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Original Research ARTICLE

Front. Phys., 05 June 2020 | https://doi.org/10.3389/fphy.2020.00181

New Soliton Applications in Earth's Magnetotail Plasma at Critical Densities

  • 1Department of Physics, College of Science and Humanities, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia
  • 2Department of Mathematics, College of Science, Taibah University, Medina, Saudi Arabia
  • 3Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
  • 4Department of Mathematics, Faculty of Science, Firat University, Elaziğ, Turkey
  • 5Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
  • 6Department of Physics, Faculty of Science, Damietta University, New Damietta, Egypt

New plasma wave solutions of the modified Kadomtsev Petviashvili (MKP) equation are presented. These solutions are written in terms of some elementary functions, including trigonometric, rational, hyperbolic, periodic, and explosive functions. The computational results indicate that these solutions are consistent with the MKP equation, and the numerical solutions indicate that new periodic, shock, and explosive forms may be applicable in layers of the Earth's magnetotail plasma. The method employed in this paper is influential and robust for application to plasma fluids. In order to depict the propagating soliton profiles in a plasma medium, the MKP equation must be solved at critical densities. In order to achieve this, the Riccati-Bernoulli sub-ODE technique has been utilized in solutions. The research findings indicate that a number of MKP solutions may be applicable to electron acoustics appearing in the magnetotail.

AMS subject classifications: 35A20, 35A99, 35C07, 35Q51, 65Z05

1. Introduction

The existence of electron acoustic solitary excitations (EAs) in plasmas has been noticed in laboratories [1, 2]. Different observations in space have confirmed propagations of EAs in magnetospheres, auroral zones, broadband electrostatic noise (BEN), heliospheric shock, and geomagnetic tails [310]. The concept of EAs was generated by Fried and Gould [11]. It is principally an acoustic-type of wave with inertia given by the mass of cold electrons and restoring force expressed by hot electron thermal pressure [12]. Abdelwahed et al. [10] inspected the modulation of characteristics of EAs in non-isothermal electron plasmas [13] using a time-fractional modified non-linear equation. Pakzad studied [14] cylindrical EAs by hot non-extensive electrons, and found through numerical simulations that the spherical amplitude is greater than the cylindrical in EAs. Non-thermal critical geometrical EA plasmas were studied using a Gardner-type equation in Shuchy et al. [15]. Contributions of solitons to science have been discussed in many research works, some of which may be listed as [1623]. The observed BEN emission bursts in auroras and the Earth's magnetotail regions indicate small and large amplitude electric fields with some explosive and rational domains at critical density. These wave structures appear to be prevalent in some parts of these regions [16, 17]. Therefore, we aim to obtain the solutions that confirm the existence of the electrostatic field in our model.

Let us consider the non-linear partial differential equation

H(φ,φx,φt,φxx,φxt,φtt,...)=0,    (1.1)

where φ(x, t) is an unknown function. Using the wave transformation

φ(x,t)=φ(ξ),      ξ=kx-ct,    (1.2)

Equation (1.1) is converted to an ODE:

E(φ,φ,φ,φ,...)=0.    (1.3)

Many models in physics, fluid mechanics, and engineering are written in the form of (1.1), and this form may be transformed into the ODE:

α1φ+α2φ3+α3φ=0,    (1.4)

(see for instance [2435], and so on). Equation (1.3) is quite significant and useful in our computations, and we employ a robust and unified method known as the Riccatti-Bernoulli (RB) sub-ODE method [36]. The RB sub-ODE method has been used as a box solver for many systems of equations arising in applied science and physics. There are other powerful analytical methods that solve such ODEs; an important example is the Lie algebra method (see [37, 38]).

Next, we describe the RB sub-ODE method briefly.

2. The RB Sub-ODE Method

According to the RB sub-ODE method [36], the solution of Equation (1.3) is

φ=aφ2-m+bφ+cφm,    (2.1)

where a, b, c, and n are constants that will be determined later. From Equation (2.1), we get

φ=ab(3m)φ2m+a2(2m)φ32m+mc2φ2m1+bc(m+1)φm+(2ac+b2)φ,    (2.2)
φ=φ[ab(3m)(2m)φ1m+a2(2m)(32m)φ22m         +m(2m1)c2φ2m2+bcn(m+1)φm1+(2ac+b2)].    (2.3)

The solitary solutions φi(ξ) of Equation (2.1) are given by

1. At m = 1

φ(ξ)=ςe(a+b+c)ξ.    (2.4)

2. At m ≠ 1, b = 0, and c = 0

φ(ξ)=(a(m-1)(ξ+ς))1m-1.    (2.5)

3. At m ≠ 1, b ≠ 0, and c = 0

φ(ξ)=(-ab+ςeb(m-1)ξ)1m-1.    (2.6)

4. At m ≠ 1,a ≠ 0, and b2 − 4ac < 0

φ(ξ)=(-b2a+4ac-b22atan((1-m)4ac-b22(ξ+ς)))11-m    (2.7)

and

φ(ξ)=(-b2a-4ac-b22acot((1-m)4ac-b22(ξ+ς)))11-m.    (2.8)

