Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma

Sankar Chattopadhyay

doi:doi:10.11648/j.ajpa.20241206.11

Research Article |

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Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma

Sankar Chattopadhyay^*

Published in American Journal of Physics and Applications (Volume 12, Issue 6)

Received: 23 October 2024 Accepted: 15 November 2024 Published: 30 December 2024

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Abstract

Propagation of large amplitude ion-acoustic solitary waves and small amplitude double layers have been investigated in a two-temperature non-isothermal electron plasma consisting of warm positive ions, warm negative ions and warm positrons by Sagdeev pseudopotential method (SPM). Small amplitude double layer solution (ϕ_DL) is also discussed theoretically in this paper for further investigation. Fully non-linear large amplitude compressive solitary waves and small amplitude compressive double layers are presented graphically by the corresponding figurs 1 – 14 under the variation of different values of the mass ratios (Q) of negative to positive ions, phase velocities (V) of solitary waves, negative ion concentrations (n_jo), positron densities (χ) and temperature ratios (σ_p) of electrons (T_e) and positrons (T_p). Consequently this large amplitude electrostatic solitary waves and small amplitude electrostatic double layers in complex plasmas have broad applications in space plasma Physics, Astrophysics, Fusion technology, Plasma – based devices and Fundamental wave research, offering tools for exploring wave-particle interactions, stability and energy transfer in multi-species plasmas.

Published in	American Journal of Physics and Applications (Volume 12, Issue 6)
DOI	10.11648/j.ajpa.20241206.11
Page(s)	112-122
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Solitary Waves, Two-Temperature Non-Isothermal Electron Plasma, Double Layers, Double Layer Width and Amplitude, Double Layer Velocity

1. Introduction

Propagation of large amplitude ion-acoustic solitary waves and small amplitude ion-acoustic double layers have been investigated theoretically in a plasma consisting of warm positive ions, warm negative ions and warm positrons along withisothermal,non-isothermal and non-thermal electronsby many authors

[1-11]

. A large number of physicists

[12-16]

investigated various types of solitary waves for warm or cold ions with magnetized or unmagnetized plasmas in presence of two-temperature or single-temperature non-isothermal electrons. Solitary waves became more interesting when negative ions are found in space plasmas. This negative ions are observed in D and F regions of the earth’s ionosphere, in Saturn’s moons and in Halley’s cometarycomae. Wong, Mamas and Arnush

[17]

discussed a method for producing plasmas with total replacement of electrons by negative ions. On the other hand, non-isothermal electrons

[18-20]

give rise many interesting results in the propagation of waves. It is also found that two-temperature Maxwellian distribution of electrons

[21, 22]

presents a better result nearer to the experimental datafor the formation of solitons and double layers. Ion-acoustic double layershave been extensively studied theoretically

[23-25]

as well as experimentally

[26-28]

by many plasma physicists. Das et al

[29]

, Watanabe

[30]

and Tagare

[31]

investigated theoretically and Cooney et al

[32]

investigated experimentally the ion-acoustic solitary waves in a multispecies plasma. Mishra et al

[33]

studied the ion-acoustic compressive and rarefactive double layers in a warm plasma by reductive perturbation method. Consequently Merlino and Loomis

[34]

discussed strong double layers experimentally for (Ar⁺, SF₆^-) plasma.Chattopadhyay et al

[35, 36]

also studied the effect of negative ions on the formation of ion-acoustic solitons for cold and warm ions in relativistic and in non-relativistic plasmas by Sagdeevpseudopotential method

[37]

.In magnetized plasma with or without negativeions, Ghosh et al and Das et al

[38, 39]

studied the ion-acoustic solitary waves by a new analytical method and reductive perturbation method. In the auroral and magnetospheric plasmas, ion-acoustic double layers have been observed for two-electron species

[40, 41]

. It is therefore interesting to investigate the ion-acoustic double layers in aplasmawhere negative ions and two-temperature electron distributions present simultaneously. Again, the electrostatic waves show a significant change and behave differently when positrons are introduced in addition to three component plasmas. Popel et al

[42]

found that the amplitude of ion-acoustic solitary waves could be considerably reduced when hot positrons were present. We are now wishing to study the ion-acoustic solitary waves and double layers when positrons are included in our system for two-temperature non-isothermal electron plasmas. In presence of warm negative ions, warm positive ions and warm positrons, the present author in recent year

[4, 18]

discussed critically the solitary waves and small amplitude double layers under the variation of different plasma parameters for two-temperature non-isothermal electron plasmas by SPM.

The aim of this paper is to discuss the large amplitude ion-acoustic compressive solitary wavesand small amplitude ion-acoustic compressive double layers in a multicomponent plasma consisting of warm positive ions, warm negative ions, warm positrons and two-temperature non-isothermal electrons by SPM. By taking some plasmas containing ion species (Ar⁺, SF₆^-), (H⁺,O₂^-)and (He⁺,O^-), the profiles of Sagdeev potential function ψ(φ) for large amplitude ion-acoustic compressive solitons and small amplitude compressive double layers are drawn under the variation of negative to positive ion mass ratios (Q),phase velocities (V) of solitary waves, negative ion concentrations

(n_{jo})

, positron density (χ) and temperature ratios (

σ_{p}

) of electrons

(T_{e})

andpositrons

(T_{p})

The plan of this paper is arranged in the following ways:

The basic set of normalized fluid equations for positive ions, negative ions along with Poisson’s equations, concentrations of two-temperature non-isothermal electrons and warm isothermal positrons are given in Sec.2. The Sagdeev potential function ψ(φ) for soliton is given in this section. By using the double layer conditions, the profiles of Sagdeev potential function ψ(φ) for small amplitude compressive double layers and double layer solutions

(ϕ_{DL})

are stated clearly in this part. In sec.3,the entire problem is discussed critically under the variation of different plasma parameters. Concluding remarks are given in sec.4.

2. Formulations

In presence of warm positrons, the set of normalized fluid equations for a collisionless, unmagnetized, non-relativistic plasma containing warm positive ions, warm negative ions and two-temperature non-isothermal electrons along with charge neutrality condition are given in the following ways:

\frac{\partial n_{i}}{\partial t}

\frac{\partial}{\partial x} (n_{i} u_{i})

= 0 (1)

\frac{\partial u_{i}}{\partial t}

u_{i} \frac{\partial u_{i}}{\partial x}

\frac{σ_{i}}{n_{i}} \frac{\partial p_{i}}{\partial x}

\frac{\partial ф}{\partial x}

(2)

\frac{\partial p_{i}}{\partial t}

u_{i} \frac{\partial p_{i}}{\partial x}

+ 3

p_{i} \frac{\partial u_{i}}{\partial x}

= 0(3)

\frac{\partial n_{j}}{\partial t}

\frac{\partial}{\partial x} (n_{j} u_{j})

= 0(4)

\frac{\partial u_{j}}{\partial t}

u_{j} \frac{\partial u_{j}}{\partial x}

\frac{σ_{j}}{{Qn}_{j}} \frac{\partial p_{j}}{\partial x}

\frac{Z}{Q} \frac{\partial ф}{\partial x}

(5)

