PHYSICAL REVIEW E, VOLUME 63, 046403

Vortex solitons: Mass, energy, and angular momentum bunchingin relativistic electron-positron plasmas
T. Tatsuno and V. I. Berezhiani
Graduate School of Frontier Sciences, The University of Tokyo, Hongo 7-3-1, Tokyo 113-0033, JapanS. M. Mahajan
Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712
共Received 12 June 2000; published 22 March 2001兲
It is shown that the interaction of large amplitude electromagnetic waves with a hot electron-positron (e-p)
plasma 共a principal constituent of the universe in the MeV epoch兲 leads to a bunching of mass, energy, andangular momentum in stable, long-lived structures. Electromagnetism in the MeV epoch, then, could provide
a possible route for seeding the observed large-scale structure of the universe.
DOI: 10.1103/PhysRevE.63.046403

PACS number共s兲: 52.40.Db, 52.35.⫺g, 95.30.Qd

I. INTRODUCTION

It is widely believed that the currently observed largescale structure of the universe 共the clusters and superclustersof galaxies兲 grew gravitationally out of small density fluctuations 关1兴. The imprint of this density variation in the early
universe was left on the cosmic microwave background radiation in the form of spatial temperature fluctuations. The
gravitational origin, however, can be only a part of the story;
the gravity can enhance, but it cannot produce these fluctuations. The quest for the physical process共es兲 which produced
the initial matter-density fluctuations has led to the emergence of the following two leading mechanisms: inflationand topological defects 关2,3兴. Of these the former is, perhaps,
the most thoroughly investigated. According to this mechanism there exists, in the evolution of the universe, an early
inflationary period in which the universe expands so rapidly
共exponentially兲 that quantum fluctuations become trapped in
the expansion. By the end of the inflation, therefore, small
irregularities covering a wide range of length scales permeate
the entire universe. Gravitational instability then acts onthese small initial irregularities, and enhances the concentration of matter from which galaxies and clusters of galaxies
eventually emerge 关4兴. The theory of such processes is, by nomeans, complete, and much needs to be done to determine
whether tiny quantum fluctuations can provide a strong
enough template for gravitational condensation to finally create the structures that we observe today. Cosmologists, usually, rely on the speculated existence of nonbaryonic dark
matter to augment the gravitational force to aid and accelerate the structure formation. The question is far from settled.
It is natural, then, to look elsewhere for the source of theseed density ‘‘fluctuations.’’ An obvious possibility is to explore if electromagnetic interactions taking place in a plasma
共known to be the source of a whole variety of linear as well
as nonlinear waves兲 can cause the required density perturbations. In the standard cosmological model of the hot Universe 共the Big Bang model兲, it is estimated that temperatures
as high as T⬃1010 K⬃1 MeV prevail up to times of ⬃1 s(t⬇1 sec) after the Big Bang. In this epoch, the main constituents of the Universe are photons, neutrinos, and antineutrinos, and e-p pairs 关1,4兴. As the plasma cools down
1063-651X/2001/63共4兲/046403共7兲/$20.00

