PHYSICS OF PLASMAS 22, 022114 (2015)
Interactions of nonlinear electron-acoustic solitary waves with vortex
electron distribution
Hilmi Demiraya)
Faculty of Arts and Sciences, Department of Mathematics, Isik University, 34980 Sile-Istanbul, Turkey
(Received 28 November 2014; accepted 25 January 2015; published online 5 February 2015)
In the present work, based on a one dimensional model, we consider the head-on-collision of
nonlinear electron-acoustic waves in a plasma composed of a cold electron fluid, hot electrons
obeying a trapped/vortex-like distribution, and stationary ions. The analysis is based on the use of
extended Poincare, Lighthill-Kuo method [C. H. Su and R. M. Mirie, J. Fluid Mech. 98, 509
(1980); R. M. Mirie and C. H. Su, J. Fluid Mech. 115, 475 (1982)]. It is shown that, for the first
order approximation, the waves propagating in opposite directions are characterized by modified
Korteweg-de Vries equations. In contrary to the results of previous investigations on this subject,
we showed that the phase shifts are functions of both amplitudes of the colliding waves. The
numerical results indicate that the waves with larger amplitude experience smaller phase shifts.
Such a result seems to be plausible from physical considerations. VC 2015 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4907790]
I. INTRODUCTION modified Poincare, Lighthill-Kuo (PLK) method, wherein
the contribution of higher order terms is also investigated.
The concept of electro-acoustic mode had been
It is well-known that one of the striking properties of
conceived by Fried and Gould1 during the numerical
solitons is their asymptotic preservation of their form when
solutions of the linear electrostatic dispersion equation in an
they undergo a collision, as first remarked by Zabusky and
unmagnetized, homogeneous plasma. Besides the well-
Kruskal.14 The unique effect due to collision is their phase
known Langmuir and ion-acoustic waves, they noticed the
shift. In a one-dimensional system, there are two distinct
existence of a heavily damped acoustic-like solution of
soliton interactions. One is the overtaking collision and the
dispersion equation. It was later shown that in the presence
other is the head-on collision.15,16 Because of the multisoli-
of two distinct groups (cold and hot) of electrons and
ton solutions of the Korteweg-de Vries(KdV) equation, the
immobile ions, one indeed obtains a weakly damped
2 overtaking collision of solitary waves can be studied byelectron-acoustic mode (Watanabe and Taniuti ), the proper-
inverse scattering transformation method17 and Zou and
ties of which significantly differ from those of the Langmuir
Su.18 For the numerical analysis of overtaking collisions of
waves.
solitary waves, it is worth of mentioning the works by Li and
To study the properties of electron-acoustic solitary
Sattinger19 and Haragus et al.20 However, for a head-on
wave structure, Dubouloz et al.3 considered a one-
collision between two solitary waves, we must examine the
dimensional, unmagnetized collisionless plasma consisting
solitary waves propagating in opposite directions, and hence,
of cold electrons, Maxwellian hot electrons and stationary
we need to employ a suitable asymptotic expansion to solve
ions. However, in practice, the hot electrons may not follow
the original conducting fluid equations. Using the extended
a Maxwellian distribution, due to the formation of phase
PLK method, Su and Mirie15 and Mirie and Su16 studied the
space holes caused by the trapping of hot electrons in a wave
head-on collision of solitary waves in a shallow water theory.
potential. Accordingly, in most space plasma the hot elec-
Huang and Velarde21 studied the head-on collision of two
trons follow the trapped/vortex-like distribution (Schamel,4,5
6 concentric cylindrical ion-acoustical solitary waves andAbbasi et al. ). Therefore, in the present work, we shall
obtained the phase shifts of right going and left going waves.
consider a plasma model consisting of a cold electron fluid,
In this context, it is worth of mentioning the works by
hot electrons obeying a non-isothermal (trapped/vortex-like)
Tantawy et al.22 for nonplanar collision in ultracold neutral
distribution, and stationary ions.
