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Asymptotic permanent profile of the ion acoustic wave driven by the Langmuir wave
Physics LenersA 168 (1992) 120-126 North-Holland
PHYSICS LETTERS A
Asymptotic permanent profile of the ion acoustic wave driven by the Langmuir wave D . J . K a u p , A. L a t i f i Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13699-5815, USA
and J. L e o n D~partment de Physique-Math~matique,
Universit~ Montpellier I1, 34095 Montpellier cdx05, France
Received 7 February 1991; revised manuscript received 13 May 1992; accepted for publication 19 June 1992 Communicated by A.P. Fordy
We study the evolution of Langmuir waves coupled to the ion acoustic wave by means of the ponderomotive force in the Karpman limit (caviton equation ). Using the spectral transform with singular dispersion relation, it is shown that the background noise (fluctuations in the ion density) is amplified and its time asymptotic behavior will be a static solution which is totally reflective for the Langmuir wave. Moreover, if the initial ion density contains a local depression, the asymptotic profile will contain a number of permanent localized density depressions (cavitons), static in the rest frame of the acoustic wave and entrained in its wake.
The interaction o f ion acoustic waves (IAWs) with electrostatic ( L a n g m u i r ) waves ( E S W s ) in the context o f a t w o - c o m p o n e n t fluid-type, homogeneous plasma, when the ion nonlinearities are weak comPared to the p o n d e r o m o t i v e force, is described by the Z a k h a r o v equation [ 1 ]. Keeping the ion nonlinearities would lead to a different m o d e l equation, see refs. [2,3]. The Z a k h a r o v system is nonintegrable but several integrable limits have been f o u n d by multiscale analysis, corresponding to different physical situations, i.e. the n o n l i n e a r Schr6dinger equation [4,5 ], the Y a j i m a - O i k a w a equation [6 ] and the K a r p m a n equation [ 1 1 ]. In this paper, we consider this last limit which is integrable by means o f the spectral transform with a singular dispersion relation [3,7] and we study the t i m e a s y m p t o t i c b e h a v i o r o f the general solution. O u r result can be s u m m a r i z e d as follows: Starting with an arbitrary initial (localized) IAW profile - whose spectrum generically consists o f a radiative part ( c o n t i n u u m ) a n d a soliton part (discrete) - we prove that 120
( i ) W h e n the ESW and the IAW are moving in the same direction then the radiative part is amplified by the p o n d e r o m o t i v e force. The L a n g m u i r wave actually builds up a barrier o f IAW which is asymptotically static and totally reflective. (ii) The soliton part asymptotically becomes a n u m b e r o f well separated static density depressions (cavitons). These cavitons will follow b e h i n d the IAW wave front (at a slower velocity) and asymptotically become static relative to the IAW. The fact that the t i m e - a s y m p t o t i c profile o f the radiative part is p e r m a n e n t and contains a sequence o f static local density depressions is a new result and it is also the first time that the spectral transform theory has led to asymptotically static solitons whatever be their initial velocities. This is in stark contrast to other similar integrable systems such as coherent pulse p r o p a g a t i o n and selfi n d u c e d transparency ( S I T ) [ 7 - 9 ]. In the attenuator case, the b e h a v i o r o f cavitons a n d coherent pulse p r o p a g a t i o n is essentially the same: the continuous spectrum decays and the solitons m o v e with differElsevier Science Publishers B.V.
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PHYSICS LETTERS A
ent velocities, related to their amplitudes. In the amplifier case, each system is uniquely different. In coherent pulse propagation, the continuous spectrum grows and eventually forms a self-similar structure known as the "~-pulse" [ 9 ]. Its width continually shrinks while all solitons are left far behind and in the tail of the n-pulse. The amplifier case for cavitons is uniquely different. Although the continuous spectrum does grow, it eventually reaches a static shape, moves as a coherent IAW wave front and has all solitons entrained in its tail at fixed locations. The integrability of the caviton equation has been shown for the first time in ref. [ 10 ] by means of the inverse scattering transform. Here we give the D-bar formulation of the problem and the basic system of our study is (for the derivation see ref. [ 10] ):
02E+(a2-q)E=O,
(1)
+oo
O,q+Ox f daE*E(a,x, t ) / z ( a ) = O .
(2)
0
E ( a , x, t) is the rf electric field, a is the (scaled) frequency, q(x, t) is the fractional change in the plasma density,/~(ot) is the profile of the source and (x, t) is the frame comoving with the IAW, which in the lab frame is moving to the right. We will solve this system (for t-,co ) for arbitrary localized initial data q(x, 0) (a background noise) and for the following asymptotic behavior of E (or, x, t),
E(a, x, t ) ~ T(a,
t ) e ~"x ,
x~
+co,
--, Id'~X+R(a, t)e -i'~x , x - - , - c o ,
To solve the system (1), (2), we consider the Schr6dinger equation
q/xx(k,x,t)+(k2-q)~u(k,x,t)=O,
k~C,
(4)
where q(x, t) is assumed to satisfy the Faddeev condition f ( 1 + Ixl ) Iq(x, 0) [
X---~ + O~ ,
~a(k, t)ei~+b(k, t)e - i ~ ,
X-~
- - ~
.