\frac{\partial p_{j}}{\partial t}

u_{j} \frac{\partial p_{j}}{\partial x}

+ 3

p_{j} \frac{\partial u_{j}}{\partial x}

= 0(6)

\frac{\partial^{2} ф}{\partial x^{2}}

n_{e}

n_{i}

+ Z

n_{j}

n_{p}

(7)

Charge neutrality condition is

1 + Z

n_{jo}

n_{io}

+ χ(8)

where

n_{e}

n_{e}

(9)

n_{p}

= χ

e^{- σ_{p} ϕ}

(10)

b_{l}

\frac{1 - β_{l}}{\sqrt{π}}

b_{h}

\frac{1 - β_{h}}{\sqrt{π}}

b_{l}^{(1)}

\frac{1 - β_{l}^{2}}{\sqrt{π}}

b_{h}^{(1)}

\frac{1 - β_{h}^{2}}{\sqrt{π}}

β_{1}

\frac{T_{el, f}}{T_{eh, f}}

β_{l}

\frac{T_{el, f}}{T_{el, t}}

β_{h}

\frac{T_{eh, f}}{T_{eh, t}}

, µ+ν = 1

σ_{p}

\frac{T_{eff}}{T_{p}}

σ_{i}

\frac{T_{i}}{T_{eff}}

σ_{j}

\frac{T_{j}}{T_{eff}}

T_{eff}

\frac{T_{el} T_{eh}}{μ T_{eh} + ν T_{el}}

;

For non-isothermal plasma 0 <

b_{l}

b_{h}

\frac{1}{\sqrt{π}}

and 0 <

b_{l}^{(1)}

b_{h}^{(1)}

\frac{1}{\sqrt{π}}

In this case,

T_{el, f}

= temperature of free electrons in low temperature,

T_{eh, f}

= temperature of free electrons in high temperature,

T_{el, t}

= temperature of trapped electrons in low temperature,

T_{eh, t}

= temperature of trapped electrons in high temperature,

T_{eff}

= effective temperature of electrons,

T_{p}

= temperature of positrons,

T_{i}

= temperature of positive ions,

T_{j}

= temperature of negative ions.

The concerned plasma parameters

n_{i}

n_{j}

n_{e}

n_{p}

u_{i}, u_{j}

p_{i}

p_{j}

σ_{i}

σ_{j}

, ϕ, Z,Q, χ,

σ_{p}

,x, t, µ, ν,

β_{1}

β_{l}

β_{h}

b_{l}

b_{h}

b_{l}^{(1)}

and

b_{h}^{(1)}

in equations (1) to (10)are respectively the density of positive ions, negative ions, electrons and positrons, the velocity of positive ion and negative ions, the pressure of positive ions and negative ions, the temperature of positive ions and negative ions, the electrostatic potential, charge of ions, mass ratio of negative to positive ions, density of positron at ϕ = 0, temperature ratio of electrons and positrons, distance, time, the unperturbed number density of low temperature and high temperature electrons, the temperature ratio of free electrons in low and high temperatures, the temperature ratio of free and trapped electrons in low temperature, the temperature ratio of free and trapped electrons in high temperatureand the normally used conventional non-isothermal parameters related with

β_{l}

and

β_{h}

In this paper the boundary conditions for our systems are

n_{i} \to n_{io}

n_{j} \to n_{jo}, u_{i} \to u_{io}

u_{j} \to u_{jo}

p_{i} \to 1

p_{j} \to 1

n_{e} \to 1

n_{p} \to χ

and ф

\to

0at

|x| \to \infty

In equations (1) to (7), the Galelian transformation η = x – V tis used where V is the velocity of the solitary waves.

Following Chattopadhyay

[4]

, the expression for Sagdeev pseudopotential function ψ(ϕ) for two-temperature non-isothermal electron plasmas with warm positive ions, warm negative ions and warm positrons, is obtained as

ψ(ϕ) = [

-

- \frac{1}{2} ϕ^{2}

\frac{8}{15} \frac{µ b_{l} + ν b_{h} β_{1}^{\frac{3}{2}}}{{(µ + ν β_{1})}^{\frac{3}{2}}} ϕ^{\frac{5}{2}} - \frac{1}{6} \frac{µ + ν β_{1}^{2}}{(µ + ν β_{1})^{2}} ϕ^{3} + \frac{16}{105} \frac{µ b_{l}^{(1)} + ν b_{h}^{(1)} β_{1}^{\frac{5}{2}}}{{(µ + ν β_{1})}^{\frac{5}{2}}} ϕ^{\frac{7}{2}} - \frac{1}{24} \frac{µ + ν β_{1}^{3}}{(µ + ν β_{1})^{3}} ϕ^{4}

+......]

\frac{1}{6} \sqrt{\frac{n_{io}^{3}}{3 σ_{i}}} [\begin{matrix} {\{{(V - u_{io} - \sqrt{\frac{3 σ_{i}}{n_{io}}})}^{2} - 2 ϕ\}}^{\frac{3}{2}} - {\{{(V - u_{io} + \sqrt{\frac{3 σ_{i}}{n_{io}}})}^{2} - 2 ϕ\}}^{\frac{3}{2}} \\ + {(V - u_{io} + \sqrt{\frac{3 σ_{i}}{n_{io}}})}^{3} - {(V - u_{io} - \sqrt{\frac{3 σ_{i}}{n_{io}}})}^{3} \end{matrix}]

\frac{1}{6} \sqrt{\frac{Q^{3} n_{io}^{3}}{3 σ_{j}}} [\begin{matrix} {\{{(V - u_{jo} - \sqrt{\frac{3 σ_{j}}{Q n_{jo}}})}^{2} + \frac{2 Zϕ}{Q}\}}^{\frac{3}{2}} - {\{{(V - u_{jo} + \sqrt{\frac{3 σ_{j}}{{Qn}_{jo}}})}^{2} + \frac{2 Zϕ}{Q}\}}^{\frac{3}{2}} \\ + {(V - u_{jo} + \sqrt{\frac{3 σ_{j}}{{Qn}_{jo}}})}^{3} - {(V - u_{jo} - \sqrt{\frac{3 σ_{j}}{{Qn}_{jo}}})}^{3} \end{matrix}]

\frac{χ}{σ_{p}}

(1-

e^{- σ_{p} ϕ}

) (11)

Where

n_{i}

\frac{1}{2} \sqrt{\frac{n_{io}^{3}}{3 σ_{i}}} [\sqrt{{(V - u_{io} + \sqrt{\frac{3 σ_{i}}{n_{io}}})}^{2} - 2 ϕ} - \sqrt{{(V - u_{io} - \sqrt{\frac{3 σ_{i}}{n_{io}}})}^{2} - 2 ϕ}]

(12)

n_{j}

\frac{1}{2} \sqrt{\frac{Q n_{jo}^{3}}{3 σ_{j}}} [\sqrt{{(V - u_{jo} + \sqrt{\frac{3 σ_{j}}{{Qn}_{jo}}})}^{2} + \frac{2 Zϕ}{Q}} - \sqrt{{(V - u_{jo} - \sqrt{\frac{3 σ_{j}}{{Qn}_{jo}}})}^{2} + \frac{2 Zϕ}{Q}}]

(13)

The requirements to yield soliton solutions from Sagdeev potential function ψ

(ϕ)

are

(i) ψ

(ϕ)

\frac{\partial ψ (ϕ)}{\partial ϕ}

= 0at φ = 0.