plasma, Xue23 on head-on collision of blood solitary waves
The propagation of small-but-finite amplitude waves in
and by Demiray24 on head on collision on solitary waves in
a one-dimensional ion-acoustic model had been studied by
fluid filled elastic tubes.
several researchers (see, for instance, Washimi and Taniuti7)
4,5 In the present work, based on a one dimensional model,and one dimensional electron-acoustic model by Schamel,
8 9 et al. 10 we consider the head-on-collision of nonlinear electron-Mamun and Shukla, El-Shawy, Devanandhan ,
et al. 11 acoustic waves in a plasma composed of a cold electronEl-Wakil , by use of the classical reductive perturba-
12 13 fluid, hot electrons obeying a trapped/vortex-like distribu-tion method (Taniuti ) and Demiray by use of the
tion, and stationary ions. The analysis is based on the use of
extended PLK method.15,16 It is shown that, for the first order
approximation, the waves propagating in opposite directions
a)E-mail: demiray@isikun.edu.tr. Fax: 90 216-712 1474. are characterized by modified Korteweg-de Vries equations.
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022114-2 Hilmi Demiray Phys. Plasmas 22, 022114 (2015)
In contrary to the results of previous investigations on this These equations will be used as we study the head-on colli-
subject, we showed that the phase shifts are functions of both sion of solitary waves.
amplitudes of the colliding waves. The result of numerical
calculations shows that the wave with larger amplitude expe-
III. EXTENDED PLK FORMALISM
rience smaller phase shifts. Such a result is consistent with
physical intuitions. We shall assume that in such a plasma two solitons A
and B, which are asymptotically far apart at the initial state
II. WAVE INTERACTIONS and travels toward each other. After some time has
elapsed, they collide with each other and then depart. In
In this work, we shall examine the head-on collision of
order to analyze the effects of collision, we shall employ
two solitary waves propagating in electron-acoustical plasma
the extended PLK perturbation method.15,16 This method is
with vortex electron distribution. The dynamics of electron-
the combination of the classical reductive perturbation
acoustical waves is governed by the following equations:
method with the technique of strained coordinates.
@nc @ According to this method, we introduce the followingþ ðucncÞ ¼ 0; (1)
@t @x stretched coordinates:
@uc þ @uc  @/ ¼ 1=2uc a 0; (2) ðx  tÞ ¼ n þ P0ðn; g; sÞ þ :::;
@t @x @x  1=22 ðx þ tÞ ¼ g þ Q@ / 1 1 0ðn; g; sÞ þ :::; (10)¼ nc þ nh  1 þ ; (3)
@x2 a a 3=2t ¼ s;
where nc is the normalized cold electron number density, nh where  is a small parameter measuring the weakness of