(5) a(k, t) and b(k, t) are the "scattering coefficients". Note that (1) and (4) are very similar. In fact, one can easily become confused if one is not careful and does not use an adequate notation to distinguish between the two problems. Equation ( 1 ) is a physical equation for E and only has meaning for ot > 0. (Note that per (3), for a < 0 , the incident rf field moves to the left and the asymptotic behavior is unphysical.) On the other hand, (4) is a well defined mathematical problem for any k, and we shall need to solve it for all k to obtain our solution. There is a link between these two problems. We note that any solution of ( 1 ) can be expressed as a linear sum of solutions of (4). In particular, for k = a > 0, we have [10,14]
E(a, x, t) - a(k,t~q/(k,x,t)
,
(6)
and
T(a, t)= a(k,1 t~) k=,~ '
(7a)
R(a, t ) - a(k,t)b(k't) k=. "
(7b)
(3)
R ( a , t) and T(a, t) are respectively the reflection and the transmission coefficients for the incident 15 wave. This asymptotic behavior corresponds to a normalized rf wave being incident from the left onto an IAW moving to the right. Thus in the lab frame, both waves are moving in the same direction. The amplitude of the scaled electrostatic field has been normalized to unity because an arbitrary amplitude can be scaled off by rescaling the time. Thus in the notation of ref. [10], we h a v e / t + ( a ) = 0 since no ESW waves are entering from the right and here we have replaced t h e / t _ ( a ) of ref. [ 10 ] with simply
~(a).
17 A u g u s t 1992
In this paper, we obtain the time-asymptotic solution of the system ( 1 ) and (2) by means of the integral equation associated to this system. The integral equation could be either the Marchenko equation obtained by the usual IST method [ 10 ] or the Cauchy-Green equation (see ( 15 ) ) obtained by the Dbar method. In both cases, one needs the explicit expression of the reflection coefficient. Indeed in the Marchenko equation, the reflection coefficient occurs through its Fourier transform as the kernel of the integral equation and in the Cauchy-Green 121
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equation, it appears simply by construction of the Dbar equation [3]. The usual IST method furnishes the time evolution of the scattering data and one has to construct (by nontrivial calculation) the time evolution of the reflection coefficient [ 10]. Here we use the method proposed in ref. [ 3,12 ] which allows one to obtain directly from the evolution equation (2) the singular dispersion relation and to write merely the time evolution of the reflection coefficient in its simplest form. We follow here the detailed study of ref. [ 3 ]. Defining f(k, x, t) such that ~u(k, x, t ) = f(k, x, t)e ikx, we can associate to the Schrrdinger equation (4) the following tT-problem [ 13 ],
~ f(k)=f(-k)r(k), (8)f(k)=l+O(1/k),
k~C,
17 August 1992
x, t) of (4). Due to the complex conjugate in (2) and the symmetries on a(a) ( a * ( a ) = a ( - a ) ), we may extend the original integral over c~from 0 to + in (2) to an integral over a from - ~ to + oo if we define # ( a ) for a < 0 to b e # ( - c ~ ) and take ½of the latter integral. The advantage of this is that one can then use contour techniques to evaluate this integral. Then using the Cauchy-Green formula, eq. (12) gives (for a = k ) +~
.f
dl #(l) fl(k)=-½1 J l-kad(l)"
Now inserting (13) into (10) and taking into account (9), we have -ik
OR(k,t)=R(k,t)
Ikl~.
(13)
/t(l) dl.
f 12_(k+ie) 2ad(l)
(8)
(14)
Let us consider first the case when q(x, 0) has no bound states (zeroes of a(k, 0)). Then r(k, x, t) is related to the reflection coefficient through
The potential q(x, t) is obtained by solving the basic integral equation
r(k, x, t) =R(k,
f(k, x, t) = 1 + ~
t)e2itc~(k I +0)
(9)
.
Following the method described in ref. [3], we assume a simple time dependence for r(k, x, t), O ~r(k)=
[fl(k)-fl(-k)]r(k),
(10)
-72-~J( -l, x, OR(l, t) C
X eZ~(k~ + i e ) ,
(15)
which solves the &problem (8). The solution q(x, t) of ( 1 ), (2) will then be given from f b y [ 3 ]
where//(k) is a given distribution obtained by the comparison of the evolution equation (2) with
q = - 2 i ~ x f ( ~ ) ( x , t) ,
qt -- _ 1~ ff d a ^ d a ~ ( a ) ~ ( - a )
where f¢~) is the coefficient of 1/k in the Laurent series expansion o f f ( k , x, t). We now have to solve the evolution (14) for R (k, t) which upon using principle value integrals can be written as
0
~--~fl( o , ,
(11)
C
where ~ is the complex conjugate of o~. Equation (11 ) is itself obtained by the compatibility condition 0 e ( 0 ~ ) = 0 ~ ( 0 / ~ , ) when the distribution fl(k) has no polynomial part (see ref. [31 ). Thus, we obtain in our case O#(a)
in ~ ( a )
2 ad(a~) 6(cq),
(12)
where d(o~) = a (-o~), ~ is the Dirac distribution and a~ represents the imaginary part of or. We remark here that we have now extended/~ (ct) to negative values of ct. The reason for this is, by (6), we can give E(ct, x, t) in terms of the solutions ~,(k, 122
(16)
O R(k)=R(k)(n#(k)_ikP kad(k)
dl
f l2 k2ad(l)j (17a)
01~(k)=l~(k)(n#(-k)
dl # ( l ) ] t2-k 2 (17b) where P represents the principal value and /~(k) = R ( - k ) . Assuming d ( k ) = a * ( k ) (which implies the reality of q(x, t)), we have also R(k) =R*(k). Now using I t ( - k ) =#(k), (17) gives
7d(
+ikPf
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a Ot
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17 August 1992 -600
(R/~) = 2n/t R_~
aa
(18)
"
--oo
Then using the basic relation a d - b b - = 1 [ 14 ] and eqs. (6) and (7), we eliminate ad from (18) and solve it to obtain
iR(k,t)12=
B(k) +D(k)
B(k)
exp[ - 2 p ( k ) n t ] ' (19)
where for ease of notation, we have introduced and defined
B(k) = IR(k, 0)12 O(k) = 1-B(k) .
