Phase-invariant solutions of nonlinear wave-wave interacting systems

Physica 81A (1975) 441-453 © North-Holland Publishing Co.

PHASE-INVARIANT SOLUTIONS OF NONLINEAR W A V E - W A V E INTERACTING S Y S T E M S P. K. C. WANG School of Engineering and Applied Science, University of California, Los Angeles, California, U.S.A. Received 29 April 1975

The existence of phase-invariant solutions for a class of nonlinear wave-wave interacting systems is studied. These solutions have the property that the phase of each wave is invariant with time or unaffected by the nonlinear interactions. Explicit results are obtained for certain two-wave and three-wave systems. It is shown that for these systems, the wave-energy transfer along the phase-invariant solutions is always unidirectional. The application of these results to the noninear interaction between two transverse waves and a longitudinal wave in a magnetized collisionless plasma is discussed.

1. Introduction The n o n l i n e a r i n t e r a c t i o n o f waves in p l a s m a s a n d o t h e r m e d i u m s has been extensively s t u d i e d 1 - 9 ) . I n the case where the initial phases o f the waves are a s s u m e d to be r a n d o m , it is possible to o b t a i n expressions for the p h a s e - a v e r a g e d solutions in closed f o r m I - 4 ) . W h e n the initial phases o f the waves are deterministic, explicit solutions in closed f o r m are o b t a i n a b l e only for certain special classes o f n o n l i n e a r w a v e - w a v e interacting systems. Recently, the existence o f ray solutions for a general class o f n o n l i n e a r systems was studied~°). The significance o f the r a y solutions for n o n l i n e a r w a v e - w a v e interactions a n d their r e l a t i o n to explosive instabilities were e x p l o r e d 11). Here, a t t e n t i o n is focused on the existence o f p h a s e - i n v a r i a n t solutions o f a class o f n o n l i n e a r w a v e - w a v e interacting systems. This class o f solutions includes the r a y solutions as a special case. Explicit results a r e given for specific n o n l i n e a r w a v e - w a v e interactions.

2. Phase-invariant solutions Let C" d e n o t e the inner p r o d u c t space o f o r d e r e d n-tuples o f c o m p l e x numbers. F o r a n y x = (xl . . . . . x,) a n d ,~ = (if1 . . . . . ft,) in C", their inner p r o d u c t is defined 441

442

P.K.C. WANG

by n

i=1

where (.)* denotes complex conjugate. Consider a system of n nonlinearly interacting waves in which the evolution o f the complex wave-amplitude vector x(t) = (x~ (t) . . . . , x,(t)) with time t is governed by the vector ordinary differential equation

dx

(t)

_

f(x(t))

(J)

dt defined on C" with initial condition x(0) = Xo ~ C ", where f = (f~, . . . , f , ) is a given C"-valued function o f x representing the nonlinear interaction terms. The equations for m a n y nonlinear wave-wave interacting systems can be transformed into the f o r m o f (1) under certain matching conditions for the wave frequencies. F o r example, the equations describing the nonlinear interaction o f three m o n o chromatic waves in plasmas s'9) given by

dyl/dt = jcolyl +/zly2Y3, dy2/dt = flo2Y2 + I.t2ylY~,

(2)

dy3/dt = jo93y3 + / ~ 3 Y l Y * , with j = ( - 1)~, can be rewritten in the f o r m o f (1) with f ( x ) = (~t*x2x3, p 2 x l x ~ , 1~3xlx*) by setting Yi = x~ exp (jco~t), i = 1, 2, 3, provided that the wave frequencies cog are all real and satisfy the matching condition o~1 = e~2 + o~3. Let I denote the time interval 0 ~< t < T < oo. By a phase-invariant solution of (1) defined on L we mean a solution o f the f o r m x~(t) = 7~(t) z~, i = 1. . . . . n, where y~ is a real nonnegative function o f t and z~ is a complex number. Evidently, such a solution has the property that the phase of each wave remains constant for all t ~ I. F o r simplicity, we shall write a phase-invariant solution as x(t) = F(t) z, where F(t) is the real diagonal matrix with diagonal elements y~(t) and z is the column vector (zl . . . . , z°) r. In the special case where ~ ( t ) = 7(t) for all i = 1, ..., n and t ~ / , we have a ray solution x(t) = V(t) z; i.e. the solution points lie along a half-ray emanating f r o m the origin o f C" in the direction z. Note that a general phase-invariant solution can be written in the f o r m : n

x(t) = ~ Vi(t) zie,,

(3)

i=1

where e~ is the unit vector in C" whose ith c o m p o n e n t is (1 + j0). Thus, the complex amplitude o f the ith wave at any t ~ I can be regarded as a point on a half-

