Self-modulation of a strong electromagnetic wave in a positron-electron plasma
induced by relativistic temperatures and phonon damping
F. T. Gratton,
1
G. Gnavi,
1
R. M. O. Galva
˜
o,
2
and L. Gomberoff
3
1
Instituto de Fı ´sica del Plasma, Consejo Nacional de Investigaciones Cientı ´ficas y Te ´cnicas and Departamento de Fı ´sica,
Universidad de Buenos Aires, Ciudad Universitaria, Pabello ´n 1, 1428 Buenos Aires, Argentina
2
Instituto de Fı ´sica, Universidade de Sa ˜ o Paulo, Caixa Postal 66318, 05389-970 Sa ˜ o Paulo, SP, Brazil
3
Departamento de Fı ´sica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile
Received 1 July 1996; revised manuscript received 4 November 1996
The modulational instability of a linearly polarized, strong, electromagnetic wave in a unmagnetized
positron-electron plasma is analyzed using relativistic two-fluid hydrodynamics to properly account for physi-
cal regimes of very high temperatures. The effect of phonon damping is also included in the treatment. The
theory can be reduced to a pair of extended Zakharov equations. The envelope modulation is then studied by
deriving the corresponding nonlinear Schro ¨dinger NLS equation, using multiscale perturbation analysis.
According to the intensity of the damping three different types of NLS are obtained. The main results are a
that relativistic temperatures modify the stability result found in the literature for low temperature, zero
damping, e
+
- e
-
plasmas, and b that phonon damping also produces substantial changes in the NLS, which
then predict unstable envelopes. This work extends previous analyses, showing that if the phonon damping is
O
0
or O
1
is the perturbation parameter, a modulational instability appears in the electron-positron case
in all ranges of temperature and wave frequencies. Thus presence of some amount of sound absorption helps
to produce an envelope decay. When the phonon damping is very small O
2
the self-modulational insta-
bility occurs in a finite band near the reduced plasma frequency, for ultrarelativistic temperatures.
S1063-651X9714302-1
PACS numbers: 82.40.Ra, 51.60.+a
I. INTRODUCTION
The literature on waves and nonlinear processes in
positron-electron plasmas has grown rapidly in recent times
in view of possible applications in the following fields. Rela-
tivistic positron-electron plasmas are encountered in pulsar
magnetospheres and in active galactic nuclei. Surveys of
these fields can be found, for instance, in Refs. 1,2pulsars
and 3,4 active galactic nuclei AGN. Interesting scenarios
for e
+
- e
-
plasmas are conjectured in the physics of the early
time Universe, i.e., 10
-4
t 1 s after the big bang 5–7. In
the laboratory, nonrelativistic electron and positron trapping
in magnetic mirror experiments are presently actively pur-
sued 8,9.
Linear and nonlinear waves in e
+
- e
-
plasmas have many
properties different from electron-ion plasmas 10, due to
the absence of high and low frequency scales associated with
the electron-ion mass difference. A survey of linear waves in
e
+
- e
-
plasmas can be found in 11, while 12 reviews
nonlinear relativistic effects in plasmas, including the e
+
- e
-
case. Other surveys on relativistic nonlinear effects in waves
for ordinary and e
+
- e
-
plasmas, related to the physical
mechanisms discussed in this paper, can be found in Refs.
13 and 14.
Propagation and nonlinear processes associated with a
strong, linearly polarized, electromagnetic wave in an un-
magnetized electron-positron plasma have been studied by
several authors see, e.g., 14–20 since pioneering work by
Chian and Kennel 15 proposed a mechanism to explain
very short intensity variations micropulses of pulsar radio
emission. They suggested that a self-modulational instability
of the electromagnetic wave may be a natural process for
amplitude modulation. Gangadhara et al. 18 examined, in-
stead, a parametric instability of a weakly relativistic, elec-
tromagnetic wave in the e
+
- e
-
plasma, to explain also the
short-time variability of the radio sources. Kates and Kaup
17 in a careful analysis of the modulational instability,
based on multiple time-space scale perturbation theory of the
nonlinear electromagnetic wave in an electron-ion plasma,
concluded that in the zero temperature limit the special case
of a e
+
- e
-
plasma is stable. When a finite classical tem-
perature is considered, only a vanishingly small frequency
interval at
p
appears, where the instability is possible
in a e
+
- e
-
plasma. The divergence with the results of 15 is
explained by the absence of the ponderomotive force and
harmonic generation effects in that reference. In an electron-
ion plasma, however, the modulational envelope instability
of the electromagnetic wave is again possible. The nonlinear
Schro
¨
dinger equation NLS derived in 16, which was sup-
portive of the instability found in 15 and which included
longitudinal density variations, was found to be incorrect in
17.
We have reexamined the problem of the self-modulational
instability of a linearly polarized, large-amplitude, electro-
magnetic wave in a unmagnetized positron-electron
plasma, using a two-fluid model and taking into account the
three nonlinear effects considered in 17, i.e., i relativistic
correction due to mass variation, ii the ponderomotive
force that produces density changes, and iii harmonic gen-
eration. Indeed, the three mentioned effects have the same
order of magnitude in a perturbation treatment and they must
all be included at the same level in the theory.
We first obtain a set of extended Zakharov-type equations
PHYSICAL REVIEW E MARCH 1997 VOLUME 55, NUMBER 3
55 1063-651X/97/553/338112/$10.00 3381 © 1997 The American Physical Society