,
(20a) (20b)
Note that if B ( k ) = 0 for some value of k, then IR (k, t) [2 will be also zero for that value of k for all t. Equation (17) can now be solved and we have
R(k, 0) exp [i0(k, t)] R(k, t) = x/B(k)+O(k) e x p [ - 2 n u ( k ) t ] ' O(k, t) = ~k P f (21b)
~
dl e2i~x 1 f~(k'xl=l-17t f l-(k+iO) f°°(- ,x)
(21a)
ln{B(k)
+D(k) exp[-2n#(k)t]} .
×exp [i0~o(/) + i arg R(/, 0) ] .
The corresponding permanent asymptotic profile of the lAW is then obtained from (16). We can evaluate the behavior at large x of the lAW profile resulting from (24). It can be shown by means of the steepest descent method that this profile is positive and exponentially localized in a given region of space. The position of this region is fixed by the argument of the input R(k, 0). A model case is given in the appendix. Let us consider now the case when the initial IAW profile contains also a density depression. Hence from (4), the spectrum of q(x, 0) will also contain a sequence of discrete eigenvalues, say k,=i(n for n = 1, ..., N. To each of these points is associated the normalization coefficients Cn and the resulting o~transform reads, instead of (9) [ 3 ],
r(k, x, t) = e 2ikx(R(k, +2n.=,~
(21b)
(24)
t)~(kl + 0 )
C.(t)~(k-k.)).
(25)
The evolution (10) then gives
O,C.(t) =
[fl(k.) - f l ( - k . )
Otk.=O and
] C.(t),
(26)
Note that i f / l ( k ) is zero for some range of k, then R(k, t) is time independent for the same range ofk. From the definition of B(k) and D(k) in (19), we immediately see that for gt> 0 we have
with fl(k) given in (13). The solution of (26) is obtained in the same way as that of R(k, t) and can be written
IR(k,t)l-,1,
C.(t) = C.(0)
t-,+~,
(22)
which was first shown in ref. [ 10 ]. Furthermore we have the remarkable property that the phase of R (k, t) also approaches a constant as t goes to infinity. Indeed, as t ~ + or:
O(k,t)--.O~(k)=-kP f ~-~lkloglR(l,O)[ _
(23)
We have used here the fact that the function IR(k) [2=R(k)R(-k) is even in k. The solution f of the integral equation (15) approaches now the solution fo~ of the following integral equation,
exp [i0. (t) ] ,
(27)
0 , ( t ) = ~ k. f ~ l dl n{B(/)
+D(l)
e x p [ - 2 / l ( / ) ~ t ] },
(28)
where now the integral is along the real axis. Note that the imaginary part of kn is always positive. We remark that when R(k, 0) is zero then one can recover the usual case where these solitons move slower than the IAW wave front. This case also occurs when the ESW moves in the opposite direction relative to the IAW, in which case p (l) in ( 28 ) would be replaced by -/1-6(/), (see ref. [10]). In either 123
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case, it is the last term in the argument of the In which would dominate at large t, giving 0°(t)=-
knf dZ 2~312_k~
[2x~(l)t-lnD(l)].
(29)
However this is not the case we are considering here. Instead, the ESW is moving in the same direction as the IAW and we take JR(k, 0) I to be nonzero for all k. In this case, instead of (29), (28) will reduce to k n f ~ l ndll R ( l , 0 N ( t ) = -x-
0)l,
(30)
where 0n is obviously independent of t. Now each and every soliton has become entrained with the IAW wave front, and will not move relative to it. The corresponding IAW profile as t ~ o e is then obtained by solving the integral equation
f~(k, x) = 1-
1
d/e2i/x
_ l_(k+i~) f ~ ( - l ' x )
× e x p [i0~ (l) + i argR (/, 0) ] - i ~ Cn(0) e x p ( i 0 T ) f ~ ( - k n , x) exp(2ik, x) n:l k~k
(31) and by using (16) to compute q~(x). For most choices of the initial data, the different coherent structures in (31 ) will be well separated and we can describe one of the static cavitons by solving (31) around the nth caviton position ( x T ) , x~= ~
1
log[C,(0)/2~,,]
1 f / 2 ~dl- ~ l ° g l R ( k ' 0 ) l "
---2re
(32)
Acknowledgement This work has been supported in part by the Office of Naval Research through Grant No. N000A-88-K0153 and by the Air Force Office of Scientific Research through Grant No. AFOSR-89-0510.