5. At m ≠ 1,a ≠ 0, and b2 − 4ac > 0

φ(ξ)=(-b2a-b2-4ac2acoth((1-m)b2-4ac2(ξ+ς)))11-m    (2.9)

and

φ(ξ)=(-b2a-b2-4ac2atanh((1-m)b2-4ac2(ξ+ς)))11-m.    (2.10)

6. At m ≠ 1, a ≠ 0, and b2 − 4ac = 0

φ(ξ)=(1a(m-1)(ξ+ς)-b2a)11-m.    (2.11)

2.0.1. Bäcklund Transformation

If φr−1(ξ) and φr(ξ)(φr(ξ) = φrr−1(ξ))) are the solutions of Equation (2.1), we have

dφr(ξ)dξ=dφr(ξ)dφr-1(ξ)dφr-1(ξ)dξ=dφr(ξ)dφr-1(ξ)(aφr-12-m+bφr-1+cφr-1m),

namely

dφr(ξ)aφr2-m+bφr+cφrm=dφr-1(ξ)aφr-12-m+bφr-1+cφr-1m.    (2.12)

Integrating Equation (2.12) once with respect to ξ, we get the Bäcklund transformation of Equation (2.1) as follows:

φr(ξ)=(-cL1+aL2(φr-1(ξ))1-mbL1+aL2+aL1(φr-1(ξ))1-m)11-m,    (2.13)

where L1 and L2 are arbitrary constants. Equation (2.13) gives the infinite solutions of Equations (2.1) and (1.1).

3. Unified Solver

In this section, we will describe the practical implementation of the concept of a unified solver.

α1φ+α2φ3+α3φ=0,    (3.1)

Substituting Equation (2.2) into Equation (3.1), we obtain

α1(ab(3m)φ2m+a2(2m)φ32m+mc2φ2m1  +bc(m+1)φm+(2ac+b2)φ)+α2φ3+α3φ=0.    (3.2)

Making m = 0, Equation (3.2) is reduced to

α1(3abu2+2a2φ3+bc+(2ac+b2)φ)+α2φ3+α3φ=0.    (3.3)

Setting each coefficient of φi(i = 0, 1, 2, 3) to zero, we get

α1bc=0,    (3.4)
α1(2ac+b2)+α3=0,    (3.5)
3α1ab=0,    (3.6)
2α1a2+α2=0.    (3.7)

Solving Equations (3.4)–(3.7) yields

b=0,    (3.8)
c=α3-2α1α2,    (3.9)
a=±-α22α1.    (3.10)

Hence, we present the following possible cases for solutions of Equations (3.1) and (1.1).

1. When b = 0 and c = 0 (α3 = 0), the solution of Equation (3.1) is

φ1(x,t)=(-α22α1(ξ+ς))-1,    (3.11)

where ς is an arbitrary constant.

2. When α3α1<0, substituting Equations (3.8)–(3.10) and (1.2) into Equations (2.7) and (2.8), the trigonometric function solutions of Equation (1.1) are then given by

φ2,3(x,t)=±α3α2 tan(-α32α1(ξ+ς))    (3.12)
φ4,5(x,t)=±α3α2 cot(-α32α1(ξ+ς)),    (3.13)

where ς is an arbitrary constant.

3. When α3α1>0, substituting Equations (3.8)–(3.10) and (1.2) into Equations (2.9) and (2.10), the hyperbolic function solutions of Equation (1.1) are,

φ6,7(x,t)=±-α3α2tanh(α32α1(ξ+ς))    (3.14)

and

φ8,9(x,t)=±-α3α2coth(α32α1(ξ+ς)),    (3.15)

where ς is an arbitrary constant.

4. Mathematical Model

We use stretched τ = ϵ3t, ξ = ϵ(x − λt), η = ϵ2y, where ϵ is an arbitrarily small number and λ is the speed of EA. Elwakil et al. [17] examined two-dimensional propagation of EAs in plasma with cold fluid of electrons and two different ion temperatures within the framework of Poisson equations:

2ϕ x2+ 2ϕ y2=(ne-nil-nih),    (4.1)
nil=μexp(- ϕνβ+μ),nih=γexp(- β ϕνβ+μ).    (4.2)

where Tl is the low ion temperature at equilibrium density μ, Th is the high ion temperature at equilibrium density γ, and  β=TlTh. The computational results indicate that the system reaches critical density μc which makes non-linearity vanish. At μ = μc, the modified KP equation was given:

ξ(τϕ+Gϕ2ξϕ+R3ξ3ϕ)+Q2η2ϕ=0

with

μc=β2λ4-λ4±(β-1)λ2β2λ4+2βλ4+λ4-12β-6β2+6β2(-3β2+6β-3),    (4.3)
G= 12λ(-3νβ22(μ+βν)2-3μ2(μ+βν)2-3λ4),    (4.4)
R=λ32,Q=λ2.

We use a similarity transformation in the form:

χ=Lξ+Mη-τ(υ1+υ2),    (4.5)
ϕ(χ)=ϕ(x,y,t)    (4.6)
τ=t,    (4.7)

where L and M are directional cosines of x and y axes.