(ii)

\frac{\partial^{2} ψ (ϕ)}{\partial ϕ^{2}} \leq

0 at φ = 0.

(iii) ψ

(ϕ)

= 0 at φ =

ϕ_{m}

andψ

(ϕ)

< 0 for 0 < φ <

ϕ_{m}

(iv)

\frac{\partial^{3} ψ (ϕ)}{\partial ϕ^{3}}

> 0 at φ = 0 for positive (compressive) potential solitons.

(v)

\frac{\partial ψ (ϕ)}{\partial ϕ}

> 0 at φ =

ϕ_{m}

for positive (compressive) potential solitons.

And the restriction on ϕ is

{- \frac{Q}{2 Z} (V - u_{jo} - \sqrt{\frac{3 σ_{j}}{Q n_{jo}}})}^{2} <

ф<

\frac{1}{2} {(V - u_{io} - \sqrt{\frac{3 σ_{i}}{n_{io}}})}^{2}

(14)

Now by using the Galelian transformation and the necessary boundary conditions, it is found from equations (7), (9), (10), (11), (12) and (13) after expansion of ψ(ϕ) in power series

\frac{d^{2} ϕ}{{dη}^{2}}

H_{1}

- H_{2} ϕ^{\frac{3}{2}}

H_{3} ϕ^{2} - H_{4} ϕ^{\frac{5}{2}}

H_{5} ϕ^{3} -

-.......=

\frac{\partial ѱ}{\partial ϕ}

(15)

And

ψ(ϕ)= -

\frac{1}{2} H_{1} ϕ^{2}

\frac{2}{5} H_{2} ϕ^{\frac{5}{2}}

\frac{1}{3} H_{3} ϕ^{3} + \frac{2}{7} H_{4} ϕ^{\frac{7}{2}}

\frac{1}{4} H_{5} ϕ^{4}

(16)

Where

H_{1}

[1 - n_{i 0} {\{(V - u_{i 0})^{2} - \frac{3 σ_{i}}{n_{i 0}}\}}^{- 1} - Z^{2} n_{j 0} {\{Q (V - u_{j 0})^{2} - \frac{3 σ_{j}}{n_{j 0}}\}}^{- 1} + χ σ_{p}]

H_{2}

\frac{4}{3} \frac{(μ b_{l} + ν b_{h} β_{1}^{\frac{3}{2}})}{{(μ + ν β_{1})}^{\frac{3}{2}}}

H_{3}

\frac{1}{2} [\begin{matrix} \frac{μ + ν β_{1}^{2}}{{(μ + ν β_{1})}^{2}} - \frac{n_{i 0}^{\frac{3}{2}}}{2 \sqrt{3 σ_{i}}} \{(V - u_{i 0} - \sqrt{\frac{3 σ_{i}}{n_{i 0}}})^{- 3} - (V - u_{i 0} + \sqrt{\frac{3 σ_{i}}{n_{i 0}}})^{- 3}\} \\ + \frac{Z^{3} n_{j 0}^{\frac{3}{2}}}{2 Q \sqrt{3 Q σ_{j}}} \{(V - u_{j 0} - \sqrt{\frac{3 σ_{j}}{Q n_{j 0}}})^{- 3} - (V - u_{j 0} + \sqrt{\frac{3 σ_{j}}{{Qn}_{j 0}}})^{- 3}\} - χ σ_{p}^{2} \end{matrix}]

H_{4}

\frac{8}{15} \frac{(μ b_{l}^{(1)} + ν b_{h}^{(1)} β_{1}^{\frac{5}{2}})}{{(μ + ν β_{1})}^{\frac{5}{2}}}

H_{5}

\frac{1}{2} [\begin{matrix} \frac{1}{3} \frac{μ + ν β_{1}^{3}}{{(μ + ν β_{1})}^{3}} - \frac{n_{i 0}^{\frac{3}{2}}}{2 \sqrt{3 σ_{i}}} \{(V - u_{i 0} - \sqrt{\frac{3 σ_{i}}{n_{i 0}}})^{- 5} - (V - u_{i 0} + \sqrt{\frac{3 σ_{i}}{n_{i 0}}})^{- 5}\} \\ + \frac{Z^{4} n_{j 0}^{\frac{3}{2}}}{2 Q^{2} \sqrt{3 σ_{jQ}}} \{(V - u_{j 0} + \sqrt{\frac{3 σ_{j}}{Q n_{j 0}}})^{- 5} - (V - u_{j 0} - \sqrt{\frac{3 σ_{j}}{Q n_{j 0}}})^{- 5}\} + \frac{χ σ_{p}^{3}}{3} \end{matrix}]

(17)

In order to get the small amplitude double layer profiles from Sagdeev potential functionψ(ϕ) and small amplitude double layer solutions

(ϕ_{DL})

, the Sagdeev potential ψ(ϕ)should satisfy the following conditions

[4]

ψ(ϕ) = 0atϕ= 0 andϕ=

ϕ_{dl}

\frac{\partial ψ}{\partial ϕ}

= 0atϕ= 0 andϕ=

ϕ_{dl}

\frac{\partial^{2} ψ}{\partial ϕ^{2}}

< 0atϕ= 0 andϕ=

ϕ_{dl}

(ϕ)

< 0 for 0 <ϕ<

ϕ_{dl}

andϕ>

ϕ_{dl}

(18)

where

ϕ_{dl}

( > 0) is some extreme value of the electrostatic potential ϕat which double layer isproduced.

Now taking terms upto

ϕ^{2}

in equation (15) and

ϕ^{3}

in equation (16), we get finally after using the above boundary conditions for double layers as

H_{1}

\frac{2}{3} H_{3} ϕ_{dl}

and

H_{2}

\frac{5}{3} H_{3} ϕ_{dl}^{\frac{1}{2}}

(19)

In terms of

H_{3}

we get finally from equations (15) and (16) after simplification as

(ϕ)

= -

\frac{1}{3} H_{3} ϕ^{2} {(\sqrt{ϕ} - \sqrt{ϕ_{dl}})}^{2}

(20)

\frac{\partial ψ (ϕ)}{\partial ϕ}

= -

\frac{1}{3} H_{3}

(\sqrt{ϕ} - \sqrt{ϕ_{dl}}) (3 \sqrt{ϕ} - 2 \sqrt{ϕ_{dl}})

(21)

ϕ_{DL}

\frac{1}{4} ϕ_{dl} {[1 - \tanh (\sqrt{\frac{H_{3} ϕ_{dl}}{24}} . η)]}^{2}

(22)

Where

H_{3}

> 0 represents the stable structure of double layer. Equations (20) to (22) give us the first order double layer profile and first order double layer solution.

3. Results and Discussions

In this section, the profiles of Sagdeev potential function ψ(ϕ) against electrostatic potential ϕfor large amplitude solitons and small amplitude double layers are drawn under the variation of different plasma parameters.

