Modulational instability of ion-acoustic waves in a plasma
with a q-nonextensive electron velocity distribution
A. S. Bains,
1,a͒
Mouloud Tribeche,
2,b͒
and T. S. Gill
1,c͒
1
Department of Physics, Guru Nanak Dev University, Amritsar 143005, India
2
Plasma Physics Group (PPG), Theoretical Physics Laboratory (TPL), Faculty of Sciences–Physics,
University of Bab-Ezzouar, U.S.T.H.B, B.P. 32, El Alia, Algiers 16111, Algeria
͑Received 17 September 2010; accepted 19 January 2011; published online 17 February 2011͒
The modulational instability ͑MI͒ of ion-acoustic waves ͑IAWs͒ in a two-component plasma is
investigated in the context of the nonextensive statistics proposed by Tsallis ͓J. Stat. Phys. 52, 479
͑1988͔͒. Using the reductive perturbation method, the nonlinear Schrödinger equation ͑NLSE͒
which governs the MI of the IAWs is obtained. The presence of the nonextensive electron
distribution is shown to influence the MI of the waves. Three different ranges of the nonextensive
q-parameter are considered and in each case the MI sets in under different conditions. Furthermore,
the effects of the q-parameter on the growth rate of MI are discussed in detail. © 2011 American
Institute of Physics. ͓doi:10.1063/1.3554658͔
I. INTRODUCTION
The dynamics of ion-acoustic waves ͑IAWs͒, which is
one of the basic wave processes in plasma, has been studied
for several decades both theoretically and experimentally.
Nonlinear theory for these waves was first considered in Ref.
1 where their basic features were studied using a mechanical
analogy. It has been established that stationary IAWs can
exist in the form of periodic or localized waves. The first
experimental observation of ion-acoustic solitons has been
made by Ikezi et al.
2
Subsequently and because of quantita-
tive discrepancies between theory and experiment, the non-
linear IAW theory has been developed to include the effects
of a finite ion temperature
3,4
and those due to a trapped elec-
tron population
5,6
and high order nonlinearity.
7
Over the last
few years, a great deal of attention has been paid to nonex-
tensive statistic mechanics based on the deviations of
Boltzmann–Gibbs–Shannon ͑BGS͒ entropic measure. A suit-
able nonextensive generalization of the BGS entropy for sta-
tistical equilibrium was first recognized by Renyi
8
and sub-
sequently proposed by Tsallis,
9
suitably extending the
standard additivity of the entropies to the nonlinear, nonex-
tensive case where one particular parameter, the entropic in-
dex q, characterizes the degree of nonextensivity of the con-
sidered system ͑q=1 corresponds to the standard, extensive,
BGS statistics͒. This nonadditive entropy of Tsallis and the
ensuing generalized statistics have been employed with suc-
cess in a wide range of phenomena characterized by
nonextensivity.
10–26
As is well-known, Maxwellian distribu-
tion in Boltzmann–Gibbs statistics is believed valid univer-
sally for the macroscopic ergodic equilibrium systems. How-
ever, for the systems with the long-range interactions, such
as plasma and gravitational systems, where the nonequilib-
rium stationary states exist, Maxwellian distribution might
be inadequate for the description of the systems. A few ex-
amples of physical systems where the standard Boltzmann–
Gibbs approach seems to be inadequate are self-gravitating
systems and some kinds of plasma turbulence. A growing
body of evidence suggests that the q-entropy may provide a
convenient frame for the analysis of many astrophysical sce-
narios, such as stellar polytropes, solar neutrino problem, and
peculiar velocity distribution of galaxy clusters. It has been
shown that the experimental results, for electrostatic plane-
wave propagation in a collisionless thermal plasma, point to
a class of Tsallis’s velocity distribution described by a non-
extensive q-parameter smaller than unity.
13
In the last few years, great attention has been given to
the study of localized solitary structures with different par-
ticle distributions.
26–32
However, most of the above cited
work confined to the small amplitude ͑Kortweg–de Vries
equation͒ or/and large amplitude ͑pseudopotential method͒
study of localized solitary waves. They describe the evolu-
tion of an unmodulated wave, a bare pulse containing no
high frequency oscillation inside the wave packet. However,
the nonlinear propagation in a dispersive media is generi-
cally subject to amplitude modulation due to carrier wave
self-interaction or due to intrinsic medium nonlinearity. Para-
digm used to study such mechanism is standard multiple
scale method leading to the derivation of a nonlinear
Schrödinger equation ͑NLSE͒ in one dimension. In the
NLSE, the nonlinearity is in balance with the group velocity
dispersion and the resulting stationary solutions have enve-
lope structures. Modulational instability ͑MI͒ of different
plasma modes in several plasma environments
33–37
have
been studied due to their relevance in stable wave propaga-
tion. In these investigations authors have studied the effects
of nonthermal and kappa distributed particles on the modu-
lational instabilities of the different plasma modes. To the
authors’ knowledge, the MI of IAWs in a nonextensive
plasma has never been reported in the plasma literature. The
aim of the present paper is therefore revisit the MI of IAWs
a͒
Electronic mail: bainsphysics@yahoo.co.in.
b͒
Electronic mail: mtribeche@usthb.dz.
c͒
Electronic mail: gillsema@yahoo.co.in.
PHYSICS OF PLASMAS 18, 022108 ͑2011͒
1070-664X/2011/18͑2͒/022108/5/$30.00 © 2011 American Institute of Physics18, 022108-1