Appendix To choose a model for R(k, 0), let us say that the incident ESW sees no IAWs until x > 0. This would correspond to the ESW incident on a plasma slab ( x > 0). Thus we would have q(x, 0) = 0 if x < 0. For such a potential, R (k, 0) would be analytic in the lower half k-plane [14]. We also know that R ( k = 0) = - 1 if no bound states are present. Thus a reasonable model for this would be R(k, 0 ) - - e x p (
Vk@i7) ,
(A.l)
where v and 7 are some constants. The constant 7 would measure the width in k-space of the continuous spectrum while ~ determines how fast the spectrum drops off in k space. For large k, R(k, O) - ~ - e-", so one wants to choose ~ large. Then with this model equation (24) becomes +oc
1 t"
dl
J l_(~+ie) f ( - l ' x ' t )
(33)
All cavitons are static in the x-frame (where one is moving with the acoustic wave). Their initial positions are determined by the (real) values C , ( 0 ) and their final positions by ( 32 ) where the value of R (k, 0) enters explicitly through (30). To summarize, when the ESW waves, which are comoving with the IAW, do dominate, then we have two new and fundamental results: 124
(i) The IAW grows until it forms a barrier which asymptotically totally reflects the Langmuir wave. (ii) This barrier also entrains all solitons and pulls them along with the barrier. The entire arrangement becomes static in the lAW frame.
f(k,x,t)=l+
Then in the neighbourhood of x T we have
q~#(x) = - 2 ~ sechZ [~n(x-xT~ ) ].
17 August 1992
×exp
2i/x+ 2 ( / + i 7 )
~
.
(A.2)
The behavior ofeq. (A.2) is evaluated by the method of steepest descent. Let vl
w(l)=2ilx+ (/+i2--~-~ 2 ' and
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vl
17 August 1992
~,(l) = 2i/x- - 2(/-i7) "
0 -4il±tl+ exp[w(/±)] q(x't)=-~x ~ l+~/±exp[w(/±)]
The stationary points are given then by
where
x<0,
Ow(l)=O=,l+-=-i,+_
~
,
(A.3)
and x>0,
O ff(l)=0~T±=iy+i
v/~.
(A.4)
q±= Ro(l ±) k / 2~l +
(A.6) ,
2x
(A.7)
w"(l ±) •
The approximation in (A.5) is valid as long as the second term of the expansion is smaller than the first one. Here, vYlxl >> 1 .
For x < 0, the path of the integration can follow the path of the steepest descent (i.e. I m [ w ( / ) ] = Im[w(/±)]). In this case, '
f(k)-,1 - ~
x
For x > 0, the integral (A.2) can be evaluated by the method of stationary phase, following the path of integration (i.e. Re[~(l) ] = R e [ ~ ( l - ) ] ). In this case,
f(k)~l+ if(-T-)exp (
-_
v[- ) 2(T- +iy)
rt T - - k
w.(u_)exp[w(l+-)l.
(A.5)
,'
,
x
~exp[~(T-)l.
(A.9)
And using (16), we obtain for
And using (16), we obtain for q(x, t)
30
(A.8)
"' ,
I
I
q(x, t)
I
25
20
15
i0
5
0
..........
-5
-I0 -BOO
i
-600
-400
I
-200
I
I
I
200
400
600
g00
Fig. 1. q(u, t=oo) with u=xvywherev=500 and y=0.01. 125
Volume 168, number 2
0 -4i/-f/exp[#(r-) q(x,t)=Ox l + f / e x p [ # ( T - ) ]
PHYSICS LETTERS A
References
] '
(A.10)
where (4x)-3/4 f/= 2n ( 7 _ xf~-y/4x) X exp ( _
2 2~'x-',/~]
47x_x/~).
(A. 11 )
Here also, the a p p r o x i m a t i o n is valid o n l y if vTx>> u 2 .
(A. 12)
I n d e e d , if x < #/7, t h e n ]'- m o v e s i n t o the l.h.p, a n d it w o u l d n o m o r e be possible to ignore the contrib u t i o n of the o t h e r t e r m s in ( A . 2 ) . I n fig. 1, we plot q(x, t) (see ( A . 6 ) a n d ( A . 1 0 ) ) with the n e w v a r i a b l e u = xuy. A c c o r d i n g to the constraints ( A . 8 ) a n d ( A . 1 2 ) , o u r s o l u t i o n is n o t valid in the region a r o u n d u = 0 , a n d m a y well be m o d e l dependent.