is the normalized hot electron number density, uc is the cold dispersion, P0 and Q0 are some unknown functions to be
electron fluid velocity, / is the electrostatic potential, and determined from the solution. Then, the following operator
the coefficient a is defined by a ¼ nh0=nc0, where nc0 and nh0 can be introduced:
are the equilibrium values of the cold and hot electron num- 
  	 
ber densities, respectively. The hot electron number density @ ¼ 1=2 @ þ @  @P0 þ @P0 @ 
n 4h (for b < 0) can be expressed by (Schamel ) @x n @g @n @g @nffiffi2ffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi
   
@Q0 @Q0 @
n ¼ Ið/Þ þ p W b/ ; (4) þ þ þ ::: ;h  Dpb @
n @g @g 	  
@
where ¼ 1=2  @ þ @ þ @ þ @P0 @P0 @
pffiffi  @t  n g  @s @n @g @nIðxÞ ¼ ½1  erf ð ðxÞ expðxÞ;x (5) þ @Q0  @Q0 @ þ ::: : (11)
WDðxÞ ¼ expð 2Þ ð 2Þ @n @g @gx exp y dy;
0
We shall assume the field quantities can be expanded into
where erf(x) is the error function. For /  1, Eq. (4) gives power series in  as
4 /2 n ¼ 2½nð1Þ þ nð2Þ þ :::;
nh ¼ 1 þ /  pffiffiffi ð1  bÞ/3=2 þ
3 p  2 u ¼ 2½uð1Þ þ uð2Þc c c þ :::; (12)8 3 pffiffiffi 1  b2 //5=2 þ þ :::: (6)
15 p 6 / ¼ 2½/ð1Þ þ /ð2Þ þ ::: :
Denoting the fluctuation of the cold electron number density Introducing the expansions (11) and (12) into Eqs. (7)–(9)
from its equilibrium value by n, i.e., nc¼ 1þ n, Eqs. (1)–(3) and setting the coefficients of like powers of  equal to
can be written as zero, the following sets of differential equations are
obtained:
@n þ @uc þ @ ðucnÞ ¼ 0; (7)
@t @x @x Oð
2Þ equations:
@uc þ @uc  @/ ¼  @n
ð1Þ @nð1Þ @uð1Þ ð1Þ
uc a 0; (8) þ þ c þ
@uc ¼ 0;
@t @x @x @n @g @n @g
2 ffiffiffi 2 ð1Þ ð1Þ ð1Þ ð1Þ@ / ¼ 1 þ  p4/ ð  bÞ/3=2 þ /  @uc þ @uc  @/a  @/a ¼ 0; nð1Þ ¼ a/ð1Þn 1 :
@x2 a  3 p 2
@n @g @n @g
 8pffiffiffi  3 (13)1  b2 /5=2 þ / þ :::: (9)
15 p 6 Oð3Þ equations:
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022114-3 Hilmi Demiray Phys. Plasmas 22, 022114 (2015)
@nð2Þ @nð2Þ @uð2Þ @uð2Þ @nð1Þ
 
@P @P @nð1Þ þ þ c þ c þ þ 0  0
@n @g @n @g @s  @n @g @n 
þ @Q @Q
ð1Þ
0  0 @n  @P @P @u
ð1Þ @Q @Q @uð1Þ0 þ 0 c  0 þ 0 c ¼ 0;
@n @g @g @n @g @n @n @g @g
 @u
ð2Þ ð2Þ
c þ @uc  @/
ð2Þ
 @/
ð2Þ @uð1Þ @P @P @uð1Þ
a a þ c þ 0  0 c (14)
@n @g  @n @g @s @n @g @n
@Q @Q @uð Þ
   
1
þ 0  0 c þ @P @P @/
ð1Þ @Q ð1Þ
a 0 þ 0 þ a 0 þ @Q0 @/ ¼ 0;
@n @g @g @n @g @n  @n @g @g
@2/ð1Þ @2/ð1Þ @2/ð1Þ nð2Þ 4ð1 ffiffiffibÞ 3=2þ þ ¼ þ /ð2Þ  p /ð1Þ2 :
@n2 @g2 @n@g a 3 p
A. Solution of the field equations
From the solution of the set (13), we have
uð1Þc ¼ a½Vðg; sÞ  Uðn; sÞ;
nð1Þ ¼ a½Uðn; sÞ þ V½g; sÞ; (15)
/ð1Þ ¼ Uðn; sÞ þ Vðg; sÞ;
where Uðn; sÞ and Vðg; sÞ are two unknown functions of their arguments whose governing equations will be obtained later.