In Figure 1, the profiles of Sagdeev potential function ψ(ϕ) against electrostatic potential ϕfor large amplitude solitary waves are drawn under the variation of the mass ratios (Q) of negative

(m_{j})

to positive

(m_{i})

ions in presence

(χ \neq 0)

of positrons.

Download: Download full-size image

Figure 1. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary wavesin two temperature non-isothermal electron plasma under the variation of the mass ratios (Q) of negative

(m_{j})

and positive

(m_{i})

ions in presence of positronsfor V = 1.601,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

n_{jo}

= 0.0501,

n_{io}

= 0.88, χ = 0.1701,

σ_{p}

= 0.4101, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z = 1 whenQ = 1.912, 3.997, 31.746.

In presence of positrons

(χ = 0.1701)

the curves

a_{2}

for Q = 1.912,

a_{3}

for Q = 3.997 and

a_{4}

for Q = 31.746 are shown in Figure 1, represent the solitary waves. The maximum value of the electrostatic potential

(ϕ_{m})

for large amplitude solitonsis increasing for increasing values of the mass ratios Q. The Sagdeev potential profiles ψ

(ϕ)

against ϕ for well shaped solitary waves denoted by

a_{2}

a_{3}

and

a_{4}

cut the ϕ axis at

ϕ_{m}

= 0.277666, 0.284871 & 0.287948 respectively in Figure 1.

Figures 2, 3, 4 and 5 show the profiles of Sagdeev potential function ψ(ϕ) against electrostatic potential ϕfor large amplitude solitary waves under the variation of the phase velocities (V) of solitary waves in presence

(χ \neq 0)

of positrons.

Download: Download full-size image

Figure 2. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary wavesin two temperature non-isothermal electron plasma under the variation of phase velocity (V) in presence of positrons for

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

n_{jo}

= 0.0501,

n_{io}

= 0.88 , χ = 0.1701,

σ_{p}

= 0, 0.4101, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z = 1, Q =3.997 when V = 1.601.

Download: Download full-size image

Figure 3. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary waves in two temperature non-isothermal electron plasma under the variation of phase velocity (V) in presence of positrons for

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

n_{jo}

= 0.0501,

n_{io}

= 0.88,χ = 0.1701,

σ_{p}

= 0.4101, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z = 1, Q =3.997 when V = 1.69445.

The Sagdeev potential function ψ

(ϕ)

against ϕ for large amplitude solitary waves denoted by

a_{5}

in presence of positrons

(χ = 0.1701)

for V = 1.601 with

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

is shown in Figure 2 whereas the curve

a_{6}

in presence of positrons

(χ = 0.1701)

for V = 1.69445 with

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

isshown in Figure 3. It is seen from Figure 3 that the curve

a_{6}

does not show any actual well shaped solitary waves for V = 1.69445 whereas the curve

a_{5}

in Figure 2 cuts the ϕ axisat

ϕ_{m}

= 0.284871showing the actual well shaped soliton nature.In Figure 3, the curve

a_{6}

does not cut the ϕ axis which shows no solitonic structure.Under this different values of V, Figures 2 and 3 present whether the well shaped nature of soliton will form or not.

Download: Download full-size image

Figure 4. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitons in two temperature non-isothermal electrons under the variation of phase velocity (V) in presence of positron for

u_{io}

= 0.5,

u_{jo}

= 0.3,

σ_{i}

\frac{1}{30}

σ_{j}

\frac{1}{25}

n_{jo}

= 0.0501,

n_{io}

= 0.88,χ = 0.1701,

σ_{p}

= 0.4101, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z = 1, Q =3.997 when V = 1.601.Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitons in two temperature non-isothermal electrons under the variation of phase velocity (V) in presence of positron for

u_{io}

= 0.5,

u_{jo}

= 0.3,

σ_{i}

\frac{1}{30}

σ_{j}

\frac{1}{25}

n_{jo}

= 0.0501,

n_{io}

= 0.88,χ = 0.1701,

σ_{p}

= 0.4101, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z = 1, Q =3.997 when V = 1.601.

Download: Download full-size image

Figure 5. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary waves in two temperature non-isothermal electron plasma under the variation of phase velocity (V) in presence of positrons for

u_{io}

= 0.5,

u_{jo}

= 0.3,

σ_{i}

\frac{1}{30}

σ_{j}

\frac{1}{25}

n_{jo}

= 0.0501,

n_{io}

= 0.88,χ = 0.1701,

σ_{p}

= 0.4101, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z = 1, Q =3.997 when V = 1.69445.Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary waves in two temperature non-isothermal electron plasma under the variation of phase velocity (V) in presence of positrons for

u_{io}

= 0.5,

u_{jo}

= 0.3,

σ_{i}

\frac{1}{30}

σ_{j}

\frac{1}{25}

n_{jo}

= 0.0501,

n_{io}

= 0.88,χ = 0.1701,

σ_{p}

= 0.4101, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z = 1, Q =3.997 when V = 1.69445.

In Figure 4, the sagdeev potential profile ψ

(ϕ)

against electrostatic potential ϕfor large amplitude solitary waves in presence of positrons

(χ = 0.1701)

is denoted by the curve

a_{7}

when the temperature of positive

(σ_{i})

and negative

(σ_{j})

ions are

σ_{i}

\frac{1}{30}

and

σ_{j}

\frac{1}{25}

with V = 1.601,

u_{io}

= 0.5,

u_{jo}

= 0.3. The curve

a_{7}

cuts the φ axis at

ϕ_{m}

= 0.17862 showing their soliton nature.

In Figure 5, the profile of sagdeev potential function ψ

(ϕ)

against electrostatic potential ϕfor large amplitude solitons in presence of positrons

(χ = 0.1701)

is denoted by

a_{8}

when the temperatures of positive

(σ_{i})

and negative

(σ_{j})

ions are

σ_{i}

\frac{1}{30}

and

σ_{j}

\frac{1}{25}

with V = 1.69445,

u_{io}

= 0.5,

u_{jo}

= 0.3. In presence of positrons

(χ = 0.1701)

the curve

a_{8}

cuts the ϕ axis at

ϕ_{m}

= 0.342575 showing the actual well shaped soliton nature.

Similarly it is further found that the profiles of sagdeev potential function ψ

(ϕ)

against electrostatic potential ϕfor large amplitude solitons in presence

(χ = 0.1701)

of positronsdo not form the actual well shape for any value of ϕ within the limit for the solitary wave condition when the temperature of positive

(σ_{i})

and negative

(σ_{j})

ions are

σ_{i}

\frac{1}{25}

and

σ_{j}

\frac{1}{20}

with V = 1.701, 1.801, 1.901& 1.99631,

u_{io}

= 0.4,

u_{jo}

= 0.2, χ = 0.1701,

σ_{p}

= 0.4101.

Figure 6 represents the Sagdeev potential profiles ψ(ϕ) against electrostatic potential ϕ for large amplitude solitary waves under the variation of the concentrations of negative ions

(n_{jo})

in presence

(χ \neq 0)

of positrons.