The MKP equation transformed to the ODE form is:

-3(v-s)ϕ+δ ϕ3+3 σd2ϕdχ2=0.    (4.8)

Equation (4.8) gives a stationary soliton in the form of

ϕc=6(v-Sδ)sech(v-Sδσδχ),    (4.9)
S=M2QL-u,    (4.10)
δ=GL,σ=RL3,    (4.11)

where u and v are traveling speeds in both directions.

5. Results and Discussion

Comparing Equation (4.8) with the general form (3.1) gives α1 = 3σ, α2 = δ, and α3 = −3(υ − s). According to the unified solver given in section 3, solutions of Equation (4.8) are expressed as follows.

5.1. Rational Function Solutions: (When υ = s)

The rational solutions of Equation (4.8) are.

ϕ1,2(x,t)=(-δ6σ(χ+ς))-1.    (5.1)

5.2. Trigonometric Function Solution: (When υ-sσ>0)

The trigonometric solutions of Equation (4.8) are

ϕ3,4(x,t)=±-3(υ-s)δ tan(υ-s2σ(χ+ς))    (5.2)

and

ϕ5,6(x,t)=±-3(υ-s)δ cot(υ-s2σ(χ+ς)).    (5.3)

5.3. Hyperbolic Function Solution: (When υ-sσ<0)

The hyperbolic solutions of Equation (4.8) are.

ϕ7,8(x,t)=±3(υs)δ tanh(sυ)2σ(χ+ς))    (5.4)

and

ϕ9,10(x,t)=±3(υ-s)δ coth(s-υ2σ(χ+ς)).    (5.5)

Two-dimensional propagation of solitary non-linear EAs has been examined in a plasma mode using parameters related to sheet layers of plasmas of the Earth's magnetotail [16, 17]. At a certain ion density value called the criticality value, the equation obtained cannot describe the mode. Hence, the new stretching produced by the MKP equation describes the critical system under investigation. Equation (4.8) represents a soliton with stationary behavior, as shown in Figure 1. At the critical point, many solitary forms are concerned with the behavior of EAs using the Riccati-Bernoulli solver for the MKP equation.

FIGURE 1
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Figure 1. Variation of ϕc against χ, β for u = 0.01, v = 0.01, L = 0.95.

Solution (5.1) is a solitary wave type called explosive type, which has rapidly increasing amplitude, as depicted in Figure 2. Solution (5.2) has a blow-up periodic shape, as shown in Figure 3. Dissipative behaviors are also produced in Figures 4, 5. In the solution of (5.4), the shock wave is propagated in the medium, as shown in Figure 4. Finally, the explosive shock profile is obtained for solution (5.5), as shown in Figure 5.

FIGURE 2
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Figure 2. Variation of rational ϕc against χ, β for u = 0.01, v = 0.01, L = 0.95.

FIGURE 3
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Figure 3. Variation of periodic ϕc against χ for β = 0.05, u = 0.02, v = 0.5, L = 0.92.

FIGURE 4
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Figure 4. Variation of shock ϕc against χ for β = 0.02, u = 0.02, v = 0.5, L = 0.5.

FIGURE 5
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Figure 5. Variation of explosive shock ϕc against χ for β = 0.02, u = 0.02, v = 0.5, L = 0.5.

6. Conclusions

We have devoted major effort to examining the adequate description of the new type solutions at critical density in plasma layers of the Earth's magnetotail. The application of perturbation theory leads to the modified MKP equation. An RB sub-ODE solver gives new solitary excitations for the MKP equation, including periodic, explosive, and shock types. The new explosive shocks represent the wave motion of plasma solitons. Moreover, these new exact solitonic and other solutions to the MKP equation supply guidelines for the classification of the new types of waves according to the model parameters and can introduce the following types: (a) solitary and hyperbolic solutions, (b) periodic solutions, (c) explosive solutions, (d) rational solutions, (e) shock waves, and (f) explosive shocks. The application of this model could be used in the verification of the broadband and observations of magnetotail electrostatic waves.

Data Availability Statement

The datasets generated for this study are available on request to the corresponding author.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Funding

This project was supported by the deanship of scientific research at Prince Sattam Bin Abdulaziz University under the research project No. 10259/01/2019.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: MKP equation, explosive solutions, Riccati-Bernoulli sub-ODE method, plasma applications, magnetotail

Citation: Abdelwahed HG, Abdelrahman MAE, Inc M and Sabry R (2020) New Soliton Applications in Earth's Magnetotail Plasma at Critical Densities. Front. Phys. 8:181. doi: 10.3389/fphy.2020.00181

Received: 04 March 2020; Accepted: 27 April 2020;
Published: 05 June 2020.

Edited by:

Manuel Asorey, University of Zaragoza, Spain

Reviewed by:

Yilun Shang, Northumbria University, United Kingdom
Babak Shiri, Neijiang Normal University, China

Copyright © 2020 Abdelwahed, Abdelrahman, Inc and Sabry. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Hesham G. Abdelwahed, h.abdelwahed@psau.edu.sa; hgomaa_eg@hotmail.com; Mustafa Inc, minc@firat.edu.tr