Phase Effects in the Nonlinear Interaction of “Negative”

206
F. ENGELMANN AND H. WILHELMSSON
Phase Effects in the Nonlinear Interaction of “Negative”-Energy Waves
F. E n g e l m a n n and H. W
il h e l m s s o n
*
Laboratori Gas Ionizzati (Associazione EURATOM -CNEN ), Frascati (Rome), Italy
(Z. Naturforsdi. 24 a, 206— 207 [1969] ; received 30 September 1968)
Nonlinear interaction of “negative”-energy waves is studied assuming well-defined phases and
perfect matching conditions. The results are compared with those of the random-phase approach,
and significant differences in the dynamics due to phase effects are pointed out.
The lowest-order nonlinear interaction of three
monochromatic waves, with well-defined phases, hav­
ing the frequencies co0 , « j , 0 ) 2 where
oj0= oj1 + o) 2
,
(2 a)
3 r..,
sign |fg-—
[ o j j 2 e { a ) j)
(2 b)
si ui~ ~ s 2 u 2 - m 12 ,
du2/d r=
s0 s1 u 0 u 1 cos 0 ,
(2 c)
50
0
sin 0 .
S0 S2
dr
]
(5)
From Eqs. (2) and (3) one obtains
,
d (» o 2)
all Sj are equal, or
5X52= —1 and s0 = arbitrary;
(6 b)
S0 U02 + S1 u 2 = m 0 1 ,
(6 c)
-)-
S2 U 22 =
//Iq o
u 0 ux u 2 sin & = T
(7)
with the m’s and r integration constants, and
(8)
of the waves has negative energy (in any frame of
reference) 4-6. Qualitatively, the evolution of the
Uj2 and 0 is readily obtained from the preceding
relations (cf. Fig. 1). For U j2
oo one has
51ä2= +1 and 50 = —5 != —s2
(or, equivalently, 5„ Sl = s„ , s = - 1).
(6 a)
,
U q"
— —2 l/ ^ 52[u0" (m01 —50 u02) (m 02 —s0 u02) —7"2] sign[cos 0].
Two different cases may occur:
II)
=
s0 s2 u0 u2 cos
$1 «So
1)
2)
Sj
th amplitude, and
d u jd x =
(3)
I)
j the phase of the
with
(4)
e(toj) being the dielectric constants.
du0/dr = —Sj s2 ux u 2 cos
dr
2> = <P0
(1)
can be shown to be described by
d<P
The Uj > 0 are the moduli of the wave amplitudes,
^
^
Uj2^ (too - r) ~ 2
(9)
depending on ^
conditions Hence>
Case I has been extensively treated 1-3 with the resuit that the U j ( r ) are oscillatory and bounded.
Physically, it corresponds to an interaction of waves
of positive energy (in an appropriate frame of re­
ference) .
We are interested here in case II, for which all
duj/dr have the same sign. Hence, all amplitudes
vary in the same sense, which is only possible if one
the increase of the wave amplitudes — after a transient period, depending on |
| and during which
the uj may even decrease — is faster than exponen­
tial ' and leads to a divergence at a finite t x . At the
same time, the phase is locked 7 to 0 = ± n for sign
^ o = — 1’ irrespective of
• W e note that, as in
case I, one can solve Eqs. (2) and (3) exactly in
terms of elliptic functions 1’ 3.
* Permanent address: Institute of Physics, University of Uppsala, Sweden.
1 J. A. A r m s t r o n g , N. B l o e m b e r g e n , J. D u c u i n g , and P. S.
P e r s h a n . P h y s . Rev. 127, 1918 [1962].
2 A. S j ö l u n d and L. S t e n f l o , Appl. Phys. Lett. 10, 201
[1967] ; Physica 33, 499 [1967].
3 R. S u g i h a r a , Phys. Fluids 11, 178 [1968].
4 B. B.
K a d o m t s e v , A. B. M i k h a i l o v s k i i , and A. V. T i m o Sov. Phys. JE T P 20, 1517 [1965].
5 V. M. D i k a s o v , L. I. R u d a k o v , and D . D . R y u t o v , S o v .
Phys. JETP 21. 608 [1965].
0 M. N. R o s e n b l u t h , B. C o p p i , and R. N. S u d a n , 3 rd Conf.
on Plasma Phys. and Contr. Nucl. Fusion Res. Novosibirsk
1968. Paper CN 24/E13.
7 Cf. also P. A. S t u r r o c k , Phys. Rev. Lett. 16. 270 [1966].
feev,
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NONLINEAR INTERACTION OF NEGATIVE-ENERGY WAVES
207
Let us consider the same interaction problem in
random-phase approximation. The corresponding
equations of motion are 5’ 6>8’ 9
d K 2)
—
- 2 —2
= Ux~ U
-
—2 —2
—0 —2
s0 Sx u 0z U .S - s0 So u 0~ u x%
[1f.\
(10)
and two similar equations for d(üx2)/df and d(ü22)/df,
from which the validity of Eqs. (6) in terms of the
üj2 is easily shown. By using them, exact solutions
for the üj2(f) are readily obtained. For üj2—> oo ,
one has in case II
üj2^ \ { r o o - t)“ 1
(11)
with rx determined by the initial conditions. Hence,
the final increase of the wave amplitudes is less “ex­
plosive” 7 here than in the case of determined phase.
On the other hand, no decrease of the amplitudes,
even transitory, is possible in the random-phase
case.
In conclusion, phase effects may have consider­
able importance for the dynamics of the considered
wave interaction.
Fig. 1. Qualitative plot of a) u0z as a function of r with the
modulus |0 O|as a parameter, assuming the initial value u002
of u02 larger than either u102 or u202, and b) 0 as a function
of u02; the arrows refer to increasing r.
H.
W i l h e l m s s o n wishes to thank Prof. B. B r u n e l l i
for the kind hospitality extended to him at the Laboratori Gas Ionizzati at Frascati.
8 V. N.
9 R. E. A a m o d t and M. L.
[1967].
T s y t o v ic h , Sov.
Phys. Uspekhi 9, 805 [1967].
S lo a n ,
Phys. Rev. Lett. 19, 1227