Introducing the solution (15) into (14), we have
 @n
ð2Þ @nð2Þ @uð2Þþ þ c þ @u
ð2Þ
c  @U @V @P @U @Q @Va  a þ a 02  2a 0 ¼ 0;
@n @g @n @g @s @s @g @n @n @g
ð2Þ ð2Þ
 @uc þ @uc  @u2  @u @Va a 2 þ a  @U þ @P0 @U þ @Q @Va 2a 2a 0 ¼ 0; (16)
@n @g @n @g @s @s @g @n @n @g
@2U @2V 4a 3=2
nð2Þ ¼ a þ a  au2 þ pffiffiffi ð1  bÞ½U þ V :
@n2 @g2 3 p
Adding and subtracting the first and second equations in Eq. (16) side by side and utilizing the third equation in Eq. (16), we
obtain
  " # " #
@ ð2Þ   @U ð1  bÞ @U 1 @
3U @ @2V 4 @U
2 uc au
3=2
2 2a þ pffiffiffi U1=2 þ þ a þ pffiffiffi ð1  bÞðU þ VÞ þ 4P@g @s p @n 2 @n3 @g @g2 03 p @n
(17)
p2affiffiffi ð1  bÞ V @U ¼ 0;
p U1=2  þ ðU"þ
1=2
VÞ @n # " #
@ 3 2ð Þ @V ð1  bÞ @V 1 @ V @ @ U 4ð1  bÞ 3=2 @V
2 u 2 1=2
@n c
þ au2  2a  pffiffiffi V  þ a   pffiffiffi ðU þ VÞ  4Q@s p @g 2 @g3 @n 0@n2 3 p @g
þ 2aðp1 ffiffiffi bÞ U @V ¼ 0; (18)
p 1=2 þ ð þ Þ1=2V U V @g
where we have utilized the following identities:
ð þ Þ1=2U V ¼ U1=2 þ V ¼ V1=2 þ U : (19)
U1=2 þ ð þ Þ1=2U V V1=2 þ ðU þ Þ1=2V
Integrating Eq. (17) w"ith respect to g and (18) with res#pect to" n, we obtain  #
ð2Þ  ¼ @U ð1 ffiffiffibÞ 3 22 u au 2ag þ p U1=2 @U 1@ Uc 2 þ  @ V 4a þ pffiffiffi ð  bÞð þ Þ3=2  ð Þ@U1 U V 4aM n;g; s  4aU ðn; sÞ;@s p @n 2 @n3 1@g2 3 p @n
" # " # (20)
@V ð1ffiffiffibÞ @V 1@3 2ð ð2Þ þ Vau Þ¼ an  p 1=2  þ @ Uþ4ð1a pffiffiffibÞð þ Þ3=2 þ @V2 uc 2 2 V U V 4aNðn;g;sÞ þ4aV g;s ; (21)@s p @g 2 @g3 @n2 1ð Þ3 p @g
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022114-4 Hilmi Demiray Phys. Plasmas 22, 022114 (2015)
where U1ðn; sÞ; V1ðg; sÞ are two new unknown functions
4 225p
2 1=2CA CA
whose governing equations will be obtained from higher U ¼ UA sech fA; UA ¼ ; fA ¼ pffiffiffi ðn CAsÞ;2
64ð1  bÞ 2 2
order expansions and Mðn; g;"sÞ and Nðn; g; sÞ are de#fined by (28)
ð Þ ¼  ð1 ffibffiffi Þ ðg Vðg0Þdg0 1=2M n; g; s P p ; 4 225pC20 B CB
2 p "U1=2 þ ½U þ Vðg0Þ1=2# V ¼ UB sech fB; UB ¼ ; fB ¼ pffiffiffi ðgþ CBsÞ;ð0 264ð1  bÞ 2 2n ð 0Þ 0
Nð ð1  bÞ U n dn (29)n; g; sÞ ¼ Q0  pffiffiffi : (22)
2 p 0 V1=2 þ ½V þ Uðn0Þ1=2 where UAðUBÞ and CAðCBÞ are the amplitude and the
! 1 solitonic velocity of the right-going(left-going)waves,As is seen from Eqs. (20) and (21), as n and g 6 the
respectively.