126
17 August 1992
[ 1] V.E. Zakharov, Zh. Eksp. Teor. Fiz. 62 (1972) 1742 [Sov. Phys. JETP 35 (1972) 908]. [2] K. Nishikawa, H. Hojo, K. Mima and H. lkezi, Phys. Rev. Lett. 33 (1974) 148. [ 3 ] J. Leon and A. Latifi, J. Phys. A 23 (1990) 1385. [4] M.V. Goldman, Rev. Mod. Phys. 56 (1984) 709. [ 5 ] A. Latifi and J. Leon, Phys. Lett. A 152 ( 1991 ) 171. [6] N. Yajima and M. Oikawa, Prog. Theor. Phys. 56 (1976) 1719. [ 7 ] M.J. Ablowitz, D.J. Kaup and A.C. Newell, J. Math. Phys. 15 (1974) 1852. [8] D.J. Kaup, Phys. Rev. A 16 (1977) 704. [9] S.V. Manakov, Zh. Eksp. Teor. Fiz. 83 (1982) 68 [Soy. Phys. JETP 56 (1982) 37]. [10] D.J. Kaup, Phys. Rev. Lett. 59 (1987) 2063; Phys. Fluids 31 (1988) 1465. [ 11 ] V.I. Karpman, Phys. Scr. 11 (1975) 263. [ 12] J. Leon, Phys. Lett. A 144 (1990) 444. [ 13 ] R. Beals and R.R. Coifman, Commun. Pure Appl. Math. 37 (1984) 39. [ 14] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Stud. Appl. Math. 53 (1974) 249.
PHYSICS LETTERS A
Asymptotic permanent profile of the ion acoustic wave driven by the Langmuir wave D . J . K a u p , A. L a t i f i Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13699-5815, USA
and J. L e o n D~partment de Physique-Math~matique,
Universit~ Montpellier I1, 34095 Montpellier cdx05, France
Received 7 February 1991; revised manuscript received 13 May 1992; accepted for publication 19 June 1992 Communicated by A.P. Fordy
We study the evolution of Langmuir waves coupled to the ion acoustic wave by means of the ponderomotive force in the Karpman limit (caviton equation ). Using the spectral transform with singular dispersion relation, it is shown that the background noise (fluctuations in the ion density) is amplified and its time asymptotic behavior will be a static solution which is totally reflective for the Langmuir wave. Moreover, if the initial ion density contains a local depression, the asymptotic profile will contain a number of permanent localized density depressions (cavitons), static in the rest frame of the acoustic wave and entrained in its wake.
The interaction o f ion acoustic waves (IAWs) with electrostatic ( L a n g m u i r ) waves ( E S W s ) in the context o f a t w o - c o m p o n e n t fluid-type, homogeneous plasma, when the ion nonlinearities are weak comPared to the p o n d e r o m o t i v e force, is described by the Z a k h a r o v equation [ 1 ]. Keeping the ion nonlinearities would lead to a different m o d e l equation, see refs. [2,3]. The Z a k h a r o v system is nonintegrable but several integrable limits have been f o u n d by multiscale analysis, corresponding to different physical situations, i.e. the n o n l i n e a r Schr6dinger equation [4,5 ], the Y a j i m a - O i k a w a equation [6 ] and the K a r p m a n equation [ 1 1 ]. In this paper, we consider this last limit which is integrable by means o f the spectral transform with a singular dispersion relation [3,7] and we study the t i m e a s y m p t o t i c b e h a v i o r o f the general solution. O u r result can be s u m m a r i z e d as follows: Starting with an arbitrary initial (localized) IAW profile - whose spectrum generically consists o f a radiative part ( c o n t i n u u m ) a n d a soliton part (discrete) - we prove that 120
( i ) W h e n the ESW and the IAW are moving in the same direction then the radiative part is amplified by the p o n d e r o m o t i v e force. The L a n g m u i r wave actually builds up a barrier o f IAW which is asymptotically static and totally reflective. (ii) The soliton part asymptotically becomes a n u m b e r o f well separated static density depressions (cavitons). These cavitons will follow b e h i n d the IAW wave front (at a slower velocity) and asymptotically become static relative to the IAW. The fact that the t i m e - a s y m p t o t i c profile o f the radiative part is p e r m a n e n t and contains a sequence o f static local density depressions is a new result and it is also the first time that the spectral transform theory has led to asymptotically static solitons whatever be their initial velocities. This is in stark contrast to other similar integrable systems such as coherent pulse p r o p a g a t i o n and selfi n d u c e d transparency ( S I T ) [ 7 - 9 ]. In the attenuator case, the b e h a v i o r o f cavitons a n d coherent pulse p r o p a g a t i o n is essentially the same: the continuous spectrum decays and the solitons m o v e with differElsevier Science Publishers B.V.
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ent velocities, related to their amplitudes. In the amplifier case, each system is uniquely different. In coherent pulse propagation, the continuous spectrum grows and eventually forms a self-similar structure known as the "~-pulse" [ 9 ]. Its width continually shrinks while all solitons are left far behind and in the tail of the n-pulse. The amplifier case for cavitons is uniquely different. Although the continuous spectrum does grow, it eventually reaches a static shape, moves as a coherent IAW wave front and has all solitons entrained in its tail at fixed locations. The integrability of the caviton equation has been shown for the first time in ref. [ 10 ] by means of the inverse scattering transform. Here we give the D-bar formulation of the problem and the basic system of our study is (for the derivation see ref. [ 10] ):
02E+(a2-q)E=O,
(1)
+oo
O,q+Ox f daE*E(a,x, t ) / z ( a ) = O .