the annihilation process e ⫹ ⫹e ⫺ → ␥ ⫹ ␥ dominates, and thee ⫹ e ⫺ pair concentration goes down. Since the equilibration
rates are fast in comparison with the changes in plasma parameters, an equilibrium e-p plasma should be present in the
MeV epoch of the early Universe. It is this plasmadominated era in which we will seek the seeds for future
structure formation.
Relativistic e-p plasmas were investigated quite extensively 关5兴. Tajima and Taniuti 关6兴 suggested that collective
processes in these plasmas could lead to interesting consequences for structure formation. In the e-p plasma of theearly Universe, localized low frequency electromagnetic
共EM兲 waves (ប ␻ ⰆT) could propagate as an envelope soliton due to the interaction with sound waves. Plasma density
variations related to these solitons could potentially be usefultoward structure formation in the Universe. However, theanalysis 关6兴 was based on a one-dimensional formulation,and the corresponding soliton solutions are likely to be unstable in higher dimensions. Berezhiani and Mahajan 关7兴 argued that in the MeV epoch of the Universe, although e-ppairs form the dominant constituent of the plasma, a minority
population of heavy ions is also present due to the baryon
asymmetry. They were able to show that, under appropriate
conditions 关when the plasma is transparent, i.e., ␻ Ⰷ ␻ e ,where ␻ ( ␻ e ) is the pulse 共plasma兲 frequency兴, the resulting
e-p –ion plasma supports the propagation of stable, nondiffracting, and nondispersing EM pulses 共light bullets兲 with alarge density bunching. It was further shown in Ref. 关8兴 thatthese bullets are exceptionally robust: they can emerge froma large variety of initial field distributions, and are remarkably stable. Note that the characteristic dimensions of suchmatter-filled light pulses are proportional to the electron toion density ratio, and tend to be considerably larger than theskin depth (␭⫽c/ ␻ e ). The implication is that when one
deals with EM structures whose characteristic dimensions 共of
the spatiotemporal inhomogeneities兲 are of the order of theskin depth, the baryon asymmetry affects can be safely neglected, and the dynamical system can be assumed as a pure
electron-positron plasma.
In the present paper we examine the propagation of strong
EM radiation in a hot pure e-p plasma, with the explicit aim

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PHYSICAL REVIEW E 63 046403

of finding soliton-type solutions. The plasma is assumed to
be transparent. We demonstrate that the dynamics of the EM
field envelope is governed by a generalized nonlinear Schrödinger equation 共NSE兲 with a defocusing nonlinearity. In one
dimension, this equation admits dark soliton solutions, whilein two dimensions, the so called vortex soliton solutions arealso possible.
Dark solitons exist as dips on a continuous-wave background field. Being stable in one dimension, they appear as
dark-stripe solitary waves in bulk medium. However, such
stripes are unstable to transverse modulations, which resultsin the induced generation of vortices with alternating polarities.
Vortex solitons, which are the most fundamental twodimensional 共2D兲 soliton solutions of NSE’s with an angular2 ␲ phase ramp, appear as local dark minima in an otherwise
bright background. Vortex solitons were recently observed inmaterials with a defocusing optical nonlinearity—the dynamics of laser beams in these materials is generally described by the NSE 关9兴. Since electromagnetic vortices carry
angular momentum that is conserved during propagation, thegeneration of vortex solitons in an e-p plasma is a potent
mechanism for creating domains with definite angular momenta, even out of an initial field distribution devoid of angular momentum. To keep the total angular momentum atzero, domains of equal and opposite angular momenta must
be created in pairs.
II. FORMULATION

We use the following set of relativistic hydrodynamic
equations in dimensionless form 关7兴:⳵A
1 ⳵d⫾ ⫾ ⫾⫿ 共 v ⫾ •“ ␾ 兲 ,共 G ␥ 兲 ⫺ ⫾ P ⫾ ⫽⫿ v ⫾ •
dt
⳵t
n ⳵t
⳵A
1
d⫾ ⫾ ⫾
共 G p 兲 ⫹ ⫾ “ P ⫾ ⫽⫿ ⫾ 关v ⫾ ⫻ 共 “⫻A兲兴 ⫿“ ␾ ,
dt
⳵t
n
⳵n⫾
⫹“• 共 n ⫾ v ⫾ 兲 ⫽0,
⳵t

⳵t
⫾

A⫺“ 2 A⫹
2

共2兲
共3兲

⫾ ⫾

⳵
“ ␾ ⫹ 共 n ⫺ v ⫺ ⫺n ⫹ v ⫹ 兲 ⫽0,⳵t⫾n ⫾ /T ⫾␥ ⫾ K 2 共 1/T ⫾ 兲

共4兲

1A⬜ ⫽ 共 x⫹iy兲 A⬜ 共 r⬜ ,z,t 兲 exp共 ikz⫺i ␻ t 兲 ⫹c.c.,
2

共5兲

共6兲

where A⬜ is a slowly varying function of r and t (kⰇ“,␻ Ⰷ ⳵ t ). The unit vectors x and y define two mutually perpendicular axes in the plane normal to the direction of wave
propagation. The Coulomb gauge condition leads to the relation A z ⫽(i/k)(“⬜ •A⬜ )ⰆA⬜ . Consequently the effects related to A z will turn out to be negligibly small. We shall now
follow standard methods to analyze the system. In the slowly
varying amplitude approximation, the transverse, highfrequency component of the equation of motion yields thesimple relation between the particle momentum and the vector potential 关7兴:p⬜⫾ G ⫾ ⫽⫿A⬜ .

