Electromagnetic solitons in fully relativistic electron-positron plasmas
with finite temperature
Nam C. Lee
Department of Physics, Pusan National University, Busan 609-735, Korea
(Received 3 March 2011; accepted 29 May 2011; published online 30 June 2011)
The existence of localized structure of electromagnetic waves in relativistic electron-positron
plasmas is investigated based on the pseudo-potential theory, without making any assumptions on
the magnitudes of the flow velocity and temperature of the medium. The conditions for the
localization of electromagnetic wave in the form of dark (dip type) soliton are found. In the small
amplitude approximation, it is found that the dip becomes deeper and narrower as the temperature
is raised. In low temperature T ( mc
2
, localized solution exists only if the equilibrium longitudinal
fluid velocity (parallel to the direction of propagation) in the wave frame is larger than the classical
thermal velocity
ffiffiffiffiffiffiffiffiffi
T=m
p
of the plasma. For ultra-relativistically high temperature T ) mc
2
,it
is shown that dark soliton can exist if the equilibrium longitudinal velocity is larger than c=
ffiffiffi
3
p
.
V
C
2011 American Institute of Physics. [doi:10.1063/1.3603309]
The nonlinear propagation of electromagnetic waves in
relativistic plasmas is a subject of considerable interest in the
context of astrophysical and laboratory problems. There
have been numerous studies on the possibility of localized
electromagnetic wave structure (soliton) in various composi-
tions of the medium, such as electron-ion,
1–3
electron-ion-
positron,
4,5
and electron-positron plasmas.
6–10
The aim of
this paper is to investigate the existence of solitary waves in
relativistic electron-positron plasmas having arbitrarily large
fluid velocity and temperature and explore the properties of
solitons analytically. The electron-positron pair creation
occurs in relativistic plasmas at high temperature (T), when
T becomes larger than twice the electron rest mass. They are
thought to be present in the high-power laser experiments
11
and in active galactic nuclei (AGN) (Ref. 12) and presum-
ably in the early period of evolution of the Universe.
13
Plasmas are usually considered to be relativistic when
the temperature is so high that the thermal energy is of order
of, or larger than, the rest energy mc
2
. However, plasmas can
be also considered as relativistic when the Lorentz relativis-
tic factor c ¼ 1=
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À u
2
p
based on the streaming speed u of
plasma (normalized with respect to the speed of light c)is
large as compared to 1. In this paper, we will study the local-
ization of electromagnetic waves in an electron-positron
plasma that is relativistic both in temperature and in stream-
ing speed, using a set of fully relativistic equations derived
from the covariant formulation of fluid plasmas.
In fluid description of fully relativistic plasmas, the con-
tinuity and momentum equations can be expressed as
6,14,15
1
c
@cn
@t
þrÁ cnuðÞ¼0 (1)
and
1
c
@hcu
@t
þ uÁr hcuðÞþ
1
cn
rp ¼ q E þ u BðÞ; (2)
for each species of plasmas, where u is the fluid velocity
normalized with respect to the speed of light c, and c
1=
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À u
2
p
is the relativistic Lorentz factor, n is the
density of the plasma, q is the charge per particle, p is the
pressure, a is the adiabatic index (4/3 for relativistic and 5/3
for nonrelativistic plasmas), and h : mc
2
+a /(a À 1)(p/n)is
the enthalpy per fluid particle. If we assume the adiabatic
relation p/n
a
¼ const, that can be deduced from the energy
equations of relativistic plasmas, h can be written as
h ¼ mc
2
þ Tn=n
0
ðÞ
aÀ1
= aÀ 1ðÞ; (3)
where T is the notation representing the temperature of
unperturbed (background) plasma defined as T : ap
0
/n
0
with the convention that Boltzmann constant k
B
¼ 1. Note
that all equations presented above apply equally well for
each species of plasma, but species specific indices are
dropped for the sake of typographic simplicity. Using the
vector and scalar potentials A and / in the Coulomb gauge,
and assuming that all quantities vary only in one spatial
coordinate z and in time t, Eqs. (1) and (2) can be simplified
as
1
c
@cn
@t
þ
@cnu
z
@z
¼ 0; (4)
1
c
@
@t
hcu
z
ðÞþ
@
@z
q/ þ hcðÞ¼0; (5)
hcu
?
þ qA
?
¼ 0; (6)
where the symbol \ refers to the component transverse to
the z direction. The system is closed by including the
Maxwell equations with fully relativistic electric charge
and current density expressed as q ¼ 4p
P
qcn and J
¼
P
qccnu; respectively, where the summation
P
runs
over all charged species. We assume that A
\
is circularly
polarized so that its components can be written as
A
x
¼ A cos h and A
y
¼ A sin h, where A and h are real func-
tions of z and t representing the amplitude and phase of the
wave. Then the Maxwell equations can be written as
1070-664X/2011/18(6)/062310/5/$30.00
V
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2011 American Institute of Physics18, 062310-1
PHYSICS OF PLASMAS 18, 062310 (2011)