(2)
0
E ( a , x, t) is the rf electric field, a is the (scaled) frequency, q(x, t) is the fractional change in the plasma density,/~(ot) is the profile of the source and (x, t) is the frame comoving with the IAW, which in the lab frame is moving to the right. We will solve this system (for t-,co ) for arbitrary localized initial data q(x, 0) (a background noise) and for the following asymptotic behavior of E (or, x, t),
E(a, x, t ) ~ T(a,
t ) e ~"x ,
x~
+co,
--, Id'~X+R(a, t)e -i'~x , x - - , - c o ,
To solve the system (1), (2), we consider the Schr6dinger equation
q/xx(k,x,t)+(k2-q)~u(k,x,t)=O,
k~C,
(4)
where q(x, t) is assumed to satisfy the Faddeev condition f ( 1 + Ixl ) Iq(x, 0) [
X---~ + O~ ,
~a(k, t)ei~+b(k, t)e - i ~ ,
X-~
- - ~
.
(5) a(k, t) and b(k, t) are the "scattering coefficients". Note that (1) and (4) are very similar. In fact, one can easily become confused if one is not careful and does not use an adequate notation to distinguish between the two problems. Equation ( 1 ) is a physical equation for E and only has meaning for ot > 0. (Note that per (3), for a < 0 , the incident rf field moves to the left and the asymptotic behavior is unphysical.) On the other hand, (4) is a well defined mathematical problem for any k, and we shall need to solve it for all k to obtain our solution. There is a link between these two problems. We note that any solution of ( 1 ) can be expressed as a linear sum of solutions of (4). In particular, for k = a > 0, we have [10,14]
E(a, x, t) - a(k,t~q/(k,x,t)
,
(6)
and
T(a, t)= a(k,1 t~) k=,~ '
(7a)
R(a, t ) - a(k,t)b(k't) k=. "
(7b)
(3)
R ( a , t) and T(a, t) are respectively the reflection and the transmission coefficients for the incident 15 wave. This asymptotic behavior corresponds to a normalized rf wave being incident from the left onto an IAW moving to the right. Thus in the lab frame, both waves are moving in the same direction. The amplitude of the scaled electrostatic field has been normalized to unity because an arbitrary amplitude can be scaled off by rescaling the time. Thus in the notation of ref. [10], we h a v e / t + ( a ) = 0 since no ESW waves are entering from the right and here we have replaced t h e / t _ ( a ) of ref. [ 10 ] with simply
~(a).
17 A u g u s t 1992
In this paper, we obtain the time-asymptotic solution of the system ( 1 ) and (2) by means of the integral equation associated to this system. The integral equation could be either the Marchenko equation obtained by the usual IST method [ 10 ] or the Cauchy-Green equation (see ( 15 ) ) obtained by the Dbar method. In both cases, one needs the explicit expression of the reflection coefficient. Indeed in the Marchenko equation, the reflection coefficient occurs through its Fourier transform as the kernel of the integral equation and in the Cauchy-Green 121
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equation, it appears simply by construction of the Dbar equation [3]. The usual IST method furnishes the time evolution of the scattering data and one has to construct (by nontrivial calculation) the time evolution of the reflection coefficient [ 10]. Here we use the method proposed in ref. [ 3,12 ] which allows one to obtain directly from the evolution equation (2) the singular dispersion relation and to write merely the time evolution of the reflection coefficient in its simplest form. We follow here the detailed study of ref. [ 3 ]. Defining f(k, x, t) such that ~u(k, x, t ) = f(k, x, t)e ikx, we can associate to the Schrrdinger equation (4) the following tT-problem [ 13 ],
~ f(k)=f(-k)r(k), (8)f(k)=l+O(1/k),
k~C,
17 August 1992
x, t) of (4). Due to the complex conjugate in (2) and the symmetries on a(a) ( a * ( a ) = a ( - a ) ), we may extend the original integral over c~from 0 to + in (2) to an integral over a from - ~ to + oo if we define # ( a ) for a < 0 to b e # ( - c ~ ) and take ½of the latter integral. The advantage of this is that one can then use contour techniques to evaluate this integral. Then using the Cauchy-Green formula, eq. (12) gives (for a = k ) +~
.f
dl #(l) fl(k)=-½1 J l-kad(l)"
Now inserting (13) into (10) and taking into account (9), we have -ik
OR(k,t)=R(k,t)
Ikl~.
(13)
/t(l) dl.
f 12_(k+ie) 2ad(l)
(8)
(14)
Let us consider first the case when q(x, 0) has no bound states (zeroes of a(k, 0)). Then r(k, x, t) is related to the reflection coefficient through
The potential q(x, t) is obtained by solving the basic integral equation
r(k, x, t) =R(k,
f(k, x, t) = 1 + ~
t)e2itc~(k I +0)
(9)
.
