The nonlinear evolution of an electron acoustic wave is shown to obey the Davey-Stewartson I equation which admits so called dromion solutions. The importance of these two dimensional localized solutions for recent satellite observations of wave structures in the day side polar cap regions is discussed and the parameter regimes for their existence is delineated

Ghosh, S. S.

Lakhina, G. S.

Sen, A.

English

Publications of the IAS Fellows

PRAMANA c Indian Academy of Sciences Vol. 55, Nos 5 & 6
— journal of Nov. & Dec. 2000
physics pp. 693–698
Dromion solutions for an electron acoustic wave and its
application to space observations
S S GHOSH, A SEN and G S LAKHINAy
Institute for Plasma Research, Bhat, Gandhinagar 382 428, India
yIndian Institute of Geomagnetism, Colaba, Mumbai 400 005, India
Email: sukti@plasma.ernet.in
Abstract. The nonlinear evolution of an electron acoustic wave is shown to obey the Davey–
Stewartson I equation which admits so called dromion solutions. The importance of these two
dimensional localized solutions for recent satellite observations of wave structures in the day side
polar cap regions is discussed and the parameter regimes for their existence is delineated.
Keywords. Davey–Stewartson equation; electron acoustic wave; space plasma.
PACS No. 52.35.Mw
1. Introduction
For the last few decades exponentially localized structures like solitons in (1+1) dimen-
sional space have been extensively studied in the context of some exactly integrable partial
differential equations [1]. Its potential application in different physical fields are also well
known [2]. However its higher dimensional generalization has remained an open field of
investigation for a long time. It was Boiti et al [3] who first discovered the corresponding
(2+1) dimensional analog which, unlike lump or algebraic solitons, is localized exponen-
tially in both the spatial directions but in contrast to solitons can exchange energy during
collisions. Since these solutions are ‘driven by boundaries’ (dromos), i.e., they possess
time dependent boundary conditions, they are called dromions [4,5] and have drawn con-
siderable attention in the field of nonlinear dynamics. A well known two dimensional p.d.e.
that admits dromion solutions is the Davey–Stewartson I equation [6] which is often called
a two dimensional generalization of the nonlinear Schro¨dinger equation. In recent times
there have been extensive investigations into the mathematical properties of dromions with
suggestions for applications in hydrodynamics, plasma physics, nonlinear optics etc. In
plasma physics while a great deal of work has been done with one dimensional structures
like solitons, these novel two dimensional dromion solutions have received limited atten-
tion [7]. Our present work is motivated by some recent satellite observations [8] of wave
structures in the polar cap boundary layers where we feel dromions can help explain the
experimental observations. The high time resolution data from the polar plasma wave in-
strument (PWI) reveal interesting low frequency electrostatic structures consisting of both
693
S S Ghosh, A Sen and G S Lakhina
large amplitude monopolar and bipolar pulses. The plasma parameters in this region typi-
cally have T
i
> T
e
[9] so that ion acoustic waves are likely to be heavily damped. Electron
acoustic waves on the other hand can exist in this region and there have been some past
theoretical studies suggesting their relevance for the polar cap region [10,11]. These stud-
ies have been confined to one dimensional structures [12] and thus cannot fully account for
these latest observations. Our aim is to demonstrate the possibility of forming two dimen-
sional nonlinear structures for the electron acoustic wave in this region. For this we carry
out a systematic reductive perturbation analysis in two dimensions of the model two fluid
equations for the nonlinear evolution of the electron acoustic wave and show that it can be
reduced to the DS-I type equations under certain conditions. We use these conditions to
map out the physical parameter regimes in which dromion solutions can exist and discuss
their relevance for the observed data.
2. Derivation of the DS-I equation
We start from the usual two fluid model description of the electron acoustic wave which is
represented by the following set of equations [13]
@n
@t
+r  (nv) = 0; (1)
@v
@t
+ (v r)v + 3nrn r+  (v  b) = 0; (2)
r
2
 = n  e
 
; (3)
where n and v are the electron density and velocities respectively and  (= !
ce
=!
pe
) is
the ratio of the electron cyclotron to the electron plasma frequency.  (= T
e
=T
i
) is the
electron to ion temperature ratio, the magnetic field b = (0; 0; 1) is normalized by the
ambient magnetic fieldB
0
and  = e=T
i
is the normalized electrostatic potential. All the
space variables are normalized by the ion Debye length (
i
), time by the electron plasma
frequency (!
pe
), velocities by the electron acoustic speed (c
s
=
p
T
i
=m
e
) and the number
densities by the ambient plasma density n
0
. Note that the ion dynamics is represented by
the Boltzmann distribution of the ion density which is substituted in the Poisson’s equation.
To carry out the reductive perturbation analysis we expand all the physical quantities as
q = q
0
+
1
X
n=1

n
1
X
l= 1
q
(n)
l
exp [{l (k  r  !t)] ; (4)
where q = fn; v
xl
; v
yl
; v
zl
; g and q
0
= f1; 0; 0; 0; 0g, and k =
 
0; k
?
; k
k

is the wave
vector. We also assume the usual normalized boundary condition that v
x;y;z
! 0; n ! 1
and ! 0 as jxj; jyj; jzj ! 1. We next introduce the following stretched variables,
 = x;  =  (y  M
y
t) ;  =  (z  M
z
t) ;  = 
2
t; (5)
694 Pramana – J. Phys., Vol. 55, Nos 5 & 6, Nov. & Dec. 2000
Electron acoustic wave
where M
y
;M
z
are the respective group velocities. Transforming all independent variables
by eq. (5) we then carry out an order by order balance of terms. In the first order of , we
recover the linear dispersion relation for the electron acoustic wave
!
4
  !
2


2
+ jkj
2
(3 +K)