⫾ 2 1/2

where p ⫽ ␥ v with factor ␥ ⫽ 关 1⫹(p ) 兴 ; d ⫾ /dt⫽ ⳵ / ⳵ t⫹ v ⫾ •“ is the comoving derivative; and G ⫾⫽K 3 (1/T ⫾ )/K 2 (1/T ⫾ ), with K n nth order modified Bessel
functions of the second kind. The superscript labels the particles, electrons (⫺), and positrons 共⫹兲, respectively. In
these equations the time and space variables are in units ofthe electron plasma frequency ␻ e ⫽(4 ␲ e 2 n 0 /m e ) 1/2, and thecollisionless skin depth c/ ␻ e , respectively, the field potentials ( ␾ , A) are in units of m e c 2 /e, and the relativistic mo-

exp共 ⫺G ⫾ /T ⫾ 兲 ⫽const,

which, at nonrelativistic temperature (T ⫾ Ⰶ1), reduces to thestandard adiabatic relation 关 n r⫾ /(T ⫾ ) 3/2⫽const兴 for an ordinary gas. In the ultrarelativistic limit 关 T ⫾ Ⰷ1 兴 , as expected,
Eq. 共5兲 describes the photon gas 关 n r⫾ /(T ⫾ ) 3 ⫽const兴 . In theultrarelativistic case, one should take into account the radiative pressure P R ⫽ ␴ T 4 ( ␴ ⫽ ␲ /45ប 3 c 3 ). For simplicity weneglect this less important effect for the current considerations. Note that in the MeV epoch, the plasma temperatureT ⫾ ⬇m e c 2 共i.e., z⬇1) and G⬇4, leading to an effective
mass of e-p pairs of m eff⬃4m e . Since the particle masses
are just a few times larger than their rest mass at these temperatures, the e-p plasma can be considered as a two component fluid rather than a photon gas.
We consider the propagation of circularly polarized EM
wave with a mean frequency ␻ , and a mean wave number kalong the z axis. The choice of circular polarization is not
restrictive; it simplifies the analysis by preventing harmonic
generation. The vector potential can be represented as

共1兲

along with the field equation 共in the Coulomb gauge “•A⫽0)⳵2

mentum vector p⫾ is in units of m e c. The particle number
density n ⫾ is normalized by the equilibrium density n 0⫾⫽n ⫾0 , and the plasma temperature (T ) is measured in units⫾ ⫾2⫾of m e c . The pressure P ⫽n r T , where n r⫾ is the density
in the rest frame of the fluid element (n r⫾ ⫽n ⫾ / ␥ ⫾ ). The
function G(z) defines the ‘‘effective’’ temperature dependent mass of the particles, and has the following limiting
expressions: G⬇1⫹5/2z for zⰇ1, and G⬇4/z if zⰆ1.
From Eqs. 共1兲–共3兲 it is straightforward to derive the adiabatic equation of state 关10兴:

共7兲

The low frequency motion of the plasma is driven by the
ponderomotive pressure 关 ⬃(p⫾ ) 2 兴 of the high frequency EM
field, and it does not depend on the sign of the particles’
charge. If we assume that in equilibrium the electron andpositron fluids have equal temperatures (T ⫾0 ⫽T 0 ), their effective masses will also be equal (G ⫾ ⫽G), and the radiationpressure will impart equal low frequency momenta to both
fluids, allowing the possibility of overall density changes