first terms cause to secularity. In order to avoid the secular-
As stated before, the distribution of the number density
ities in the solution, the coefficients of n and g must vanish,
nh is valid for b < 0. Then, Eqs. (30) show that, for fixed
i.e.,
solitonic velocities, the wave amplitudes UA and UB decrease
@U ð1 ffiffiffibÞ @U 1 @3U with increasing jbj. This result is qualitatively in agreementþ p U1=2 þ ¼ 0; (23)
3 with the result found in Ref. 9.@s p @n 2 @n
Hence, up to Oð2Þ, the trajectories of two solitary
@V ð1 ffiffiffibÞ = @V 1 @3V waves for weak head-on collision are p V1 2  ¼ 0: (24)
@s p @g 2 @g3
1=2ðx  tÞ ¼ n þ P ðn; g; sÞ þ Oð20 Þ;
(30)
These evolution equations are known as the modified 1=2ðx þ tÞ ¼ g þ Q0ðn; g; sÞ þ Oð2Þ:
KdV equations. Moreover, the terms Mðn; g; sÞ@U=@n and
Nðn; g; sÞ@V=@g in Eqs. (20) and (21) may not cause to secu- To obtain the phase shifts after head-on collision of two
larity for this order but it may cause secularity at the higher solitary waves, we shall assume that the solitary waves
order expansion. Therefore, these coefficients must also van- characterized by UA and UB are asymptotically far from
ish, which results in the follow"ing equations: # each other at the initial time (t ¼ 1), the solitary wave Uð Ais at n ¼ 0; g ¼ 1, and the solitary wave UB is at
¼ ¼ ð1pffibffiffi Þ g Vðg0Þdg0M 0; or P ; (25) g ¼ 0; n ¼ þ1, respectively. After the collision (t¼þ1),0
2 p U1=2 þ ½U þ Vðg0Þ1=20
ð " #
the solitary wave UB is far to the right of solitary wave UA,
i.e., the solitary wave UB is at n ¼ 0; g ¼ þ1, and the soli-
¼ ¼ð1pffibffiffiÞ n Uðn0Þdn0 tary wave UA is at g ¼ 0; n ¼ 1. Following Su andN 0; or Q0 : (26)0 1=2 Mirie15 and Xue,232 p 0 V1=2 þ½VþUðn Þ the corresponding phase shifts DA and DB
may be obtained as follows:
These equations make it possible to determine the trajectory
1=2 1=2
functions P0ðn; g; sÞ and Q0ðn; g; sÞ. Here, we should note DA ¼  ðx  tÞjn¼0; g¼1   ðx  tÞjn¼0; g¼1
that, as opposed to previous studies on head-on collision ¼ ½P0ðn; g; sÞjn¼0; g¼1  P0ðn; g; sÞjn¼0; g¼1; (31)
problems, the trajectory functions P0 and Q0 depend both on
1=2
Uðn; sÞ and Vðg; sÞ. This means that phase shifts will be a DB ¼  ðx þ tÞj 1=2n¼1; g¼0   ðx þ tÞjn¼1; g¼0
function of both amplitudes of the colliding waves. ¼ ½Q0ðn; g; sÞjn¼1; g¼0  Q0ðn; g; sÞjn¼1; g¼0: (32)
From th e solution of!Eqs. (17) and (18), we obtain Utilizing Eqs. (25) anðd (26), the phase shifts are obtained as¼ 1 @2U 2þ @ V þ 2ð1u pffiffiffibÞ ð þ Þ3=22 U V4 @n2 @g2 3 p ¼ ð1  bDA  pffiffiffi Þ 1 Vðg0Þdg0 ; (33)
2 p 1 U 1=2 þ ½U þ Vðg0Þ1=2þ ðn sÞ þ A AU1 ; V1!ðg; sÞ;
ð Þ a @
2U @2V 
  ¼  ð ð1  bÞ 1 Uðn0Þdn02 ¼   ð Þ  DB  pffiffiffi : (34)uc a U 1=21 n; s V1ðg; sÞ ; (27) 2 p 1=2 04  2 @g2 ! 1 ½ ð Þ@n UB þ UB þ U n
3a @2U @2V 2að1 ffiffiffibÞ Introducing the solutions (28) and (29) into (33) and (34),ð2Þ ¼ þ þ p ð þ Þ3=2n U V
2 @g2 p the phase shifts are given by4 
 @n 3
 a U1ðn; sÞ þ V1ðg; sÞ : DA ¼
15 1=2
 pffiffiffiCB IþðkÞ; (35)
2 2
¼  1p5D ffiffiffi 1=2 C IB ðA kÞ; (36)
B. Solitary waves and the phase shifts 2 2
The evolution equations (23) and (24) assume the soli- where we have made the following substitutions and
tary wave solution of the form definitions
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022114-5 Hilmi Demiray Phys. Plasmas 22, 022114 (2015)
C
tanh f BA;Bð¼ z; k ¼ ;CA1 2
Iþ
ð Þ
ð Þ ¼ h k 1  z dzk i ;1=2
0 2
ð 1 þ 1 þ k2ð1  z2Þ1
 ðð Þ ¼ h 1  z2ÞdzI k i : (37)1=2
0
k þ k2 þ ð 21  z2Þ
As might be seen from Eqs. (35) and (36), for fixed