Download: Download full-size image

Figure 6. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary wavesin two temperature non-isothermalelectron plasma under the variation ofthe negative ion concentrations

(n_{jo})

in presence of positrons for

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

, χ = 0.1701,

σ_{p}

= 0.4101, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1, Q = 1.912, V = 1.601,

n_{io}

= 0.8299, 0.88 when

n_{jo}

= 0, 0.0501.Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary wavesin two temperature non-isothermalelectron plasma under the variation ofthe negative ion concentrations

(n_{jo})

in presence of positrons for

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

, χ = 0.1701,

σ_{p}

= 0.4101, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1, Q = 1.912, V = 1.601,

n_{io}

= 0.8299, 0.88 when

n_{jo}

= 0, 0.0501.

In presence of positrons

(χ = 0.1701)

, the sagdeev potential profiles ψ(ϕ) against the electrostatic potential ϕfor large amplitude solitary waves are denoted by

c_{1}

for

n_{jo}

= 0 and

c_{2}

for

n_{jo}

= 0.0501when V = 1.601,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

. The curves

c_{1}

and

c_{2}

cut the ϕ axis at

ϕ_{m}

= 0.297416 and

ϕ_{m}

= 0.277660 respectively showing their well shaped soliton nature. This shows the effect of concentrations of negative ions on Sagdeev potential function ψ(ϕ)in presence of positrons.

Download: Download full-size image

Figure 7. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary wavesin two temperature non-isothermal electron plasma under the variation of the negative ion concentrations

(n_{jo})

in presence of positrons for

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

, χ = 0.1701,

σ_{p}

= 0.4101,µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1, 0; Q = 1.912,V = 1.601,

n_{io}

=0.84, 0.9 when

n_{jo}

= 0.0101, 0.0901.

In Figure 7, the sagdeev potential function ψ(ϕ) against the electrostatic potential ϕ for large amplitude solitary wavesin presence

(χ = 0.1701)

of positronsare denoted by

c_{3}

for

n_{jo}

= 0.0101 and

c_{4}

for

n_{jo}

= 0.0901 whenV = 1.601,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

. It is evident from this figure that the curve

c_{3}

for

n_{jo}

= 0.0101 cuts the ϕ axis at a larger distance than the curve

c_{4}

for

n_{jo}

= 0.0901 with χ = 0.1701 where the curve

c_{4}

cuts the ϕ axis at

ϕ_{m}

= 0.259325. The Figures 6 and 7 show the characteristics of the sagdeev potential function under the variation of the concentration of negative ions

(n_{jo})

in presence

(χ = 0.1701)

of positrons.

In Figure 8, the profiles of Sagdeev potential function ψ(ϕ) against the electrostatic potential ϕ for large amplitude solitary waves are drawn under the variation of the concentrations of positrons(χ). For different values of the concentrations of positrons

(χ)

, the sagdeev potential function ψ

(ϕ)

cuts the ϕ axis at different values. The sagdeev potential function ψ

(ϕ)

at χ = 0.0801 denoted by

c_{5}

cuts the ϕ axis at

ϕ_{m}

= 0.213413 and at χ = 0.1201, the sagdeev potential function ψ

(ϕ)

denoted by

c_{6}

cuts the ϕ axis at

ϕ_{m}

= 0.246681. The curve

c_{7}

represents the sagdeev potential function ψ

(ϕ)

cuts the ϕ axis at

ϕ_{m}

= 0.277660 for χ = 0.1701 which is larger than all other values.

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Figure 8. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary waves in two temperature non-isothermal electron plasma under the variation of the concentrations

(χ)

of positrons for V = 1.601,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

n_{jo}

= 0.0501,

n_{io}

=1.0501, 0.97, 0.93,0.88,

σ_{p}

= 0, 0.4101,µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1,Q = 1.912 when χ =0.0801, 0.1201,0.1701.

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Figure 9. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitons in two temperature non-isothermal electrons under the variation of the temperature ratios

(σ_{p})

of electrons and positrons for V = 1.601,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

n_{j o}

= 0.0501,

n_{io}

=0.88, χ =0.1701,µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1,Q = 1.912 when

σ_{p}

= 0.4101, 1.Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitons in two temperature non-isothermal electrons under the variation of the temperature ratios

(σ_{p})

of electrons and positrons for V = 1.601,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

n_{j o}

= 0.0501,

n_{io}

=0.88, χ =0.1701,µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1,Q = 1.912 when

σ_{p}

= 0.4101, 1.

In Figure 9, the Sagdeev potential profiles ψ(ϕ) against the electrostatic potential ϕ for large amplitude solitons are drawn under the variation of the different temperature ratios

(σ_{p})

of electron

(T_{e})

and positron

(T_{p})

. It is observed from Figures 9 and 10 that the Sagdeev potential profiles cut the ϕ axis at larger distances as long as the temperature ratios

(σ_{p})

ofelectron

(T_{e})

and positron

(T_{p})

is increasing. The Sagdeev potential profiles ψ(ϕ) against ϕfor solitary waves are denoted respectively by the curves

c_{9}

for

σ_{p}

= 0.4101,

c_{10}

for

σ_{p}

= 1 in Figure 9 and

c_{12}

for

σ_{p}

= 1.301,

c_{13}

for

σ_{p}

= 1.9501 in Figure 10. In Figure 9, the Sagdeev potential curves for actual well shaped solitary waves denoted by

c_{9}

for

σ_{p}

= 0.4101 cuts the ϕ axis at

ϕ_{m}

= 0.277660,

c_{10}

for

σ_{p}

= 1 cuts the ϕ axis at

ϕ_{m}

= 0.307871 but in Figure 10, for

σ_{p}

= 1.301 and 1.9501 no solitary waves for actual well shaped are found and thus it is concluded that for

σ_{p}

> 1 no proper solitary waves are found in two-temperature non-isothermal electron plasmas in presence of negative ions.

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Figure 10. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitons in two temperature non-isothermal electrons under the variation of the temperature ratios of electrons and positrons

(σ_{p})

for V = 1.601,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

n_{jo}

= 0.0501,

n_{io}

=0.88,χ =0.1701, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1, Q = 1.912 when

σ_{p}

= 1.301, 1.9501.Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitons in two temperature non-isothermal electrons under the variation of the temperature ratios of electrons and positrons

(σ_{p})

for V = 1.601,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{20}

σ_{j}

\frac{1}{25}

n_{jo}

= 0.0501,

n_{io}

=0.88,χ =0.1701, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1, Q = 1.912 when

σ_{p}

= 1.301, 1.9501.

In the next part, we are now presenting graphically the profiles of small amplitude double layers in presence

(χ = 0.1701)

of positrons under the variation of some plasma parameters.

In Figure 11, the profiles of Sagdeev potential function ψ(ϕ) against the electrostatic potential ϕ for small amplitude double layers are drawn in presence

(χ = 0.1701)

of positrons under the variation of the phase velocity (V) of solitary waves. In presence of positrons (χ = 0.1701), the Sagdeev potential profiles ψ(ϕ) against ϕ for small amplitude double layers are denoted by

p_{1}

for V = 1.6501,

p_{2}

for V = 1.6801. It is evident from the figure that for increasing values of V, the maximum values of the electrostatic potential (

ϕ_{dl}

) where the respective curves cut the ϕ axis are larger.