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PHYSICAL REVIEW E 63 046403

without producing charge separation. The charge neutrality
conditions n ⫺ ⫽n ⫹ ⫽N, ␾ ⫽0 will be assumed in the rest of
this paper. It is also evident that the symmetry between thetwo fluids keeps their temperatures always equal (T ⫾ ⫽T) if
they were equal initially.
A considerable simplification results when we invoke thewide beam approximation 关11兴. We assume that the longitudinal variation of the field envelope is much stronger than
the transverse variation, i.e., L z , the characteristic length
along the propagation direction, is much shorter than L⬜ , thecharacteristic length in the transverse plane. This approximation, coupled with charge neutrality, allows us to extract,from Eqs. 共1兲 and 共2兲, the following, leading order description for the low frequency response: the equation of motion

which is readily solved for an explicit expression for thelongitudinal momentum in terms of the transverse vector potential,

1 ⳵ NT1 ⳵ 兩 A⬜ 兩 2
d
Gp⫹
,
⫽⫺
dt
N ⳵z ␥
2␥G ⳵z共9兲冋␥ ⫽ 1⫹

G2

⫹p

2

册

1/2

.

共10兲

Substituting Eqs. 共6兲 and 共7兲 into Eq. 共4兲, we find that theslowly varying amplitude A⬜ must satisfy
2i␻ 共 ⳵ t ⫹ v g ⳵ z 兲 A⬜ ⫹“⬜2 A⬜ ⫹ 共 ⳵ z2 ⫺ ⳵ 2t 兲 A⬜ ⫹ 共 ␻ 2 ⫺k 2 兲 A⬜
⫺

2N
A ⫽0,
␥G ⬜

共11兲

where v g denotes the group velocity of the carrier waves;v g ⫽d␻ /dk⫽k/ ␻ .
We are still not quite done with simplifying assumptions.
We seek solutions which vary slowly with time in a frame
comoving with the wave, that is, in a frame propagating withthe group velocity v g . The transformations ␰ ⫽z⫺ v g t and␶ ⫽t, with the condition v g ⳵ ␰ Ⰷ ⳵ ␶ , help implement this approximation. Equations 共8兲 and 共9兲 can now be combined toderive⳵关 G 共 ␥ ⫺ v g p 兲兴 ⫽0;
⳵␰

共12兲

the implied constant of motion is to be determined from theboundary conditions. We demand p and A⬜ to be zero at
infinite ␰ , but allow them to be finite as r⬜ →⬁. Integrating
Eq. 共12兲 leads to (T 0 is the particle temperature at infinity兲,G 共 T 兲共 ␥ ⫺ v g p 兲 ⫽G 0 共 T 0 兲 ,

共13兲

册

共14兲

where ␥ g ⫽1/(1⫺ v 2g ) 1/2 is the ‘‘effective relativistic factor’’associated with the group velocity of the wave; it is not to be
confused with the particle ␥ . The continuity equation can besimilarly integrated to determine the particle density 关afterusing Eq. 共14兲 for p]:
Nv g␥ g⫽ 2 2.
␥ G 共 ␥ g G 0 ⫺G 2 ⫺ 兩 A⬜ 兩 2 兲 1/2

共15兲

Substituting Eq. 共15兲 into Eq. 共11兲, we obtain the following nonlinear Schrödinger equation for the complex amplitude A⬜ :2i ␻⳵ ␶ A⬜ ⫹“⬜2 A⬜ ⫹Here we have used the condition that the ponderomotive
pressure gives equal longitudinal momenta to both electronsand positrons (p z⫾ ⫽p). Note that the assumed circular polarization of the EM field insures that the relativistic factor ␥does not depend on the ‘‘fast’’ time (1/␻ ) scale; it can be
written as兩 A⬜ 兩 2

冋

G0
1
1⫺
共 ␥ 2 G 2 ⫺G 2 ⫺ 兩 A⬜ 兩 2 兲 1/2 ,G
G 0v g␥ g g 0

共8兲

and the ‘‘energy’’ conservation equation
1 ⳵ NT1 ⳵ 兩 A⬜ 兩 2
d
G␥⫺
.
⫽
dt
N ⳵t ␥
2␥G ⳵tp⫽ v g ␥ 2g