solitonic velocities and for this order perturbation approx-
imations, the phase shifts are independent of the parame-
ter b, but it may depend on it for the higher order
expansions.
In all the previous studies on head-on collision problems
existing in the current literature (see, for instance, Refs. 15
and 23), the phase shift D of right-going wave is only a FIG. 2. The variation of phase shift parameter IðkÞ with k ¼ CB=CA A.
function the amplitude UB of the left-going wave, whereas
the phase shift DB of left-going wave is only a function of Solving these differential equations under the requirement of
the amplitude UA of the right-going wave. The results of the non-secularity of the solutions, we have obtained the evolu-
present work reveal that the phase shifts should be a function tion equations for both right and left-going waves as the
of both amplitudes of colliding waves. These couplings are modified Korteweg-deVries equations and some additional
characterized by the factors IþðkÞ and IðkÞ. The variations relations making possible to determine the unknown trajec-
of these factors are depicted in Figures 1 and 2. Figure 1 tory functions. The results of the solitary wave solution
shows that, for fixed UB, the phase shift DA is a decreasing of these evolution equations reveal that, for fixed (given)
function of the amplitude UA. This means that the wave with solitonic velocities, the wave amplitudes decrease with
larger amplitude experiences smaller phase shift. Similar ex- increasing parameter jbj. In contrary to the results of previ-
planation may be given for Figure 2. Such a result is to be ous studies on the same subject, here we showed that the
expected from the physical considerations. phase shifts depend on both amplitudes of colliding waves. It
is further shown that, for fixed (given) solitonic velocities,
IV. RESULTS AND DISCUSSIONS the phase shifts are independent of the parameter b. The nu-
In this work, employing the extended PLK method we merical results indicate that the solitary waves with larger
have studied the head-on collision of two solitary waves in a amplitude experience smaller phase shifts. Such a result is to
plasma composed of a cold electron fluid, hot electrons be expected from physical intuitions.
obeying a trapped/vortex-like distribution, and stationary
ions. By expanding the field variables and the trajectory V. CONCLUSIONS
functions into power series of the smallness parameter , we In this work, our effort is devoted to the mathematical
have obtained a set of partial differential equations involving analysis of head-on collision of two solitary waves in a
both the unknown field variables and the trajectory functions. plasma whose field equations are nonlinear of fractional
order, via use of the extended PLK method. It is shown that,
for the first order approximation in the perturbation expan-
sion, the waves propagating in opposite directions are char-
acterized by the modified Korteweg-de Vries equations. In
contrary to the results of previous studies on head-on colli-
sions of solitary waves, here we showed that the phase shifts
are functions of both amplitudes of colliding waves.
Furthermore, for fixed (given) solitonic velocities, our results
reveal that the phase shifts are independent of the parameter
b, but the wave amplitudes decrease with increasing jbj. The
numerical results indicate that the solitary waves with larger
amplitude experience smaller phase shifts.
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