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Figure 11. Profile of Sagdeev potential functionψ(ϕ) against ϕ for small amplitude double layers in presence of positrons under the variation of phase velocity (V) of solitary waves for

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{25}

σ_{j}

\frac{1}{20}

n_{jo}

= 0.0501,

n_{io}

=0.88, χ =0.1701,

σ_{p}

= 0.4101,µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1,Q = 1.912 when V = 1.6501, V = 1.6801.

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Figure 12. Profile of Sagdeev potential function ψ(ϕ) against ϕ for small amplitude double layers in presence of positronsunder the variation of the concentration of negative ions

(n_{jo})

forV = 1.6501,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{25}

σ_{j}

\frac{1}{20}

, χ =0.1701,

σ_{p}

= 0.4101,µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1,Q = 1.912,

n_{io}

=0.87 when

n_{jo}

= 0.0401. Profile of Sagdeev potential function ψ(ϕ) against ϕ for small amplitude double layers in presence of positronsunder the variation of the concentration of negative ions

(n_{jo})

forV = 1.6501,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{25}

σ_{j}

\frac{1}{20}

, χ =0.1701,

σ_{p}

= 0.4101,µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1,Q = 1.912,

n_{io}

=0.87 when

n_{jo}

= 0.0401.

Figures 12 and 13 show the profiles of Sagdeev potential function ψ(ϕ) against the electrostatic potential ϕ for small amplitude double layers in presence of positrons

(χ = 0.1701)

under the variation of the concentrations of negative ions

(n_{jo})

.In Figure 12, when the concentration of negative ions is

n_{jo}

= 0.0401, the Sagdeev potential profile ψ(ϕ) against ϕ for small amplitude double layers is represented by the curve

p_{3}

for χ = 0.1701 whereas in Figure 13, when the concentration of negative ions is

n_{jo}

= 0.0501, the Sagdeev potential profile ψ(ϕ) against ϕ for small amplitude double layers is denoted by the curve

p_{4}

for χ = 0.1701. In Figure 12, it is observed that the curve

p_{3}

in presence of positron (χ = 0.1701) cuts the ϕ axis at

ϕ_{dl}

= 0.020621 while in Figure 13, the Sagdeev potential profile ψ(ϕ) against ϕ for small amplitude double layers denoted by the curve

p_{4}

for χ = 0.1701 cuts the ϕ axis at

ϕ_{dl}

= 0.342005 when the concentration of negative ions is

n_{jo}

= 0.0501.

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Figure 13. Profile of Sagdeev potential function ψ(ϕ) against ϕ for small amplitude double layersin presence of positronsunder the variation of the concentration of negative ions

(n_{jo})

forV = 1.6501,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{25}

σ_{j}

\frac{1}{20}

, χ =0.1701,

σ_{p}

= 0.4101,µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1,Q = 1.912,

n_{io}

=0.88 when

n_{jo}

= 0.0501.Profile of Sagdeev potential function ψ(ϕ) against ϕ for small amplitude double layersin presence of positronsunder the variation of the concentration of negative ions

(n_{jo})

forV = 1.6501,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{25}

σ_{j}

\frac{1}{20}

, χ =0.1701,

σ_{p}

= 0.4101,µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1,Q = 1.912,

n_{io}

=0.88 when

n_{jo}

= 0.0501.

Figure 14 shows the profiles of Sagdeev potential function ψ(ϕ) against the electrostatic potential ϕ for small amplitude double layers under the variation of the temperature ratios

(σ_{p})

of electrons

(T_{e})

and positrons

(T_{p})

. The Sagdeev potential profiles ψ(ϕ) against ϕ for small amplitude double layers in presence of positron (χ = 0.1701)denoted by

p_{6}

for

σ_{p}

= 0.1501 and

p_{7}

for

σ_{p}

= 0.4101 cut the ϕ axis at

ϕ_{dl}

= 0.322136 and 0.342005 respectively. In this case, it is found that as

σ_{p}

increases

ϕ_{dl}

also increases.

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Figure 14. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for small amplitude double layers in presence of positrons under the variation of the temperature ratios

(σ_{p})

of electrons and positronsfor V = 1.6501,

u_{io}

= 0.4,

u_{jo}

= 0.2,

σ_{i}

\frac{1}{25}

σ_{j}

\frac{1}{20}

n_{jo}

= 0.0501,

n_{io}

=0.88, χ =0.1701, µ = 0.15, ν = 0.85,

b_{l}

= 0.15,

b_{h}

= 0.4,

β_{1}

= 0.25,

b_{l}^{(1)}

= 0.24,

b_{h}^{(1)}

= 0.51, Z =1,Q = 1.912 when

σ_{p}

= 0.1501, 0.4101.

From Figure 14, it is observed that in presence of positron

(χ = 0.17)

, the Sagdeev potential profiles ψ

(ϕ)

against ϕ for small amplitude double layers cut the ϕ axis at larger values of

ϕ_{d l}

for higher values of

σ_{p}

while for the smaller values of

σ_{p}

the respective Sagdeev potential curves for small amplitude double layerscut the ϕ axis at smaller values of

ϕ_{dl}

4.Conclusion

In this paper, the present author investigated the large amplitude ion-acoustic compressive solitary waves and small amplitude ion-acoustic compressive double layers in two-temperature non-isothermal electron plasmas consisting of warm negative ions, warm positive ions and warm positronsby Sagdeev pseudopotential method under the variation of different plasma parameters. In presence

(χ \neq 0)

of positrons, the nature of Sagdeev potential function ψ

(ϕ)

for large amplitude ion-acoustic compressive solitary waves are shown in Figures 1 to 10 under the variation of the mass ratios (Q) of negative to positive ions, phase velocities (V) of solitary waves, concentrations

(n_{jo})

of negative ions,concentrations of positrons

(χ)

and temperature ratios

(σ_{p})

of electrons

(T_{e})

and positrons

(T_{p})

. In Figures 11 to 14, the profiles of Sagdeev potential profiles ψ

(ϕ)

against electrostatic potential ϕfor small amplitude double layers are presented graphically under the variation of the phase velocity (V) of solitary waves, concentration of negative ions

(n_{jo})

and temperature ratios

(σ_{p})

of electrons

(T_{e})

and positrons

(T_{p})

. In presence of positrons, the author studied and discussed the effect of positronsfor large amplitude solitons and small amplitude double layers. Generally solitons are important in understanding energy transport in space plasmas and have been considered in plasma–based devices for signal transmission without dissipation. On the other hand, double layers are relevant in the context of space weather phenomena and are important for understanding energy dissipation and particle acceleration mechanisms in astrophysical plasmas. The present study has been used in the experimental investigation of strong double layers for (Ar⁺,SF₆^-)plasma and will be helpful for the investigation of ion-acoustic waves in space and in laboratory experiments.

The author’s future plan is to solve the fully non-linear differential equations for relativistic warm positive and negative ions, two-temperature isothermal electrons and warm isothermal positrons to investigate the ion-acoustic compressive (rarefactive) solitary waves, double layers and supersolitons.