⫹

冉

1

␥ 2g

⳵ ␰2 A⬜

冊

2v g␥ gG 0
1⫺ 2 2A⬜ ⫽0,
G0
共 ␥ g G 0 ⫺G 2 ⫺ 兩 A⬜ 兩 2 兲 1/2

共16兲

where the wave frequency ␻ satisfies the dispersion relation
␻ 2 ⫽k 2 ⫹2/G 0 , implying that the parameter ␥ g⫽ ␻ 冑G 0 /2 关 ␥ g ⫽( ␻ / ␻ e ) 冑G 0 /2 in physical quantities兴. A set
comprising Eq. 共16兲, and the equation of state 共5兲 关in which
relation 共15兲 could be easily incorporated兴 constitutes a complete description of the dynamics of strong EM waves inrelativistic e-p plasma in the wide beam approximation.
We remind the reader that Eq. 共16兲 was derived under theassumption ⳵ ␰ Ⰷ“⬜ 共i.e., L z ⰆL⬜ ). In spite of this, for ahighly transparent plasma ( ␥ g Ⰷ1) the second, ‘‘diffractive,’’ term can be of the same order or even greater than thethird, ‘‘dispersive,’’ term. For this paper, we will not attempt
the general solutions of this quite complicated set of equations; we will simply deal with waves for which the plasma
is so highly transparent that the diffractive term dominates.
Using ␥ g G 0 ⰇG, and neglecting the dispersive term, theNSE simplifies to1i⳵ ␶ A⬜ ⫹ “⬜2 A⬜ ⫺2
2

冋冉

1⫺

兩 A⬜ 兩 2

␥ 2g G 20

冊 册
⫺1/2

⫺1 A⬜ ⫽0,

共17兲

where the following renormalizations are used: ␶ /2 ␻ G 0→ ␶ and r⬜ / 冑2G 0 →r⬜ .
The vector potential 兩 A⬜ 兩 is restricted from above by thecondition 兩 A⬜ 兩 ⬍ ␥ g G 0 . This restriction is necessary for thevalidity of the hydrodynamic treatment for the particles. For
larger amplitudes, the electromagnetic waves are overturned,
causing a multistream motion of the plasma requiring a kinetic description. Note that despite the upper bound on the046403-3

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PHYSICAL REVIEW E 63 046403

amplitude of the vector potential, the EM field can be still
relativistically strong, i.e., the normalized 兩 A⬜ 兩 Ⰷ1, since␥ g Ⰷ1.
III. STATIONARY SOLUTIONSIn the NSE derived above the diffractive and nonlinear
terms have opposite signs and as a consequence Eq. 共17兲
does not admit transversely localized solutions 共also calledbright solitons兲. Any localized initial EM field, therefore,
will undergo transverse spreading during propagation. The
NSE with a defocusing nonlinearity can, however, support
stationary structures with asymptotically 共at infinity兲 nonvanishing fields. Dark solitons in one dimensions, and vortex
solitons in two dimensions, are the fundamental representatives of such solutions. In the extreme low amplitude limit,兩 A⬜ 兩 Ⰶ ␥ g G 0 , Eq. 共17兲 reduces to a NSE with a cubic nonlinearity. In one-dimensional geometry we have1
⳵ A⬜ 1 ⳵ 2 A⬜
⫹
⫺ 2 2 兩 A⬜ 兩 2 A⬜ ⫽0.
i
2
⳵␶
2 ⳵x
␥ gG 0共18兲This equation is exactly integrable via the inverse scattering
method 关12兴, and its one-soliton solution can be written as
关9兴
2A⬜ 共 x, ␶ 兲 ⫽ ␥ g G 0 A 0 共 ␣ tanh ⌰⫹i ␤ 兲 e ⫺iA 0 ␶ ,共19兲⌰⫽ ␣ A 0 共 x⫺ ␤ A 0 ␶ 兲 .

