Cylindrical and spherical electron acoustic solitary waves with nonextensive
Hamid Reza Pakzad
Department of Physics, Bojnourd Branch, Islamic Azad University, Bojnourd, Iran
(Received 5 May 2011; accepted 12 July 2011; published online 11 August 2011)
Nonlinear propagation of cylindrical and spherical electron-acoustic solitons in an unmagnetized
plasma consisting cold electron ﬂuid, hot electrons obeying a nonextensive distribution and
stationary ions, are investigated. For this purpose, the standard reductive perturbation method is
employed to derive the cylindrical/spherical Korteweg-de-Vries equation, which governs the
dynamics of electron-acoustic solitons. The effects of nonplanar geometry and nonextensive hot
electrons on the behavior of cylindrical and spherical electron acoustic solitons are also studied by
2011 American Institute of Physics. [doi:10.1063/1.3622207]
Electron acoustic waves (EAWs) are one of the basic
wave processes in plasmas and they have been studied for
several decades both theoretically and experimentally.
EAWs can exist in two-temperature (cold and hot) electron
plasmas. The evidence of two populations of electrons in
laboratory and space plasmas has already been reported. The
EAWs are typically high frequency waves in comparison
with the ion plasma frequency. Therefore, ions remain sta-
tionary and form a neutralized background. The phase speed
of the EAW lies between the cold and hot electron thermal
velocities, so that the Landau damping effects are ignored
for the consistency of ﬂuid theory in two electron population
plasmas. Watanabe et al.
used a linear electrostatic Vlasov
dispersion equation to show that electron acoustic waves can
be destabilized in such a plasma. Later on, Yu and Shukla
and also Gary et al.
obtained a dispersion relation for
EAWs in a two (electron-ion) and three (two-temperature
electrons and ions) components plasmas, respectively.
The electron-acoustic solitary wave (EASW) is a local-
ized nonlinear wave which arises due to a delicate balance
between nonlinearity and dispersion effects. EASWs have
been extensively studied theoretically
as well as numeri-
They have been observed in experiments with pure
and in laser-produced plasmas.
lot of related numerical simulations have been reported.
Nonlinear propagation of EASWs has been investigated by
This propagation has been studied
in unmagnetized two-temperature electron plasmas
well as magnetized plasmas.
Mamun et al.
modiﬁed Korteweg-de Vries (mKdV) equation in a collision-
less plasma having cold ﬂuid electrons, hot non-isothermal
electrons obeying a vortex-like distribution, and stationary
ions. They investigated the effect of electron trapping on
small but ﬁnite amplitude acoustic solitary waves.
On the other hand, space plasma observations indicate
clearly the presence of ion and electron populations which
are far away from their thermodynamic equilibrium.
Over the last two decades, a great deal of attention has been
paid to nonextensive statistical mechanics based on the devi-
ations of Boltzmann–Gibbs–Shannon (B-G-S) entropic mea-
sure. A suitable nonextensive generalization of the B-G-S
entropy for statistical equilibrium was ﬁrst recognized by
and subsequently proposed by Tsallis,
extending the standard additivity of the entropies to the non-
linear, nonextensive case where one particular parameter, the
entropic index q, characterizes the degree of nonextensivity
of the considered system (q ¼ 1 corresponds to the standard,
extensive, B-G-S statistics). Indeed, there are many physical
systems which cannot be explained using the classical statis-
tical description correctly. Some of them can be described
by suitable framework of nonextensive statistics. As is well-
known, the Maxwellian distribution in Boltzmann–Gibbs
statistics is believed valid universally for the macroscopic
ergodic equilibrium systems. However, for systems with
long-range interactions, such as plasmas (Coulombian long-
range interaction) and gravitational systems, with non-
equilibrium stationary states, the Maxwellian distribution
might be inadequate for complete description of the features.
The parameter “q” that underpins the generalized entropy of
Tsallis is linked to the underlying dynamics of the system
and measures the amount of its non-extensivity. Nonexten-
sive systems (in statistical mechanics and thermodynamics)
are systems for which the whole entropy is different from the
sum of the entropies of the respective parts. In other words,
the generalized entropy of the whole is greater than the sum
of the entropies of the parts if q < 1 (superextensivity),
whereas the generalized entropy of the system is smaller
than the sum of the entropies of the parts for q > 1 (subexten-
sivity). Nonextensive statistics was successfully applied to a
number of astrophysical and cosmological scenarios. Those
include stellar polytropes,
the solar neutrino problem,
culiar velocity distributions of galaxies,
and systems with
long range interactions and also fractal like space-times. Cos-
mological implications were discussed in Ref. 33 and plasma
oscillations in a collisionless thermal plasma (which has been
recently analyzed) were provided from q-statistics.
other hand, kappa-distributions are highly favored in any
kind of space plasma modeling
where a reasonable physical
background was not apparent. A comprehensive discussion of
kappa distributions in view of experimentally favored non-
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2011 American Institute of Physics18, 082105-1
PHYSICS OF PLASMAS 18, 082105 (2011)