J. Plasma Fusion Res. SERIES, Vol.2 (1999) 363-367

Structure of Multi-Dimensional Soliton and Generationof Gaviton in the Nonlinear Beam-Plasma System
MORI Ichiro, MORIMOTO Toshifumil, KAWAKAMI Retsuo ANd TOMINAGA KiKUO
Faculty of Engineering The IJniversity of Tokushima, Minami-josanjima 770 Tokushima JapanlTakuma Radio Technical College, Kohda, Takuma, 761-Il, Kagawa, Japan(Received: 8 December 1998 / Accepted: 24 February 1999)

Abstract
Results of experimental analysis with respect to the mechanism for generation of soliton, and alsothat of the cavitons are discussed. The appearance of the solitons coincide actually with the ion waves (orthe cavitons). A new theory is obtained by using renormalization technique with Gaussian type Green'sfunction. Our theory shows that the beam electron can not emit the elementary excitation (soliton) byitself, but can emit by coherent interaction between electrons through intermediation of the elementaryexcitation (phonon), under the presence of ions.

Keywords:
soliton, caviton, beam-plasma

Figure I shows the spatial plasma density, n", and1. lntroduction
One dimensional envelope solitons appeared in thealso the magnetic field density distribution. The plasma

field of applied or plasma physics, which are rather

density was measured by probe and calibrated initiallyby micro-wave experiment with small horn at center
position. The scale of 5 on vertical axis corresponds tothe density of 1.5 x lOrsm-3. At the gas pressure

idealized one, has been almost analyzed and completed.

In the actual experimental plasma, however, soliton ismulti-dimensional (three dimensional) and restricted byboundary such as density gradient or magnetic fieldof =2.1 x l0-aTorr, the plasma density increases

gradient. In experimental stand point, these nonhomogenity play a role of driving force to make thesoliton. In this paper, we discuss the ground forsuddenly by ionization of large amplitude solitary wave,

we call this stage as second stage. The solitary wave
appears as burst with time width 150-500nsec having anenvelope ofhyperbolic secant curve at low gas pressure,
whose carrier frequency is = 400MHz. Phase of thecarrier is confirmed to be continuous with 2GHz
oscilloscope. The solitary waves are constantly
generated as an intermittent burst in suspended time ofgeneration of soliton in the mirror magnetic field.

2. Experiments
2.1.Plasma density
We used the field. S0Gauss in center and mirror
ratio of 1.4. A stainless vessel for plasma region is 16cmin diameter and 42cm in length. The region is initiallyevacuated to 10-6Torr and is fed with argon gas from
10-a to 7 x lO-aTorr. A beam of 8mm in diameter,

every 4=8psec. It is also found that the density cavities
are formed by pressure of the successive bursts and byretarded responce of ions, as shown on the curve of gas
pressure 2.2 x l0-4 at the position 20-25cm measured

1.9keV-lSmA, is injected into plasma. Any plasma

from beam entrance.

density of 1014-5 x l0rsm-3 in conter is possible.
Corresponding author's e-mail: mori@ ee.tokushima-u.ac.jp363@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchMoi I. et al., Structure of Multi-Dimensional Soliton and Generation of Caviton in the Nonlinear Beam-Plasma System
K=l&n,o6(U5(taidlco'6co)'6c(l)
Eo
Eo(d1JEo).9oEo)G

Distance from beam entrance (cm)
Wave number K,, (radlm)

Fig. 1 Spatial distributions of the plasma density and themagnetic field density are presented. The scale of
5 on veftical axis corresponds to the density of 1.5x 10r5m-3. Pressure balance: leE2l2l + (ttH2lZl + p =const. is shown. where the p, E are the plasmapressure, electric field by soliton, respectively.

Fig. 2 Examples of dispersion relation for hybrid wave

at several point are shown. The burst-wavegeneration by non-convective i nteraction between
the wave branch (a) and the beam's space charge
wave (SSCW) can be expected.

2.2. Dispersion relataon
From the plasma density and the magnetic fieldThus we obtained a twelveth order algebraic equation
for rrr, and solved al for a known set of (ft11, /c1). We
assumed that the hybrid wave propagates in such a waythat energy, ha, and momentum, h k, of the elementarydensity mentioned above, one can obtain dispersion
relation of hybrid wave at each point, then an original
dispersion relation [] is asymptotically expanded andget finally the following equation:

excitation were conserved everywhere, then therequirements ,) = const., kl1 + k21 = k2 = const. wereused.

Figure 2 represents the dispersion at differentk?,+*r-+l*rt. ;1

otl"k?rkt rr,

za'a?",lf2

Ja!"ki,v'?.rxz

a4

position, where the two wave groups, upper hybrid wave
and lower hybrid wave are seen in the same sign: (a),(b), (c). In the figure, beam waves such as slow spacecharge wave (SSCW), fast space charge wave (FSCW),
slow cyclotron wave (SCW), fast cycloton wave (FCW)
are presented.