共20兲

where

Here A 0 is a measure of the asymptotic fields at the spatial
infinity, and ␣ and ␤ are constants with ␣ 2 ⫹ ␤ 2 ⫽1. The
solution, with a nonzero value at the center of the dip, istermed the ‘‘gray soliton’’ to distinguish it from the ‘‘blacksoliton’’ 共zero amplitude at the dip兲 corresponding to ␤ ⫽0.
Dark solitons of this class of NSE’s do not have any threshold values for their excitation, unlike bright solitons 共of theappropriate equations兲, which do. In other words, dark solitons can be created by an arbitrary small initial dip on ahomogeneous background.
In two transverse dimensions, a dark soliton represents adark stripe imposed on a homogeneous bright background. It
is well known that such a stripe is unstable to transverse,
long wavelength modulations 关13兴. The instability causes the
stripe to split into a sequence of vortex solitons of alternative
polarities 关14兴. The vortices are dark holes on a bright background, with a nested phase dislocation of the order m⫽⫾1,⫾2, . . . at their core.
Vortex soliton solutions of the NSE were first suggested
by Pitaevskii 关15兴 as topological excitations in an imperfect
Bose gas in the superfluids. The ability of some electromagnetic systems 共like the e-p plasma兲 to simulate fluid dynamical phenomena 共like vortex formation兲 can be demonstrated
by applying the Madelung transformation A⬜ ⫽ 冑␳ exp(i␺) to
the defining equations. The transformation converts theoriginal set to one that is similar to the fluid hydrodynamic
equations with a fluid ‘‘density’’ ␳ and fluid ‘‘velocity’’ v⫽“⬜ ␺ . Vortices can exist despite the potential nature of the‘‘fluid’’ flow. Indeed, the Madelung transform is singular atpoints where ␳ ⫽0; these are just branch points where thereal and imaginary parts of the field become zero, while thevelocity circulation 养 v •dl⫽2 ␲ m, where the integration isdone on a closed path enclosing the singular point, and theinteger m is known as the topological ‘‘charge’’ of the vortex. Thus the vortex soliton is a topological structure; it candisappear only when annihilated by a vortex soliton of the
opposite charge. The development of the transverse instability of a dark soliton has close parallels in hydrodynamics: for
instance, the Kelvin-Helmholtz instability, which occurs
when the boundary between two flows develops so-called‘‘vortex streets’’ 关14兴. Since a dark solitary stripe does notcarry any topological charge, it is evident that vortices haveto be born with equal and opposite topological charges.
It is straightforward to show that e-p plasmas can support
large amplitude dark solitons as well. In the general case
共amplitude large, but subject to the condition 兩 A⬜ 兩 ⬍ ␥ g G 0 ),we cannot construct analytic solutions even in one dimension. It is possible, however, to extract the general propertiesof the solution by using reasonably simple techniques, especially when A⬜ has the time dependence
A 共 x, ␶ 兲 ⫽Â 共 x 兲 exp共 ⫺i␭ ␶ 兲 ,共21兲

where A⫽A⬜ / ␥ g G 0 is the normalized amplitude, and is always less than unity. Here ␭ is so-called nonlinear frequency
shift. This time dependence implies that the amplitude squareis stationary 共what follows, therefore, are classed as stationary solutions兲, and the dip of the wave does not propagate in
the comoving frame with quite the group velocity of thelinear wave.
The 1D equation 共18兲 now can be cast in the formd2
dx 2

Â⫹V ⬘ 共 Â 兲 ⫽0,

共22兲

where the prime on V denotes the derivative with respect toÂ, and
V 共 Â 兲 ⫽ 共 ␭⫹2 兲 Â 2 ⫹4 冑1⫺Â 2 ⫺4

共23兲

denotes the potential. The resemblance of Eq. 共22兲 to the oneobeyed by a Newtonian particle in a nonlinear potential suggests an obvious method for analysis. One can easily provethat a bounded solution exists provided the nonlinear frequency shift is positive (␭⬎0). The profile of the potential,
shown in Fig. 1 for ␭⫽1, reveals that the dark soliton solution may reside in the potential well. Equating d 2 Â/dx 2 withzero, we estimate the upper bound on Â:Â ub⫽

冑 冉 冊
1⫺

2 2
.
␭⫹2

共24兲

The lowermost value of the amplitude 兩 A 兩 is zero, that is, werecover the dark soliton. Note that for small values of ␭(Ⰶ1), Â ub→0, while Â ub→1 for ␭Ⰷ1.