PHASE-INVARIANT SOLUTIONS OF NONLINEAR SYSTEMS

443

ray from the origin of C" in the direction z~ei. Also, we recall that in the r a n d o m phase approximation, it is assumed that the characteristic time associated with the wave-amplitude variations due to nonlinear effects is much longer than the phaseshift time. Consequently, the nonlinear interactions only alter slightly the phases of the waves or the correlation between the phases of the waves at any fixed t > 0 is small. Here, we shall show that in nonlinear interaction of waves with deterministic phases, there exist solutions along which complete decoupling of the phases of the waves is attained. In other words, the nonlinear interactions only modify the amplitude of each wave without disturbing its phase. In what follows, we shall assume that the interaction terms f~ in (1) have one or both of the following properties: (P-l) There exists a vector £, = (Z1 . . . . . ,~n)T ~ C" with 1~[ :~ 0 for i = 1, ..., n such that each f~(F:~) can be written in the f o r m : ~i( .r~ ) = gi (71 . . . . . 7 n ) f / ( ~ )

for all real v e c t o r s (71 . . . . . 7n), w h e r e / " is the diagonal matrix with diagonal elements 7i. and each g~ is a real-valued function of its arguments; (P-2) The function f is h o m o g e n e o u s with degree k > 0, i.e. f ( c x ) = c k f ( x ) for some k > 0 and all x E C" and real numbers e. For nonlinear w a v e - w a v e interactions describable by (1) with f~ being h o m o geneous polynomials having the same degree k for all i = 1. . . . . n, property (P-2) is satisfied. F o r the transformed system (2), b o t h properties (P-l) and (P-2) are satisfied. Now, assuming that f has p r o p e r t y (P-l), we seek phase-invariant solutions of (1) in the f o r m x(t) = F(t) ~. Let f~(~') = Ifi(~')[ exp (j~o~), and ~, = I~,[exp(j0,), i = 1. . . . . n. Substituting x(t) into (1), we obtain the following set of equations for 7i(t):

d7, (t)/dt = g, (71(t) . . . . . 7,(0) If,(~')l

le, l-1

exp ( j (~0, - 0,)),

i = 1. . . . . n,

(4)

Evidently, if the phase conditions Wi -

0i

=

nfz,

i =

1 .....

n

are satisfied for some (nl . . . . ,n,), ni = 0, _ l , + 2 . . . . . solution exists when the set of equations

(5) then a phase-invariant

d7, (t)/dt = s, If,(~)l I~,1- I g, ( 7 , ( 0 . . . . ,7.(t)), 7i(0) = 71o > / 0 ,

i = I ..... n

(6)

444

P.K.C. WANG

has a n o n n e g a t i v e solution 7~(t), i = 1. . . . , n defined on some time interval I for s o m e (71o, . - . , 7,o), where s~ = cos (n~=). In the special case where the n o n l i n e a r t r a n s f o r m a t i o n f o r t C" into itself has the p r o p e r t y that there exist a ~, = ( ~ , . . . , £,)T e C" a n d real n u m b e r s 2~, i = 1, . . . , n such that ~ ~ 0 + j 0 a n d f-(~') = 2~t for all i = l . . . . . n, then (1) has a p h a s e - i n v a r i a n t solution on I i f the set o f e q u a t i o n s dT, (t)/dt -- 2,g,

(v,(t),

7g(0) = 7io ~> 0,

. . . , 7,(t)),

i = 1. . . . . n

(7)