2.3. Non-convective instability and group*'rn'rv'rr1ro2 + a2"r) a2oo

velocity dispersion(a2 - a?q) a?q

In beam-plasma system, wave generation by non-, o3"k1 1 otl"k\vl1

convective interaction between the upper hybrid wave
and the beam's slow space charge wave (SSCW) at theintersecting point of both dispersion curves, can be- al") al,

point between the curve (a) and the SSCW is given by' a, - al"- t @, _ arA ai".1+2

k2uk2rv'.r,, (3a'surely expected. In fig.2, for example, the intersecting{krr =103.3 (rad/m), a=2.505 x lOe (radlsec)), then thegenerating backward wave at 36cm point can propagate
to 23cm point by decreasing ft1 and increasing ft11 by therestrictions o= 2.505 x 10e = 400MHz = constant and(af - at)'
atv2rtto2r"k^, vT, to?

,lT-2

lrLl -0.

lkl = l2ott = constant.
Figure 3 shows dispersion relation at 36cm point.

(a2 -4rtJr-)
4a2"") aft"364

Moi l. et al., Structure of Multi-Dimensional Soliton and Generation of Caviton in the Nonlinear Beam-Plasma SystemVg = do/dkr
1xt o6llx107lqo(ErgE-oJ(t.eooE(Uo:Oloc)oEec

Distance from beam entrance (cm)

Wave number K (rad/m)

Fig. 4 Calculated group velocities from the data of fig.1

Fig.3 Group velocity dispersion at 36cm point ispresented. The region of the negative groupvelocity shows abnormal dispersion and theburst-wave generation is expected as in theoptical solitons in fibers. We assumed that themomentum of an elementary excitation, D& is

at which main field current, l-, is 3A, arepresented as the function of position. The
generated wave at 36cm point propagate
oppositely against the beam, and their groupvelocity lessen near the position of cavities.
Possible mechanism of energy accumulation of
the soliton may thus be occured by this slowing

conserved at any position.

down.

3. The Nonlinear Theory
3.1. Soliton generation
Weexamine to describe an emissinof a solitary
wave with the electron beam by using theThe horizontal axis is lftl and not a ft11, because the ft11 isslowly increasing function with ar and decided only byresonance condition of the beam: a - k1. v,1 - ImIo = 0,
where 11 is the beam velocity and the Xo is a resonancebroadening. The al has maximum at lftl = 440 (rad/m).
For the large ltcl, group velocity,0roldlkl, changes itsdirection. Backward group velocity defined by vr- dalrenormalization technique which was developed byAl'tshul, Karpmann [2], Dupree [3], Weinstock [4],Kono, Ichikawa [5]. For the soliton, one must put

d/cn, is also presented in the figure and some shift of theemphasis on coherent interaction of same wave number,zero-group-velocity point compared with the former
owing to k1, is seen. The fact that the large wave lengthi.e., /c1 - ft, while the theory of Weinstock dealed withthe case ltrl >>lftl and if we use the case /cq = /c in thediscussion of Kono, Ichikawa, their diffusion coefficientdegenerate to that of the quasi-linear theory. Then weused a Gausian-type Green's function that was proposed
initially by Horton [6] and define a diffusion tensor D,;

is related to the large v' is quite similar to thepropagation of optical soliton in a fiber, therefore, thelarge wave length region corresponds to negative groupvelocity dispersion and medium has abnormal
dispersion. At the 36cm point, plasma frequency isinstead of scalar function D. then the Green function issmaller than the cyclotron frequency, ape 1 ace, while at
23cm point, opposite relation is held, a)ps) ace.It seems

described as:that the higher density region is suitable to normalGo(k, v, t' k', v', t')= - i

dispersion.

Figure 4 represents the group velocities at eachpoint. The parameter 1m = 3.0A-3.3A is a main
magnetic field current. The group velocity increases
at =36cm point and decreases at =23cm point. This
result fits the fact that any waves can not enter the highdensity region except the digging caviton by solitonpressure. Thus we can justify the many experimental
results including soliton.

\#r, k j D ij (k,rr r)k, (v - v'),k, (v -v'),".0[-4kikjDijk,v)t- io.r * r'>,]u4o365_k,k,tD,.,(k,v) jt

12Mori l. et al., Structure of Multi-Dimensional Soliton and Generation of Caviton in the Nonlinear Beam-Plasma Systemr

Fourier transformation of the above Geen's

f ,{k.v. a7= J dv'G"(k.v. (D: k', v', c'/1

function becomes:' f'2' (*', v', af ) ,G (k,v, a;k',v', of )k,k,_?
=-2i {47T ( -; ). o,, (k,v)lk'(-b)',
a ,t!-t,i2'i(-al+c)'|

n=t mtz"(k,v, a\=1ff;,41-(4)+ E(k,, a,1)

fr {G"(k- k ,,v. k'v .K-'b--t+ ,)l 64( 6,.,,t (1)r^_IIl,{i"ea'E(-k,. - ar,,.t.3t.
dv,k,(v-v'),k,(v-v')ir.JDt)(s)

