5, a1=1, a3=0, a4=1, k1=1, r1=0.4, q1=0.4, α=0.8, β=1, γ=1, ω=1.5, and a2=0.1 (solid line), a2=0.09 (dotted line), a2=0.08 (dashed line).
(a) (b) (c)Examination of Figure2(a) shows that the propagation trajectory of the soliton exhibits a periodic oscillation. Figure 2(a) shows the curve propagation trajectory with constant amplitude and constantly changing velocity, where the velocity changes with time. Furthermore, Figure 2(b) shows the two-soliton interaction with constantly changing velocity. When T→0, the trajectory is sinusoidal with periodic oscillation. Otherwise, when T is far from the origin, the trajectory is parabolic-like. It can be seen from Figure 2(c) that the soliton generates a peak at the time of the interaction. Based on this, we conclude that, in addition to the periodic oscillation of the solitons in the local region, the large-scale propagation trajectories for such a structure show parabolic-type curves. Thus, if the variable coefficients are taken to have other forms, the corresponding characteristic curves will present different behaviors.Figure 2
Interaction of the two solitons of ion-acoustic solitary waves with (a):a1=1, a2=0.1, a3=1, a4=1, k1=1.6, r1=1, q1=1, k2=-1.9, r2=1, q2=1, α=0.9, β=0.8, γ=0.9, ω=1.9; (b): a1=1, a2=0.1, a3=1, a4=1, k1=1.6, r1=1, q1=1, k2=-1.9, r2=1, q2=1, α=0.7, β=1, γ=1, ω=2.8; (c): a1=1, a2=0.1, a3=1, a4=1, k1=1.6, r1=1, q1=1, k2=-1.9, r2=1, q2=1, α=0.8, β=1, γ=1, ω=2.8.
(a) (b) (c)Remark 4.
The present study describes the propagation and interaction of two small but finite amplitude ion-acoustic solitary waves in a weakly relativistic 3D magnetized plasma. Our conclusions can be considered as a generalization of the model suggested by Nejoh [53] and Pakira and Chowdhury [54] by including the effect of different coefficients. It has been found that the initial superposed solitons travel different distances over a period of time for the different choices of a1 and a2 and that the solitonic amplitude increases with an increase in the a2/a1 ratio. Moreover, the time fractional order ω and space fractional orders α, β, and γ play an important role in higher-order perturbation theory in the variation of the soliton amplitude. We believe that our research may be of basic interest for particle trapping experiments. Compared to other solitary waves, the unique features of the ion-acoustic solitary waves are the existence of an ultra-low frequency regime for wave propagation and the high charging of the grains, which can fluctuate because of the collection of plasma currents onto dust surfaces. It is difficult to carry out a reasonable comparison between previous studies and the present work. Nevertheless, due to the flexibility provided by the nonextensive method, we suggest that the quantitative discrepancies between the theory and experiment can be reduced. Thus, the application of our model may be particularly interesting in some plasma environments, such as space-plasmas, laser-plasma interactions, and the plasma sheet boundary layer of the earth’s magnetosphere.


## 6. Conclusions

In this paper, based on the basic system of equations of ion-acoustic solitary waves, we have obtained a new 3D Schamel-KdV equation by applying multi-scale analysis and the perturbation method. Then, based on the newly developed model and using the semi-inverse method and the fractional variational principle, a new 3D TSF-Schamel-KdV equation is obtained. We study the conservation laws and soliton solutions of the 3D TSF-Schamel-KdV equation. By theory and image analysis, the following conclusions can be obtained:(1) Based on the basic system of equations and using multi-scale analysis and the perturbation method, we have obtained a new 3D Schamel-KdV equation. This equation is more suitable than other models for the study of ion-acoustic solitary waves. Furthermore, based on the new integer-order model and using the semi-inverse method and the fractional variational principle, we obtain the 3D TSF-Schamel-KdV equation. The fractional model opens the door to the study of ion-acoustic solitary waves.(2) Using the Riemann-Liouville fractional derivative, we study the conservation laws of the 3D TSF-Schamel-KdV equation. Then, we discuss the soliton solutions of the new fractional model. Using the multi-soliton solutions, we study the characteristics of motion of ion-acoustic solitary waves.



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