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PHYSICAL REVIEW E 63 046403

FIG. 1. The form of the potential V(Â) in Eq. 共22兲 is shown for␭⫽1.

FIG. 2. The form of the solitonlike solutions for the normalized
potential Â is shown for m⫽1, 2, and 3 in the case of ␭⫽1.

Similarly one can show the existence of vortex soliton
solutions in two dimensions. We shall again seek stationary
solutions in 2D polar coordinates (r, ␪ ). The ansatz
A⫽Â 共 r 兲 exp共 im ␪ ⫺i␭ ␶ 兲 ,共25兲

with Â(r) real, and with the perpendicular Laplacian operator given by
“⬜2 ⫽

d2
dr

⫹
2

m2
1 d
⫺ 2,
r dr r共26兲

converts Eq. 共17兲 to the ordinary differential equation 关similar to Eq. 共22兲兴d2
drÂ⫹V ⬘ 共 Â 兲 ⫽⫺2

1 dÂ m 2
⫹ 2 Â,
r dr
r

共27兲

where the potential V(Â) is the same as the one-dimensional
expression given by Eq. 共23兲. If we were to extend the ‘‘particle in a potential’’ analogy further, Eq. 共27兲 could beviewed as the nonconservative motion of a particle. Since theright-hand side approaches zero in the limit r→⬁, Eq. 共27兲gives precisely the 1D asymptotic value 关Eq. 共24兲 for thevector potential Â]. The behavior at the origin (r⫽0) is
totally different; the regular singular point at the origin r⫽0 forces the acceptable Â to vanish for m⭓1 as r m . The
numerical solutions of the two-dimensional nonlinear Schrödinger equation for m⫽1, 2, and 3 are shown in Fig. 2. As
expected, the solitonlike solutions evidently go to zero as r m
for small r, and reach an m-independent asymptotic value
predicted by Eq. 共24兲.
More general aspects of the dynamics of the EM field canbe studied mainly through numerical simulations of Eq.
共17兲—this is beyond the intended scope of this paper. We
content ourselves here by making a few qualitative remarks,
and pointing out directions for future efforts. The nonlinearity in Eq. 共17兲 is a faster growing function 共faster than the
cubic兲 of the field amplitude, and does not exhibit saturation.
One would expect, then, that the development of vortex
chain structures from the dark stripe soliton instability for thegeneral system will be faster in comparison with the lowamplitude 共cubic nonlinearity兲 case. This is, indeed, confirmed by a numerical solution of Eq. 共17兲; the detailed re-sults of the simulation will be discussed in a later publication. The stability of the vortex soliton solution is anotherissue that has to be dealt with. It is usually believed, however, that vortices with m⫽⫾1 are topologically stable,
whereas vortices with larger value of the ‘‘charge’’ m may
decay into ‘‘single-charge’’ vortices.
In three dimensions the vortices form the so-called vortex
line 共it looks like a pancake with a hole in the center, hanging and moving along the wire-vortex line兲. Effects related to
the finite group velocity dispersion may lead to a transverse
instability of the vortex line. All these interesting effects areleft for future studies.
We would like to emphasize that the electromagnetic
fields associated with dark and vortex solitons are asymptotically nonvanishing 共at infinity兲. Due to the generally accepted requirement that in physical system, the fields be localized in all directions, these objects have received muchless attention than their localized cousins. However, in recent
experiments studying laser field dynamics in different kindsof optical media, it was demonstrated that dark and vortex
solitons can be readily created as superimpositions upon alocalized field background 关9兴. This background can be just a
few times wider than the soliton width. During propagation
the background spreads out, reducing its own intensity. In
light of these experiments let us try to put in perspective thecurrent study of dark and vortex solitons in e-p plasmas in
the early Universe. Because the typical scale length of these
solitons is the collisionless skin depth, we would need a supporting background spanning several skin depths. Thisshould pose no problem, because the ambient uniform fieldbackground could easily foot the bill. The next scale lengthon which we encounter ‘‘bulletlike’’ electromagnetic structures 共which owe their origin to the baryon asymmetry兲 isconsiderably larger than the skin depth. Thus the dark andvortex solitons can propagate in a slowly changing background 共spreading and decreasing in intensity with the diffractive spreading rate of the soliton decreasing as the background expands兲, adiabatically maintaining their properties,
until they hit baryon-asymmetry scale lengths.
Topological considerations will insure the preservation of
the singular points during propagation. In propagating vortex
chains, the vortices can move away from one another, reducing the possibility of their mutual annihilation. The propagation introduces elements similar to Hubble expansion—the046403-5