The eqs. (l)-(5) construct a set of fundamental
equation to analyze the soliton emission. The soliton
comes from 3'd order nonlinear, while the diffusion
coefficient in velocity space, Du, and the self energypart,2o, have 2nd order nonlinear since they include a4k,k,.D,,(k,v)

k,k,.D,,(k,v)'". c=k'y-i:o,t2

factor: f,E (kv a)l2.Go(k -kr,v,k.v - i:.o- ar1). For
the 3'd order solution, it is sufficient to substitute the Du

where the function Kn_t@) is the modified Bessel
function and m = 0 term is very important and K_ (x)or the Io for old quantities and get new Green's
function. Thus the E (k, a),f" (k,v, a) are obtained in
3'd order, howeverb the soliton comes also fromcoherent interaction, kt = k, then the limits ft1 -+ k, andot + k . y must be taken. 'The Green's function eq. (l')
is sternly normal and regular and not vanish even if at ftI= Ktr@) = ^[tr/(2i)'"xp(-x). The terms m * oareproportional b k2^ then they vanish when weconsider the limit ft -+ 0. For the quantity kik1D11lk2, weuse D; simply as scalar since the k,k,lk2 is O(l), theniI4ttD,, lv_v'l(\I,-l.r/ier't'expl- lv-v'l+c)1. (1')
r_I t/D,,I\,

G-(m =0) =-= 0 in contrast to the linear case.
To solve the 3'd order solutions, we used only m =0-th term and assumed firstly that lE (kv a)12 = lEl2 =const. in the expressions of D4 (k, v), and 2"(k,v, a),then the E (k, t), fd (k, v, t) are solved as initial value
problems through inverse transformation. Results would

We denote that /J0) (k', v', rlo'), 2o G, v, ro) are theinitial beam distribution and the collision frequency

be presented elsewhere [7].

between electrons and the elementary excitations or self-3.2.lnteraction with ions & anomalouscollision
The quantity lE (kr a)|') . Go\k - k1, v. k . v - i>,oenergy term respectively.

D,i(k,r=rhf

d,a, k,, k,,- ro1) means that two electrons which have (a, k) and(-at, -k), interact trough exchanging the elementary

4l ;;tE&,@)r

E(k. @)=-4?

excitations (phonons) munder the presence of ions, since

the coherent interaction brings about an increase ofelectron's /c=0-component and it causes ion-wave
generation from the balance of ion's ft-0-component.

* t a,I a,'

Our theory also show that, the beam electron can notemit the soliton by itself and any retarded response of
Eo is required for emission. We obtained also a relation
between the D11 &, v), and the electron-vs.-wave
collision frequency Io (/c, r, al) as:

lo+ G"(k,v, ot; k',v', of)J Qnt'.fY',&" v', aJ) ,(3)366N{oril. et al., Structure of Multi-Dimensional Soliton and Generation of Caviton in the Nonlinear Beam-Plasma System(6)

Since band width is {\normalsize Acrl = 300KHz,ion
thewave is time delayed about A/ - l.7ps from awhere the y has a varue of 0-1. The eq. (6) is arelation A,aLt 2 half-cycle. The pointed peak shown by
an arrow represents envelope soliton and the followed
wide peaks are lMHz-component of ion waves. There4t
ReIo=y''- lv-v'l' ^,anomalous collision frequency obtained by Tsytovich inqusilinear theory [8], if we exchange our conelational
length in velocity space, ly - r'1, with r.
Figure 5 shows the time sequence between thedetected envelope of solitons and the lMHz componentsignal of ion waves (or cavitons) obtained by HP8554L-85528 spectrum-analyzer.

are also seen that the ion waves are surrounded by apedestal consisting of dispersive wave whose width isalmost same as that of the envelope soliton. Thus thewaves would be in the strong coupling state and theresults are quite similar to the theory. Our previous
paper [9] supports also the situation.

ReferencesI.
Mori,Phys.
Fluids28,3269 (1985).
lll[2] I.M. Al'tshul and V.I. Karpman, Sov. Phys. JETP

tcocoo.

elE'6I22,36r (1966).

t3l T.H. Dupree, Phys. Fluids 9, 1773 (1966).
[4] J. Weinstock, Phys. Fluids 12, 1045 (1969).
t5l M. Kono and Y.H. Ichikawa, Prog. Theor. Phys.

o(,Noto49,7s4 (1973).
t6l C.W. Horton, "Long time Prediction in Dinamics",
John Wiley p.311 (1983).
[7] I. Mori, T. Morimoto, R. Kawakami and K.
Tominaga, to be published in Nuclear Instument
and Method in Physics Research B (1999).
t8l V.N. Tsytovitch, "Nonlinear Effects in Plasmas"'6coao.Eo.Il3coTime (/s)(Trans. by M. Hamberger), p.170 New York-Fig. 5 The strong couplings between the soliton and theion wave are presented. The pointed peaks shownby the arrows represent envelope solitons andLondon; Plenum Press (1970).
[9] I. Mori and K. Ohya, Phys. Rev. Lett. 59, 1825
(1987).

wide peaks are lMHz component of ion waves.
The ion waves are surrounded by dispersive
waves whose width are almost same as that ofsolitons.

36'.1