Abbreviations

SPM	Sagdeev Pseudopotential Method

Acknowledgments

The present author would like to thank Dr. S.N. Paul for his valuable suggestions and discussions in the preparation of this paper to its present form.

Funding

Thus this research work did not receive any specific grant from any funding agencies in the public, commercial, or not-for-profit sectors.

Conflicts of Interest

The author declares no conflicts of interest.

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Chattopadhyay, S. (2024). Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma. American Journal of Physics and Applications, 12(6), 112-122. https://doi.org/10.11648/j.ajpa.20241206.11

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Chattopadhyay, S. Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma. Am. J. Phys. Appl. 2024, 12(6), 112-122. doi: 10.11648/j.ajpa.20241206.11

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Chattopadhyay S. Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma. Am J Phys Appl. 2024;12(6):112-122. doi: 10.11648/j.ajpa.20241206.11

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@article{10.11648/j.ajpa.20241206.11,
  author = {Sankar Chattopadhyay},
  title = {Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma},
  journal = {American Journal of Physics and Applications},
  volume = {12},
  number = {6},
  pages = {112-122},
  doi = {10.11648/j.ajpa.20241206.11},
  url = {https://doi.org/10.11648/j.ajpa.20241206.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20241206.11},
  abstract = {Propagation of large amplitude ion-acoustic solitary waves and small amplitude double layers have been investigated in a two-temperature non-isothermal electron plasma consisting of warm positive ions, warm negative ions and warm positrons by Sagdeev pseudopotential method (SPM). Small amplitude double layer solution (ϕDL) is also discussed theoretically in this paper for further investigation. Fully non-linear large amplitude compressive solitary waves and small amplitude compressive double layers are presented graphically by the corresponding figurs 1 – 14 under the variation of different values of the mass ratios (Q) of negative to positive ions, phase velocities (V) of solitary waves, negative ion concentrations (njo), positron densities (χ) and temperature ratios (σp) of electrons (Te) and positrons (Tp). Consequently this large amplitude electrostatic solitary waves and small amplitude electrostatic double layers in complex plasmas have broad applications in space plasma Physics, Astrophysics, Fusion technology, Plasma – based devices and Fundamental wave research, offering tools for exploring wave-particle interactions, stability and energy transfer in multi-species plasmas.
},
 year = {2024}
}

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TY  - JOUR
T1  - Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma
AU  - Sankar Chattopadhyay
Y1  - 2024/12/30
PY  - 2024
N1  - https://doi.org/10.11648/j.ajpa.20241206.11
DO  - 10.11648/j.ajpa.20241206.11
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 112
EP  - 122
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20241206.11
AB  - Propagation of large amplitude ion-acoustic solitary waves and small amplitude double layers have been investigated in a two-temperature non-isothermal electron plasma consisting of warm positive ions, warm negative ions and warm positrons by Sagdeev pseudopotential method (SPM). Small amplitude double layer solution (ϕDL) is also discussed theoretically in this paper for further investigation. Fully non-linear large amplitude compressive solitary waves and small amplitude compressive double layers are presented graphically by the corresponding figurs 1 – 14 under the variation of different values of the mass ratios (Q) of negative to positive ions, phase velocities (V) of solitary waves, negative ion concentrations (njo), positron densities (χ) and temperature ratios (σp) of electrons (Te) and positrons (Tp). Consequently this large amplitude electrostatic solitary waves and small amplitude electrostatic double layers in complex plasmas have broad applications in space plasma Physics, Astrophysics, Fusion technology, Plasma – based devices and Fundamental wave research, offering tools for exploring wave-particle interactions, stability and energy transfer in multi-species plasmas.

VL  - 12
IS  - 6
ER  -

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Sankar Chattopadhyay

Department of Mathematics, Sishu Bikash Academy, Kolkata, India

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http://orcid.org/0000-0003-4347-5762

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Figure 1

Figure 1. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary wavesin two temperature non-isothermal electron plasma under the variation of the mass ratios (Q) of negative $(m_{j})$ and positive $(m_{i})$ ions in presence of positronsfor V = 1.601, $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{20}$ , $σ_{j}$ = $\frac{1}{25}$ , $n_{jo}$ = 0.0501, $n_{io}$ = 0.88, χ = 0.1701, $σ_{p}$ = 0.4101, µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z = 1 whenQ = 1.912, 3.997, 31.746.
Figure 2

Figure 2. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary wavesin two temperature non-isothermal electron plasma under the variation of phase velocity (V) in presence of positrons for $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{20}$ , $σ_{j}$ = $\frac{1}{25}$ , $n_{jo}$ = 0.0501, $n_{io}$ = 0.88 , χ = 0.1701, $σ_{p}$ = 0, 0.4101, µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z = 1, Q =3.997 when V = 1.601.
Figure 3

Figure 3. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary waves in two temperature non-isothermal electron plasma under the variation of phase velocity (V) in presence of positrons for $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{20}$ , $σ_{j}$ = $\frac{1}{25}$ , $n_{jo}$ = 0.0501, $n_{io}$ = 0.88,χ = 0.1701, $σ_{p}$ = 0.4101, µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z = 1, Q =3.997 when V = 1.69445.
Figure 4

Figure 4. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitons in two temperature non-isothermal electrons under the variation of phase velocity (V) in presence of positron for $u_{io}$ = 0.5, $u_{jo}$ = 0.3, $σ_{i}$ = $\frac{1}{30}$ , $σ_{j}$ = $\frac{1}{25}$ , $n_{jo}$ = 0.0501, $n_{io}$ = 0.88,χ = 0.1701, $σ_{p}$ = 0.4101, µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z = 1, Q =3.997 when V = 1.601.
Figure 5

Figure 5. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary waves in two temperature non-isothermal electron plasma under the variation of phase velocity (V) in presence of positrons for $u_{io}$ = 0.5, $u_{jo}$ = 0.3, $σ_{i}$ = $\frac{1}{30}$ , $σ_{j}$ = $\frac{1}{25}$ , $n_{jo}$ = 0.0501, $n_{io}$ = 0.88,χ = 0.1701, $σ_{p}$ = 0.4101, µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z = 1, Q =3.997 when V = 1.69445.
Figure 6

Figure 6. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary wavesin two temperature non-isothermalelectron plasma under the variation ofthe negative ion concentrations $(n_{jo})$ in presence of positrons for $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{20}$ , $σ_{j}$ = $\frac{1}{25}$ , χ = 0.1701, $σ_{p}$ = 0.4101, µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z =1, Q = 1.912, V = 1.601, $n_{io}$ = 0.8299, 0.88 when $n_{jo}$ = 0, 0.0501.
Figure 7

Figure 7. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary wavesin two temperature non-isothermal electron plasma under the variation of the negative ion concentrations $(n_{jo})$ in presence of positrons for $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{20}$ , $σ_{j}$ = $\frac{1}{25}$ , χ = 0.1701, $σ_{p}$ = 0.4101,µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z =1, 0; Q = 1.912,V = 1.601, $n_{io}$ =0.84, 0.9 when $n_{jo}$ = 0.0101, 0.0901.
Figure 8