has a n o n n e g a t i v e solution on I for some (71o . . . . . 7,o)- In the case where 2~ = 2 ¢ 0 for all i = I, . . . , n, ~, c o r r e s p o n d s to an i n v a r i a n t direction or an eigenvector o f f associated with the eigenvalue 2. W h e n the f~'s are h o m o g e n e o u s p o l y n o m i a l s o f the x f s , the c o r r e s p o n d i n g g~'s are h o m o g e n e o u s p o l y n o m i a l s o f the 7 f s which satisfy a local Lipschitz c o n d i t i o n a b o u t the initial p o i n t (7~o . . . . . 7,o), F r o m a well-known result in the t h e o r y o f o r d i n a r y differential equations~2), (6) or (7) has a u n i q u e solution defined on s o m e time interval I_~ [0, ~ ) . F o r p h a s e - i n v a r i a n c e on some time interval I ' , each 7~(t) m u s t be n o n n e g a t i v e for every t e I ' . In view o f the f o r e g o i n g r e m a r k a n d the c o n t i n u i t y o f solutions 7~(t), if Vio > 0 for i = 1, . . . , n a n d the phase conditions (5) are satisfied, then there exists a time interval I ' ~ I on which the existence o f a n o n n e g a t i v e solution 7~(t) or a p h a s e - i n v a r i a n t solution is ensured. Similar conclusion can be m a d e ifTg o/> 0 for i = I, . . . , n, p r o v i d e d t h a t d T / d t at t = 0 is positive for those i's c o r r e s p o n d i n g to 7~o = 0. N o w , assume that f has p r o p e r t y (P-2). W e seek p h a s e - i n v a r i a n t r a y solutions o f the f o r m x(t) = 7 ( 0 z with z ¢ 0 e C". U s i n g p r o p e r t y (P-2), we have

d x (t)/dt = d 7 (t)/dt z = f ( 7 ( t ) z) = yk(t) f ( z ) .

(8)

I f z is an i n v a r i a n t direction o f f (i.e. f ( z ) = 2z, z ~ 0 e C" for some n o n z e r o real 2), then 7 ( 0 must satisfy

d 7 (t)/dt = 27 k (t),

7( 0 ) = 7o >~ O,

(9)

where Z = (f(z), z)/(z, z). The solution o f (9) is given by v ( 0 = ~,o {l - (k -

1)27~-'t}-,/,k-1,,

(10)

where we take {...}-1/~k-1) to be the nonnegative real r o o t when k > 1. Clearly, for k > 1, (f(z), z) > 0 a n d 7o > 0, the foregoing p h a s e - i n v a r i a n t solution is explosive or 7(t) ~ ~ as t ~ T - , where ~ is the explosion time given b y

= 7~- ~ l ( ( k ,

1) 2).

(1 I)

PHASE-INVARIANT

SOLUTIONS OF NONLINEAR

SYSTEMS

445

Finally, we consider the wave-energy transfer along the phase-invariant solutions of (1). Since the energy of the ith wave at time t is directly proportional to ]Xi(t)[ 2, the rate of energy transfer associated with the ith wave along a phaseinvariant solution x~(t) = 7~(t) zi with 7~(t) governed by (6) is given by

(12)

d [x,(t)12/dt = 2s,7, (t) ]z,I ]f~(z)l g, (y~(t) . . . . . y.(t)).

It is apparent that the direction of the wave-energy transfer for the ith wave at any fixed t is determined by the sign of stg~ (71(t) . . . . . 7.(t)). For the case where (1) has a phase-invariant ray solution x(t) = 7(t) z with 7(t) given by (10), we have

d Ix,(t)lZ/dt = 2 Re {x*(t)f~(x(t))} = 2 Re {7(0 z*f~(y(t) z)} = 27 k+' (t) Re {z*f~(z)} = 227 k+l (t)Iz~l 2.

(13)

Thus, the direction of the wave-energy transfer for the ith wave at any time t depends only on the sign of 2 or (f(z), z) which is invariant with t. This implies that the energy transferred to or from the ith wave along a phase-invariant ray solution is unidirectional or irreversible. Expression (13) can be used to derive lower bounds for lx~(t)111).

3. Specific wave-wave interacting systems In this section, the existence of phase-invariant solutions for nonlinear twowave and three-wave interactions will be studied. The applications of the results to certain nonlinear wave-wave interactions in a plasma will be discussed.

3.1. Two-wave interactions Consider a nonlinear two-wave interacting system of the form:

dxa/dt

=

#lx*xz,

dx2/dt

= # 2. X 12,

(14)

where/~1 and #2 are complex coupling coefficients. The above equations describe the nonlinear interaction of a plane electromagnetic wave (with real frequency 0) 1 and complex amplitude y~) with its second harmonic (with frequency ~o2 = 20)2 and complex amplitude Y2) in a nonlinear dielectric mediumS), where 35 = x~ exp (jo)it). We shall seek phase-invariant solutions of (14) in the form xi(t) = 7~(t) z~ with z~ = iz, I exp (jOt) and 7z(t) ~> 0, i = 1, 2. Let #~ = I#~l exp(j~b~). Then, under the following phase conditions 20t - 02 + q~, = n,rc,

n, ~ {0, _ 1 , __2. . . . },

i = 1, 2,

(15)

446

P.K.C. WANG

the equations for V,(t) corresponding to (6) are given by d)'l/dt = sla1)'lT2, )'1(O) = ~ 1 0

dyz/dt = sza2)'~,

~ O,

(16)

)'2(0) = ) ' 2 0 ~ O,

where

~ = Iml Iz21,

~'= = 1~21 [zl1211Zzl,

si = cos (n?:),

i = 1, 2.