Following the method described in ref. [3], we assume a simple time dependence for r(k, x, t), O ~r(k)=
[fl(k)-fl(-k)]r(k),
(10)
-72-~J( -l, x, OR(l, t) C
X eZ~(k~ + i e ) ,
(15)
which solves the &problem (8). The solution q(x, t) of ( 1 ), (2) will then be given from f b y [ 3 ]
where//(k) is a given distribution obtained by the comparison of the evolution equation (2) with
q = - 2 i ~ x f ( ~ ) ( x , t) ,
qt -- _ 1~ ff d a ^ d a ~ ( a ) ~ ( - a )
where f¢~) is the coefficient of 1/k in the Laurent series expansion o f f ( k , x, t). We now have to solve the evolution (14) for R (k, t) which upon using principle value integrals can be written as
0
~--~fl( o , ,
(11)
C
where ~ is the complex conjugate of o~. Equation (11 ) is itself obtained by the compatibility condition 0 e ( 0 ~ ) = 0 ~ ( 0 / ~ , ) when the distribution fl(k) has no polynomial part (see ref. [31 ). Thus, we obtain in our case O#(a)
in ~ ( a )
2 ad(a~) 6(cq),
(12)
where d(o~) = a (-o~), ~ is the Dirac distribution and a~ represents the imaginary part of or. We remark here that we have now extended/~ (ct) to negative values of ct. The reason for this is, by (6), we can give E(ct, x, t) in terms of the solutions ~,(k, 122
(16)
O R(k)=R(k)(n#(k)_ikP kad(k)
dl
f l2 k2ad(l)j (17a)
01~(k)=l~(k)(n#(-k)
dl # ( l ) ] t2-k 2 (17b) where P represents the principal value and /~(k) = R ( - k ) . Assuming d ( k ) = a * ( k ) (which implies the reality of q(x, t)), we have also R(k) =R*(k). Now using I t ( - k ) =#(k), (17) gives
7d(
+ikPf
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a Ot
PHYSICS LETTERSA
17 August 1992 -600
(R/~) = 2n/t R_~
aa
(18)
"
--oo
Then using the basic relation a d - b b - = 1 [ 14 ] and eqs. (6) and (7), we eliminate ad from (18) and solve it to obtain
iR(k,t)12=
B(k) +D(k)
B(k)
exp[ - 2 p ( k ) n t ] ' (19)
where for ease of notation, we have introduced and defined
B(k) = IR(k, 0)12 O(k) = 1-B(k) .
,
(20a) (20b)
Note that if B ( k ) = 0 for some value of k, then IR (k, t) [2 will be also zero for that value of k for all t. Equation (17) can now be solved and we have
R(k, 0) exp [i0(k, t)] R(k, t) = x/B(k)+O(k) e x p [ - 2 n u ( k ) t ] ' O(k, t) = ~k P f (21b)
~
dl e2i~x 1 f~(k'xl=l-17t f l-(k+iO) f°°(- ,x)
(21a)
ln{B(k)
+D(k) exp[-2n#(k)t]} .
×exp [i0~o(/) + i arg R(/, 0) ] .
The corresponding permanent asymptotic profile of the lAW is then obtained from (16). We can evaluate the behavior at large x of the lAW profile resulting from (24). It can be shown by means of the steepest descent method that this profile is positive and exponentially localized in a given region of space. The position of this region is fixed by the argument of the input R(k, 0). A model case is given in the appendix. Let us consider now the case when the initial IAW profile contains also a density depression. Hence from (4), the spectrum of q(x, 0) will also contain a sequence of discrete eigenvalues, say k,=i(n for n = 1, ..., N. To each of these points is associated the normalization coefficients Cn and the resulting o~transform reads, instead of (9) [ 3 ],
r(k, x, t) = e 2ikx(R(k, +2n.=,~
(21b)
(24)
t)~(kl + 0 )
C.(t)~(k-k.)).
(25)
The evolution (10) then gives
O,C.(t) =
[fl(k.) - f l ( - k . )
Otk.=O and
] C.(t),
(26)
Note that i f / l ( k ) is zero for some range of k, then R(k, t) is time independent for the same range ofk. From the definition of B(k) and D(k) in (19), we immediately see that for gt> 0 we have
with fl(k) given in (13). The solution of (26) is obtained in the same way as that of R(k, t) and can be written
IR(k,t)l-,1,
C.(t) = C.(0)
t-,+~,
(22)
which was first shown in ref. [ 10 ]. Furthermore we have the remarkable property that the phase of R (k, t) also approaches a constant as t goes to infinity. Indeed, as t ~ + or:
O(k,t)--.O~(k)=-kP f ~-~lkloglR(l,O)[ _
(23)
We have used here the fact that the function IR(k) [2=R(k)R(-k) is even in k. The solution f of the integral equation (15) approaches now the solution fo~ of the following integral equation,
exp [i0. (t) ] ,
(27)
0 , ( t ) = ~ k. f ~ l dl n{B(/)
+D(l)
e x p [ - 2 / l ( / ) ~ t ] },
(28)
where now the integral is along the real axis. Note that the imaginary part of kn is always positive. We remark that when R(k, 0) is zero then one can recover the usual case where these solitons move slower than the IAW wave front. This case also occurs when the ESW moves in the opposite direction relative to the IAW, in which case p (l) in ( 28 ) would be replaced by -/1-6(/), (see ref. [10]). In either 123
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case, it is the last term in the argument of the In which would dominate at large t, giving 0°(t)=-
knf dZ 2~312_k~
[2x~(l)t-lnD(l)].