+ 
2
k
2
k
(3 +K) = 0; (6)
where K = 1=1 + jkj2 and ! is the frequency of the wave. In the second order we get
the expressions for the group velocities. In the third order, we get two generalized coupled
equations. In the case of pure perpendicular propagation (i.e., for k
k
= M
z
= 0) and
restricting ourselves to two dimensions (with @

! 0) these equations have the following
reduced form
{A

+ d

A

+ d

A

 
 
d
2
Q+ d
1
jAj
2

A = 0; (7)
 M
2
y
Q

+ (3 + 1)Q

+
h
6 + (jkjK)
2
i
jAj
2

= 0; (8)
where A = n(1)
1
and Q = n(2)
0
. The various coefficients, namely d
p;j
s are complicated
algebraic functions of , k
?
and  and are listed in the Appendix. Due to the symmetry of
the system, all the cross-derivative terms in eqs (7), (8) vanish. The above two eqs (7) and
(8) reduce to the DS-I type equations when the following conditions are satisfied [14],
d

=d

> 0; d
1
> 0: (9)
In the limit of the one dimensional approximation (i.e.,  ! 0), the DS-I equation further
collapses to the well known nonlinear Schro¨dinger equation. In general, the DS-I equation
admits both dromion and breather solutions which can be obtained either numerically [15]
or for some specific values of the coefficients even in an analytic fashion [16,17]. For
example, an idealized form of DS-I equation (under a specific transformation and for the
specific values of the transformed coefficients) [14] is given by
{A

+
1
2
(A

+A

) 
 
Q+ jAj
2

A = 0; (10)
Q

  Q

+ 2jAj
2

= 0; (11)
where  and  are given in the Appendix. The above equation can be solved by the Hirota
bi-linear method. A simple analytic solution can be written down as [16]
Q
0
= 2p sech
2
(
0
) sech
2
[p
0
+ tanh (
0
)  2 ln 2p] ; (12)
where 0 denotes the transformed variable and p is some arbitrary real constant. Figure 1
illustrates this solution for p = 5:5.
For our analysis we worked with the more generalized equations (7), (8) and utilized
condition (9) to determine numerically the parameter regime where they can acquire DS-I
equation.
3. Results and discussion
Our numerical results are summarized in figure 2 which displays the regions in (k; )
space where condition (9) is satisfied. The region enclosed between the dashed curves
Pramana – J. Phys., Vol. 55, Nos 5 & 6, Nov. & Dec. 2000 695
S S Ghosh, A Sen and G S Lakhina
Figure 1. Analytical solution for a dromion.
Figure 2. Existence domain of DS-I equation for different .
is for  = 0, that within the dash–dot curves is for  = 0:1 and the largest enclosed re-
gion between solid curves is for  = 1. Thus increasing electron temperature enlarges the
domain of parameter space where the evolution of the electron acoustic wave is governed
by the DS-I type equation. Correspondingly the region for the existence of dromion like
or breather type solutions also increases. We have next checked to see if these parametric
regions include the physical parameter range corresponding to the experimental observa-
tions. The typical measured parameters in the region of the satellite observations (at a
height of 5 to 7 earth radii) are T
i
= 100 eV, T
e
= 10 eV (i.e.,  = 0:1), n ' 1 cm 3
(f
pe
' 10 kHz), f
ce
' 5:5 kHz (i.e.  = 0:5) [9]. For these values figure 2 suggests the
possibility of dromion-like solutions for k  0:35 which corresponds to a wave of dura-
tion 0.15 ms (f (!)  6:69 kHz). This is consistent with the observed frequency range of
the wave structures. For a more quantitative understanding of the shape and finer features
of the structure one needs to carry out a detailed numerical investigation of the evolution
equation with appropriate initial conditions. Such an investigation is in progress and will
be reported elsewhere. We are also examining the more general situation of oblique propa-
gation (finite k
k
). Further it should also be pointed out that similar analyses can be carried
696 Pramana – J. Phys., Vol. 55, Nos 5 & 6, Nov. & Dec. 2000
Electron acoustic wave
out for other low frequency waves in this region, e.g. the lower hybrid waves, which can
yield such two dimensional structures. Dromions thus offer a rich and new paradigm for
understanding plasma wave phenomena in the ionospheric plasma and need to be investi-
gated more intensely.
Appendix A: Coefficients for the equations
a =
!
!
2
  
2
; a
!
=
2!
4!
2
  
2
;  =
1
1 + 4jkj
2
;
1
h
= 1 + a
2
k
2
?

1 +

2
!
2

(3 +K) ; f
!
= aa
!
(3 +K) ;
d

= a h
h
 
3 +K
2

  4k
2
?
n
K
3
+ a
2
h
 
3 +K
2

2
oi
;
d

=
h
ak
2
?
 
1  a
2
k
4
?
K
2

;
d
1
= a
1
k
2
?
[hf
k
  2 (3 +K)] ;
d
2
= a k
2
?
h
h
6 + (jkjK)
2
i
;
where
f
k
= f
2
!
k
2
?
!
2
+ 2
2
!a
!
+ 2f
!
k
2
?

1 +
2
2
!
2



2 (3 + ) f
n
+
 
3   K
2

+
 
6 +K   K
2

f
n
  K
2
;
f
n
=
! + a
!
k
2
?

a
2!
 

2
+ 2!
2

(3 +K) +
 
3   K
2

!   2a
!
k
2
?
(3 + )
;
 =
d

d

M
2
y
(3 + 1)
;  =
d
2
d
1
h
6 + (jkjK)
2
i
(3 + 1)
:
Acknowledgments
One of the authors, S S Ghosh, would like to thank CSIR for its financial assistance for the
work.
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Dromion solutions for an electron acoustic wave and its application to space observations

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