(17)

Note that the conditions in (15) imply that ~b~ and q~z must differ by an integral multiple of rc and there exists an infinite number of pairs (0~, 0z) or (z~, Zz) satisfying these conditions. Now, the first integral of (16) can be obtained by integration:

v~(t)

~"" )4(0

=

v~o

S2
-

~,o~,

- - ) ' 2 0

,

-

0.

(18)

S2~;2

F o r s~ = - s 2 , the nonnegative solutions of (16) are portions of ellipses restricted to the first quadrant of the (711 72)-plane, and they are bounded by y12(t) ~< 2 ° + ~, ~,2o, 2 -

-

2 )'~(t) ~< )'~o + ~x2 ~,~o

~2

-

(19)

-

~X1

for all t 1> 0. F o r s~ = s2 and 6 ¢ 0, the solutions of (16) are portions of hyperbolas in the first quadrant of the ()'a, y2)-plane. An explicit expression for y2(t) can be obtained by first substituting )'](t) given by (18) into the second equation in (16) and then integrating the result: 72(0

f(~2

d~ = SlcHt. + ~ (s2~2)/(s,c,,))

(20)

~20

F o r A~ = 6 (sza2)/(Slal) > O, we have )'2(t) =

)'20 + A+ t a n ( s a ~ A+t) Y2o A+ 1 tan (sloq A+t)

(21)

1 -

and for A 2 = - 6 (s2a2)/(sl~O > O, )'2(0 = d _ tanh {tanh -1 (yao/A-) - sjc~, A _ t } .

(22)

PHASE-INVARIANT SOLUTIONS OF NONLINEAR SYSTEMS

447

Evidently, for y2(t) given by (21) with s, = 1 and ~'zo > 0, there exists a finite time r, given by r~ = (o¢1 A+) -~ tan -~ (A+/yzo)

(23)

such that ~'2(t) --+ oO as t --+ *~-. Also, for y2(t) given by (22) and sa = 1, there exists a finite time r2 given by (24)

r2 = (~t A - ) -* t a n h - t ( Y z o / A - )

such thai: y2("g2) ~- 0. The expressions for yl(t) can be obtained by substituting y2(t) given by (21) or (22) into (18). F o r the special case where ~ = 0 and st = s2, we have Yl(t) = (~x~/0¢2)~ Yi(t) which corresponds to a ray solution x(t) = y2(t) ((o~/~2) ~ zl, 22) r, where ",'2(0 = ~,2o/(1 -

(25)

slo, t r 2 o t ) .

F o r s, = 1 and ~2o > 0, ~'2(t) --+ oO as t + ( ~ t y 2 o ) - t ; and for st = - 1 , y2(t) --+0 as t --+ oo. This ray solution was obtained earlier in ref. 11. . Finally, along a phase-invariant solution xi(t) = ~,~(t) z~, (12) has the explicit form :

d

Ix,(t)12/dt

= 2s, I/t~[ Izt[ 2 Iz21 y~(t)~,2(t),

(26)

i = 1, 2.

Thus, the direction of the wave-energy transfer for the ith wave depends only on the sign o f s~ which is invariant with time. This implies that the wave-energy transfer is unidirectional along the phase-invariant solutions o f (14).

3.2. Three-wave interactions Now, we consider the existence o f phase-invariant solutions for a three-wave interacting system given by (2). Introducing the transformation yi = x~ exp ( j ~ t ) , i = 1, 2, 3, we have the following equations for x~(t):

dxl/dt = [z~x2x3,

dx2/dt = IZ2XaX *,

dx3/dt = Ig3X1X~.

(27)

Again, we seek phase-invariant solutions o f the f o r m xi(t) = 7i(t) z~ = 7~(t)Iz~l × exp (jO~), i = 1, 2, 3. Following the same notations for the two-wave interactions, it can be readily deduced that under the phase conditions: 0, -- 02 - 03 + ~b, = n,r:,

n , e {0, -q-l, ___2. . . . },

i = 1,2,3,

(28)

448

P.K.C.

