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Evolution of ion-acoustic shock waves in magnetized plasma with hybrid Cairns–Tsallis distributed electrons

  • Biswajit Sahu ORCID logo EMAIL logo and Rabindranath Maity
Published/Copyright: September 8, 2022
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 77 Issue 12

Abstract

The propagation of nonlinear electrostatic ion-acoustic (IA) shock waves in presence of external magnetic field having Cairns–Tsallis distributed electrons and ion kinematic viscosity is investigated. In the linear regime, the dispersion relation of the ion acoustic shock wave is found to be modified by the external magnetic field. Adopting reductive perturbation approach, it is shown that the dynamics of shocks is modeled by a hybrid Ostrovsky–Burgers’ equation. The influence of relevant physical parameters such as nonthermality and nonextensivity of electrons, magnetic field strength, and ion kinematic viscosity on the time evolution of the shock structure is numerically examined. It is observed the present plasma system supports both compressive and rarefactive shock waves. Furthermore, the analysis is performed through dynamical system approach to elucidate the various aspects of the phase-space shock dynamics.

Keywords: Cairns–Tsallis distribution; magnetized plasmas; Ostrovsky–Burgers’equation; shock waves
Corresponding author: Biswajit Sahu, Department of Mathematics, West Bengal State University, Barasat, Kolkata, 700 126, India, E-mail:

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

The distribution function in the presence of a nonzero potential can simply be found by replacing v 2 v t e 2 in Eq. (4) by v 2 v t e 2 − 2 e ϕ k B T e . Therefore distribution function (4) reads as

(A1) f e ( v ) = C q , α 1 + α v 2 v t e 2 − 2 e ϕ k B T e 2 × 1 − ( q − 1 ) v 2 2 v t e 2 − e ϕ k B T e 1 / ( q − 1 ) .

Here, we have given the detailed derivation of the electron density.

Case I: −1 < q < 1.

n e ( ϕ ) = ∫ − ∞ ∞ f e ( v ) d v

⇒ n e ( ϕ ) = ∫ − ∞ ∞ C q , α 1 + α m e v 2 k B T e − 2 e ϕ k B T e 2 × 1 + ( q − 1 ) e ϕ k B T e − ( q − 1 ) m e v 2 2 k B T e 1 ( q − 1 ) d v = ∫ − ∞ ∞ C q , α 1 + ( q − 1 ) e ϕ k B T e 1 ( q − 1 ) . × 1 + 4 α e 2 ϕ 2 k B 2 T e 2 − 4 α m e e ϕ k B 2 T e 2 v 2 + α m e 2 k B 2 T e 2 v 4 × 1 − ( q − 1 ) m e v 2 2 k b T e 1 + ( q − 1 ) e ϕ k B T e 1 ( q − 1 ) d v

Using the following variable change

x = v 1 + ( q − 1 ) e ϕ k B T e 1 2

we obtain

n e ( ϕ ) = C q , α 1 + ( q − 1 ) e ϕ k B T e 1 ( q − 1 ) + 1 2 × 2 ∫ 0 ∞ 1 + 4 α e 2 ϕ 2 k B 2 T e 2 − 4 α m e e ϕ k B 2 T e 2 1 + ( q − 1 ) e ϕ k B T e x 2   + α m e 2 k B 2 T e 2 1 + ( q − 1 ) e ϕ k B T e 2 x 4 × 1 − ( q − 1 ) m e 2 k B T e x 2 1 2 d x

⇒ n e ( ϕ ) = C q , α 1 + ( q − 1 ) e ϕ k B T e 1 ( q − 1 ) + 1 2 × 2 π k B T e m e . Γ 1 1 − q − 5 2 Γ 1 1 − q ( 1 − q ) 5 / 2 × 3 α + 1 1 − q − 3 2 1 1 − q − 5 2 ( 1 − q ) 2 × 1 − 16 q α 3 − 14 q + 15 q 2 + 12 α e ϕ k B T e   + 16 ( 2 q − 1 ) q α 3 − 14 q + 15 q 2 + 12 α e 2 ϕ 2 k B 2 T e 2

Using the expression of C q,α given in Eq. (5), we obtain

n e ( ϕ ) = n e 0 1 + ( q − 1 ) e ϕ k B T e q + 1 2 ( q − 1 ) × 1 + F 1 e ϕ k B T e + F 2 e ϕ k B T e 2 .

Case II: q > 1.

n e ( ϕ ) = ∫ − v max + v max f e ( v ) d v

Proceeding as in case I, we also obtain.

n e ( ϕ ) = n e 0 1 + ( q − 1 ) e ϕ k B T e q + 1 2 ( q − 1 ) × 1 + F 1 e ϕ k B T e + F 2 e ϕ k B T e 2 .

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Received: 2022-04-23
Accepted: 2022-08-17
Published Online: 2022-09-08
Published in Print: 2022-12-16

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. General
  3. The electrostatic wave modes and formation of dust voids in an externally magnetized cylindrical dusty plasma
  4. Atomic, Molecular & Chemical Physics
  5. Accurate theoretical calculation of relativistic atomic data of Zn-like, Ga-like and Ge-like Re ions
  6. Dynamical Systems & Nonlinear Phenomena
  7. Evolution of ion-acoustic shock waves in magnetized plasma with hybrid Cairns–Tsallis distributed electrons
  8. Free vibration analysis of rotating sandwich beams with FG-CNTRC face sheets in thermal environments with general boundary conditions
  9. Phase synchronization of Wien bridge oscillator-based Josephson junction connected by hybrid synapse
  10. Gravitation & Cosmology
  11. Formulation of axion-electrodynamics with Dirac fields
  12. Solid State Physics & Materials Science
  13. Tunable properties of the defect mode of a ternary photonic crystal with a high TC superconductor and semiconductor layers
  14. Trace cadmium ion detection using optical fiber Mach–Zehnder interferometer coated with PVA/TEOS/APTES
https://doi.org/10.1515/zna-2022-0117
Keywords for this article
Cairns–Tsallis distribution; magnetized plasmas; Ostrovsky–Burgers’equation; shock waves

Articles in the same Issue

  1. Frontmatter
  2. General
  3. The electrostatic wave modes and formation of dust voids in an externally magnetized cylindrical dusty plasma
  4. Atomic, Molecular & Chemical Physics
  5. Accurate theoretical calculation of relativistic atomic data of Zn-like, Ga-like and Ge-like Re ions
  6. Dynamical Systems & Nonlinear Phenomena
  7. Evolution of ion-acoustic shock waves in magnetized plasma with hybrid Cairns–Tsallis distributed electrons
  8. Free vibration analysis of rotating sandwich beams with FG-CNTRC face sheets in thermal environments with general boundary conditions
  9. Phase synchronization of Wien bridge oscillator-based Josephson junction connected by hybrid synapse
  10. Gravitation & Cosmology
  11. Formulation of axion-electrodynamics with Dirac fields
  12. Solid State Physics & Materials Science
  13. Tunable properties of the defect mode of a ternary photonic crystal with a high TC superconductor and semiconductor layers
  14. Trace cadmium ion detection using optical fiber Mach–Zehnder interferometer coated with PVA/TEOS/APTES
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