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structures ‘‘run away’’ from one another. These highly
speculative remarks need careful investigation. It is possiblethat the spreading of the background field may just affect thevortex distribution, and only a cosmological expansion will
drive them apart.
What is extremely significant is that during the evolutionof the fields, the integrals of motion should be preserved. It iseasy to prove, by direct calculations, that Eq. 共17兲 conserves
the angular momentum M:
共 M兲 z ⫽i2冕dr⬜ 关 r⬜ ⫻ 共 A⬜* “⬜ A⬜ ⫺c.c.兲兴 z .

共28兲

Equation 共28兲 for the angular momentum is a paraxial approximation for the orbital angular momentum, ME ⫽ 兰 dr关 r⫻(E⫻B) 兴 , of the EM field 关16兴. The angular momentum
carried by the vortices is M z ⫽mN, where N is another conserved quantity known as the ‘‘photon number’’ N⫽ 兰 dr⬜ 兩 A⬜ 兩 2 关17兴.
It follows, then, that relativistic e-p plasmas are capableof sustaining electromagnetic vortexlike structures, and thatthese structures have domains in which the EM fields carrynonzero angular momenta, although the total angular momentum of the entire system is zero. If this angular momentum could, somehow, be locally transferred to the surrounding medium, we would have a rather effective mechanism
for imparting angular momentum to different domains ofmatter in the early universe. In our next publication we willshow that when baryon asymmetry effects are incorporated,the medium can, indeed, acquire angular momentum fromthe EM field vortices.

type solutions for this equation. The transverse instability ofdark soliton stripes leads to the formation of a vortex chain
such that the EM fields in each vortex carry angular momentum. Such objects could play an important role in cosmology
as sources of the structure formation in the MeV epoch of theevolution of the Universe. In commonly adopted cosmological scenarios about the origin of the rotation of galaxies,
structures grow in a hierarchy by the gravitational assemblyof clumps out of subclumps. The origin of the angular momentum of galaxies, if they were formed from initial fluctuations in a Friedman Universe, was suggested 共by Hoyle 关18兴兲
to be due to tidal interactions between the condensing system
关4兴. However, it is still not clear whether this mechanism
gives an adequate solution 关19兴. We hope that the suggested
mechanism of angular momentum generation in the MeV
epoch of the Universe is an interesting alternative to explore
and examine. Electromagnetism, operating through the versatile substrate of the e-p plasma, seems to readily generatethese highly interesting, long-lived objects—the carriers oflarge amounts of mass, energy, and angular momentum.
Since an initial localization of mass, energy, and angularmomentum is precisely the seed that gravity needs for eventual structure formation, electromagnetism may have provided a key element in the construction of a large-scale mapof the observable Universe.
Results of this paper can also be applied to astrophysicalobjects like pulsars, and active galactic nuclei—the e-p pairs
are thought to be a major constituent of the plasma emanating both from the pulsars, and from the inner region of theaccretion disks surrounding the central black holes.

IV. CONCLUSIONSACKNOWLEDGMENTS

We have investigated the dynamics of the highly relativistic ( ␥ g Ⰷ1) nonlinear propagation of electromagnetic
waves in unmagnetized hot electron-positron plasmas. The
system is described by a nonlinear Schrödinger equation 共17兲with an inverse square root type 共nonsaturating兲 nonlinearity.
We have shown the possibility of dark and vortex soliton

This work was supported in part by a Grant-in-Aid fromthe Japanese Ministry of Education, Science, and Culture,
No. 12780353. The work of V.I.B. was partially supportedby the INTAS Georgian CALL-97, and S.M.M.’s work was
supported by the U.S. Department of Energy Contract No.
DE-FG03-96ER-54346.

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