(29)
However this is not the case we are considering here. Instead, the ESW is moving in the same direction as the IAW and we take JR(k, 0) I to be nonzero for all k. In this case, instead of (29), (28) will reduce to k n f ~ l ndll R ( l , 0 N ( t ) = -x-
0)l,
(30)
where 0n is obviously independent of t. Now each and every soliton has become entrained with the IAW wave front, and will not move relative to it. The corresponding IAW profile as t ~ o e is then obtained by solving the integral equation
f~(k, x) = 1-
1
d/e2i/x
_ l_(k+i~) f ~ ( - l ' x )
× e x p [i0~ (l) + i argR (/, 0) ] - i ~ Cn(0) e x p ( i 0 T ) f ~ ( - k n , x) exp(2ik, x) n:l k~k
(31) and by using (16) to compute q~(x). For most choices of the initial data, the different coherent structures in (31 ) will be well separated and we can describe one of the static cavitons by solving (31) around the nth caviton position ( x T ) , x~= ~
1
log[C,(0)/2~,,]
1 f / 2 ~dl- ~ l ° g l R ( k ' 0 ) l "
---2re
(32)
Acknowledgement This work has been supported in part by the Office of Naval Research through Grant No. N000A-88-K0153 and by the Air Force Office of Scientific Research through Grant No. AFOSR-89-0510.
Appendix To choose a model for R(k, 0), let us say that the incident ESW sees no IAWs until x > 0. This would correspond to the ESW incident on a plasma slab ( x > 0). Thus we would have q(x, 0) = 0 if x < 0. For such a potential, R (k, 0) would be analytic in the lower half k-plane [14]. We also know that R ( k = 0) = - 1 if no bound states are present. Thus a reasonable model for this would be R(k, 0 ) - - e x p (
Vk@i7) ,
(A.l)
where v and 7 are some constants. The constant 7 would measure the width in k-space of the continuous spectrum while ~ determines how fast the spectrum drops off in k space. For large k, R(k, O) - ~ - e-", so one wants to choose ~ large. Then with this model equation (24) becomes +oc
1 t"
dl
J l_(~+ie) f ( - l ' x ' t )
(33)
All cavitons are static in the x-frame (where one is moving with the acoustic wave). Their initial positions are determined by the (real) values C , ( 0 ) and their final positions by ( 32 ) where the value of R (k, 0) enters explicitly through (30). To summarize, when the ESW waves, which are comoving with the IAW, do dominate, then we have two new and fundamental results: 124
(i) The IAW grows until it forms a barrier which asymptotically totally reflects the Langmuir wave. (ii) This barrier also entrains all solitons and pulls them along with the barrier. The entire arrangement becomes static in the lAW frame.
f(k,x,t)=l+
Then in the neighbourhood of x T we have
q~#(x) = - 2 ~ sechZ [~n(x-xT~ ) ].
17 August 1992
×exp
2i/x+ 2 ( / + i 7 )
~
.
(A.2)
The behavior ofeq. (A.2) is evaluated by the method of steepest descent. Let vl
w(l)=2ilx+ (/+i2--~-~ 2 ' and
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PHYSICSLETTERSA
vl
17 August 1992
~,(l) = 2i/x- - 2(/-i7) "
0 -4il±tl+ exp[w(/±)] q(x't)=-~x ~ l+~/±exp[w(/±)]
The stationary points are given then by
where
x<0,
Ow(l)=O=,l+-=-i,+_
~
,
(A.3)
and x>0,
O ff(l)=0~T±=iy+i
v/~.
(A.4)
q±= Ro(l ±) k / 2~l +
(A.6) ,
2x
(A.7)
w"(l ±) •
The approximation in (A.5) is valid as long as the second term of the expansion is smaller than the first one. Here, vYlxl >> 1 .
For x < 0, the path of the integration can follow the path of the steepest descent (i.e. I m [ w ( / ) ] = Im[w(/±)]). In this case, '
f(k)-,1 - ~
x
For x > 0, the integral (A.2) can be evaluated by the method of stationary phase, following the path of integration (i.e. Re[~(l) ] = R e [ ~ ( l - ) ] ). In this case,
f(k)~l+ if(-T-)exp (
-_
v[- ) 2(T- +iy)
rt T - - k
w.(u_)exp[w(l+-)l.
(A.5)
,'
,
x
~exp[~(T-)l.
(A.9)
And using (16), we obtain for
And using (16), we obtain for q(x, t)
30
(A.8)
"' ,
I
I
q(x, t)
I
25
20
15
i0
5
0
..........
-5
-I0 -BOO
i
-600
-400
I
-200
I
I
I
200
400
600
g00
Fig. 1. q(u, t=oo) with u=xvywherev=500 and y=0.01. 125
Volume 168, number 2
0 -4i/-f/exp[#(r-) q(x,t)=Ox l + f / e x p [ # ( T - ) ]
PHYSICS LETTERS A
References
] '
(A.10)
where (4x)-3/4 f/= 2n ( 7 _ xf~-y/4x) X exp ( _
2 2~'x-',/~]
47x_x/~).
(A. 11 )
Here also, the a p p r o x i m a t i o n is valid o n l y if vTx>> u 2 .
(A. 12)
I n d e e d , if x < #/7, t h e n ]'- m o v e s i n t o the l.h.p, a n d it w o u l d n o m o r e be possible to ignore the contrib u t i o n of the o t h e r t e r m s in ( A . 2 ) . I n fig. 1, we plot q(x, t) (see ( A . 6 ) a n d ( A . 1 0 ) ) with the n e w v a r i a b l e u = xuy. A c c o r d i n g to the constraints ( A . 8 ) a n d ( A . 1 2 ) , o u r s o l u t i o n is n o t valid in the region a r o u n d u = 0 , a n d m a y well be m o d e l dependent.
126
17 August 1992
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