Positron nonextensivity effect on the propagation of dust ion acoustic Gardner waves

Akbar Nazari-Golshan; Vahid Fallahi

doi:10.1515/zna-2021-0012

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Positron nonextensivity effect on the propagation of dust ion acoustic Gardner waves

Akbar Nazari-Golshan und Vahid Fallahi

Veröffentlicht/Copyright: 23. August 2021

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Aus der Zeitschrift Zeitschrift für Naturforschung A Band 76 Heft 11

Abstract

Propagation of dust ion-acoustic (DIA) Gardner wave in a dusty electron–positron–ion (e–p–i) plasma is investigated. This plasma consists of q-distributed electrons and positrons, warm ions, and dust grains. The effects of the electron nonextensivity, positron nonextensivity, and fractional parameter on the properties of DIA Gardner wave are investigated. Space fractional Gardner (SFG) equation is derived using the semi inverse technique. An efficient modified G′/G-expansion method is presented to solve the SFG equation. It is found that the amplitude of the DIA Gardner wave increases with an increase in space fractional parameter β and spatial parameter ζ . On other hands, the DIA Gardner wave shape can be modulated using the space fractional parameter β . Our results may help understand the astrophysical environments such as star magnetospheres, solar flares, and galactic nuclei.

Keywords: dust ion-acoustic wave; dusty e–p–i plasma; fractional Gardner equation; modified G′/G-expansion method

Corresponding author: Akbar Nazari-Golshan, Physics Department, Shahed University, P.O.B: 18155/159, Tehran, Iran, E-mail: nazarigolshan@yahoo.com

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Derivation of SFG equation

To formulate the space fractional Gardner (SFG) equation, introducing a potential function, U ζ , τ , as ϕ ζ , τ = U ζ ζ , τ = U ζ , and substituting it into Eq. (8), then the Gardner equation is obtained as:

(A1) U ζ τ + A 1 U ζ U ζ ζ + A 2 U ζ 2 U ζ ζ + A 3 U ζ ζ ζ ζ = 0 ,

The functional of Eq. (A1) is given by:

(A2) H U = ∫ d ζ ∫ d τ U t 1 U ζ τ + t 2 A 1 U ζ U ζ ζ + t 3 A 2 U ζ 2 U ζ ζ + t 4 A 3 U ζ ζ ζ ζ

where t ₁, t ₂, t ₃, and t ₄ are constants.

Integrating Eq. (A2) by parts using U ζ R = U ζ ζ R = U ζ ζ ζ R = U ζ T = 0 , we obtain:

(A3) H U = ∫ d ζ ∫ d τ − t 1 U ζ U τ − 1 2 t 2 A 1 U ζ 3 − 1 3 t 3 A 2 U ζ 4 + t 4 A 3 U ζ ζ 2

By taking the first variation of Eq. (A3) gives:

t 1 = 1 / 2 , t 2 = 1 / 3 , t 3 = 1 / 4 , t 4 = 1 / 2 .

Therefore, the Lagrangian of Gardner equation is presented as:

(A4) F U τ , U ζ , U ζ ζ = − 1 2 U ζ U τ − A 1 6 U ζ 3 − A 2 12 U ζ 4 + A 3 2 U ζ ζ 2 ,

Similarity, Lagrangian of the fractional Gardner equation is written as:

(A5) K U τ , D ζ β 0 U , D ζ β β 0 U = − 1 2 U τ D ζ β 0 U − A 1 6 D ζ β 0 U 3 − A 2 12 D ζ β 0 U 4 + A 3 2 D ζ β β 0 U 2 , 0 < β ≤ 1 ,

In Eq. (A5), D ζ β 0 is given by Eq. (11).

Thus, the functional of SFG equation is given by:

(A6) J U = ∫ R d ζ ∫ T d τ K U τ , D ζ β 0 U , D ζ β β 0 U ,

Taking the first variations of Eq. (A6), according to the semi inverse and Agrawal techniques [49–51], respect to U, δ J U , and optimizing it , δ J U = 0, leads to:

(A7) ∂ ∂ τ ∂ K ∂ U τ − D ζ β 0 ∂ K ∂ D ζ β 0 U + D ζ β β 0 ∂ K ∂ D ζ β β 0 U = 0 .

Substituting Eq. (A5) into Eq. (A7) and applying ϕ = D ζ β 0 U , lead to

(A8) ∂ ϕ ∂ τ + A 1 ϕ D ζ β 0 ϕ + A 2 ϕ 2 D ζ β 0 ϕ + A 3 D ζ β β β 0 ϕ = 0 ⋅ 0 < β ≤ 1 .

Equation (A8) is called SFG equation that describes the DIA waves.

Appendix B: An efficient modified G′/G-expansion method

A nonlinear space fractional PDE has the following form:

(B1) Q U , U t , U t t , D x β 0 U , U D x β 0 U , D x β β 0 U , … = 0 0 < β ≤ 1 .

where Q is a polynomial.(U, U _tt, … ) and ( D x β 0 U , U t D x β 0 U , … ) are derivatives and space fractional derivatives of U respectively.

The mentioned method has the main steps as follow:

Step 1: A fractional complex transform [52, 53] is proposed to derive an integer order ODEs:

(B2) U x , t = U ρ ρ = f x β Γ ( 1 + β ) − g t

Substituting Eq. (B2) into Eq. (B1) gives an ordinary differential equation as follows:

(B3) Q U , ∂ U ∂ ρ , ∂ 2 U ∂ ρ 2 , ∂ 3 U ∂ ρ 3 , … = 0

Step 2: In modified G′/G-expansion method, the solution of Eq. (B3) can be written as:

(B4) U ρ = ∑ j = 0 n c j ∂ G ( ρ ) / ∂ ρ G ( ρ ) j , c n ≠ 0 ,

where c _j, j = 0, 1, 2, … , n are constants. While, G(ρ) satisfies the following ODE

(B5) ∂ 2 G ( ρ ) ∂ ρ 2 + d 1 ∂ G ( ρ ) ∂ ρ + d 2 G ( ρ ) = 0 ,

where d ₁, d ₂ are constants.

Step 3: By balancing the highest order derivatives with the nonlinear term in Eq. (B3), n is obtained. Then, using the general solution of Eq. (B5), substituting Eq. (B4) into Eq. (B3), collecting all coefficients ∂ G ( ρ ) / ∂ ρ G ( ρ ) j with the same power together, c _j is obtained. Finally, by substituting c _j, d ₁, d ₂, f and g, the solution of Eq. (B1) can be obtained.

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Received: 2021-01-16

Revised: 2021-07-05

Accepted: 2021-08-03

Published Online: 2021-08-23

Published in Print: 2021-11-25

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Schlagwörter für diesen Artikel

dust ion-acoustic wave; dusty e–p–i plasma; fractional Gardner equation; modified G′/G-expansion method