Figure 8. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitary waves in two temperature non-isothermal electron plasma under the variation of the concentrations $(χ)$ of positrons for V = 1.601, $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{20}$ , $σ_{j}$ = $\frac{1}{25}$ , $n_{jo}$ = 0.0501, $n_{io}$ =1.0501, 0.97, 0.93,0.88, $σ_{p}$ = 0, 0.4101,µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z =1,Q = 1.912 when χ =0.0801, 0.1201,0.1701.
Figure 9

Figure 9. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitons in two temperature non-isothermal electrons under the variation of the temperature ratios $(σ_{p})$ of electrons and positrons for V = 1.601, $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{20}$ , $σ_{j}$ = $\frac{1}{25}$ , $n_{j o}$ = 0.0501, $n_{io}$ =0.88, χ =0.1701,µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z =1,Q = 1.912 when $σ_{p}$ = 0.4101, 1.
Figure 10

Figure 10. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for large amplitude solitons in two temperature non-isothermal electrons under the variation of the temperature ratios of electrons and positrons $(σ_{p})$ for V = 1.601, $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{20}$ , $σ_{j}$ = $\frac{1}{25}$ , $n_{jo}$ = 0.0501, $n_{io}$ =0.88,χ =0.1701, µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z =1, Q = 1.912 when $σ_{p}$ = 1.301, 1.9501.
Figure 11

Figure 11. Profile of Sagdeev potential functionψ(ϕ) against ϕ for small amplitude double layers in presence of positrons under the variation of phase velocity (V) of solitary waves for $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{25}$ , $σ_{j}$ = $\frac{1}{20}$ , $n_{jo}$ = 0.0501, $n_{io}$ =0.88, χ =0.1701, $σ_{p}$ = 0.4101,µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z =1,Q = 1.912 when V = 1.6501, V = 1.6801.
Figure 12

Figure 12. Profile of Sagdeev potential function ψ(ϕ) against ϕ for small amplitude double layers in presence of positronsunder the variation of the concentration of negative ions $(n_{jo})$ forV = 1.6501, $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{25}$ , $σ_{j}$ = $\frac{1}{20}$ , χ =0.1701, $σ_{p}$ = 0.4101,µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z =1,Q = 1.912, $n_{io}$ =0.87 when $n_{jo}$ = 0.0401.
Figure 13

Figure 13. Profile of Sagdeev potential function ψ(ϕ) against ϕ for small amplitude double layersin presence of positronsunder the variation of the concentration of negative ions $(n_{jo})$ forV = 1.6501, $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{25}$ , $σ_{j}$ = $\frac{1}{20}$ , χ =0.1701, $σ_{p}$ = 0.4101,µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z =1,Q = 1.912, $n_{io}$ =0.88 when $n_{jo}$ = 0.0501.
Figure 14

Figure 14. Profiles of Sagdeev potential function ψ(ϕ) against ϕ for small amplitude double layers in presence of positrons under the variation of the temperature ratios $(σ_{p})$ of electrons and positronsfor V = 1.6501, $u_{io}$ = 0.4, $u_{jo}$ = 0.2, $σ_{i}$ = $\frac{1}{25}$ , $σ_{j}$ = $\frac{1}{20}$ , $n_{jo}$ = 0.0501, $n_{io}$ =0.88, χ =0.1701, µ = 0.15, ν = 0.85, $b_{l}$ = 0.15, $b_{h}$ = 0.4, $β_{1}$ = 0.25, $b_{l}^{(1)}$ = 0.24, $b_{h}^{(1)}$ = 0.51, Z =1,Q = 1.912 when $σ_{p}$ = 0.1501, 0.4101.

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APA Style

Chattopadhyay, S. (2024). Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma. American Journal of Physics and Applications, 12(6), 112-122. https://doi.org/10.11648/j.ajpa.20241206.11

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ACS Style

Chattopadhyay, S. Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma. Am. J. Phys. Appl. 2024, 12(6), 112-122. doi: 10.11648/j.ajpa.20241206.11

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AMA Style

Chattopadhyay S. Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma. Am J Phys Appl. 2024;12(6):112-122. doi: 10.11648/j.ajpa.20241206.11

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@article{10.11648/j.ajpa.20241206.11,
  author = {Sankar Chattopadhyay},
  title = {Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma},
  journal = {American Journal of Physics and Applications},
  volume = {12},
  number = {6},
  pages = {112-122},
  doi = {10.11648/j.ajpa.20241206.11},
  url = {https://doi.org/10.11648/j.ajpa.20241206.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20241206.11},
  abstract = {Propagation of large amplitude ion-acoustic solitary waves and small amplitude double layers have been investigated in a two-temperature non-isothermal electron plasma consisting of warm positive ions, warm negative ions and warm positrons by Sagdeev pseudopotential method (SPM). Small amplitude double layer solution (ϕDL) is also discussed theoretically in this paper for further investigation. Fully non-linear large amplitude compressive solitary waves and small amplitude compressive double layers are presented graphically by the corresponding figurs 1 – 14 under the variation of different values of the mass ratios (Q) of negative to positive ions, phase velocities (V) of solitary waves, negative ion concentrations (njo), positron densities (χ) and temperature ratios (σp) of electrons (Te) and positrons (Tp). Consequently this large amplitude electrostatic solitary waves and small amplitude electrostatic double layers in complex plasmas have broad applications in space plasma Physics, Astrophysics, Fusion technology, Plasma – based devices and Fundamental wave research, offering tools for exploring wave-particle interactions, stability and energy transfer in multi-species plasmas.
},
 year = {2024}
}

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TY  - JOUR
T1  - Formation of Large Amplitude Solitons and Small Amplitude Double Layers in Two-temperature Non-isothermal Electron Plasma
AU  - Sankar Chattopadhyay
Y1  - 2024/12/30
PY  - 2024
N1  - https://doi.org/10.11648/j.ajpa.20241206.11
DO  - 10.11648/j.ajpa.20241206.11
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 112
EP  - 122
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20241206.11
AB  - Propagation of large amplitude ion-acoustic solitary waves and small amplitude double layers have been investigated in a two-temperature non-isothermal electron plasma consisting of warm positive ions, warm negative ions and warm positrons by Sagdeev pseudopotential method (SPM). Small amplitude double layer solution (ϕDL) is also discussed theoretically in this paper for further investigation. Fully non-linear large amplitude compressive solitary waves and small amplitude compressive double layers are presented graphically by the corresponding figurs 1 – 14 under the variation of different values of the mass ratios (Q) of negative to positive ions, phase velocities (V) of solitary waves, negative ion concentrations (njo), positron densities (χ) and temperature ratios (σp) of electrons (Te) and positrons (Tp). Consequently this large amplitude electrostatic solitary waves and small amplitude electrostatic double layers in complex plasmas have broad applications in space plasma Physics, Astrophysics, Fusion technology, Plasma – based devices and Fundamental wave research, offering tools for exploring wave-particle interactions, stability and energy transfer in multi-species plasmas.

VL  - 12
IS  - 6
ER  -

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