WANG

the equations for 7~(t) corresponding to (6) are given by

dTj/dt = s1~17273, 7i(0)

=

dv2/dt

7io >~ O,

dv3/dt

$2~271~3 '

=

$3~3~172 '

=

i = 1, 2, 3,

(29)

where ,~', :

Iz~l/Iz, I,

Iff,I Iz~l

si = cos (n~=),

~

= I,u~l Iz, I

Iz31/Iz~l,

*~ = Iff~l IZl[ Iz~l/l=~l,

i = 1, 2, 3.

(30)

The first integral for (29) is given by

- '5'10~I

-

v,o)

=

(~,](t).

--

$2 ~'~2

730)

(31)

S3'/X 3

O1"

1

72(t )

1

S10¢ 1

72(t)

1

$2~4 2

1 --

2 710

$1~; 1

I --

)j2(t)

S 3 !2,~3

- - 7 2 0

2

1 --

S2,.'X 2

2

- - 7 3 0

~ =

(32) •

S3(X 3

When s2 = s3 = - s l , the solution (7~(t), 72(0, 73(0) for any fixed t e I lies on the surface o f an ellipsoid restricted to the positive octant g~+ = {(y~ ,72,73):?~ >~ 0, i = 1, 2, 3} in the three-dimensional real-coordinate space R 3. This implies that 7~(t), i = 1, 2, 3 are b o u n d e d for all t 7> 0. For other combinations o f s]s, (?~(t), 72(t), 73(0) lies on the surface of a hyperboloid with elliptic cross section or on a hyperbolic surface restricted to (6~+. F r o m (31), we can express any pair of 7~(t) in terms of the remaining 7j(t), i.e. 7i(t) =

72o + .s'i~, (7~(t) - 7~o) -~ >/ O,

i # j.

(33)

It is apparent that if 7j(t) is bounded, then the remaining 7i(t), i ~ j, are also bounded. Also, if 7j(t) is unbounded, then 7~(t) is u n b o u n d e d (resp. bounded) when s~ -- s). (resp. si = -.s)). A n expression for 7i(t) can be obtained by substituting (33) into the equation for 7j(t) in (29) and integrating the result: ~'j(t)

d~

f [l~*s rtTio2 7jo

= s~c~fl. --

~'golj

(34)

PHASE-INVARIANT SOLUTIONS OF NONLINEAR SYSTEMS

449

F o r special choices o f p a r a m e t e r s Y~o, s~ a n d a~, i = 1, 2, 3, 7j(t) c a n b e w r i t t e n explicitly as a f u n c t i o n o f t. F o r example, f o r j = 1 a n d st = s2 = sa = 1,

y~(t) - [ylo/Cn (~'3o0¢1t,k) for Ly,o/cn (9'20 (c~t~3) ~ t, k')

y2o = 0,

(35a)

for

(35b)

y3o = 0,

where cn is the J a c o b i elliptic cosine f u n c t i o n a n d the c o n d i t i o n s k-=-

1

cx3Yto 2 ~1Y30

< 1,

k'-

1

7 . oqy20

< 1

(36)

are a s s u m e d to be satisfied. F o r ~/lO > 0, ~/1(t) given b y (35) is explosive with respective e x p l o s i o n times x a n d z' for (35a) a n d (35b) given b y T -----:(0gl~30) - 1

K(k),

./.t = y[O~ (Oqa3)¢ K ( k ' ) ,

(37)

where K is the c o m p l e t e elliptic integral :x/2

K(k) = f

d~ (1 - k 2 sin 2 ~)~"

(38)

o

F o r ,h = s2 = - s 3 = 1, 730 = 0 a n d k = Y ~ o Y ~ (~2/oq) ÷ < 1, we have an oscillatory s o l u t i o n given b y

7t(t) = 71o cn (Y2o (oqo¢3)~ t, k).

(39)

S o l u t i o n o f a similar f o r m is o b t a i n e d when st = s3 = - $ 2 = 1, Y2o = 0 and k = ~',,,r;o' (o,3fo,1) +" < 1. T h e f o r e g o i n g solutions c o r r e s p o n d to the cases where one o f the yto'S o r the initial wave a m p l i t u d e s is zero. F o r 7~o ¢ 0, i = 1, 2, 3, explicit solutions can also be o b t a i n e d for certain special cases, for example, for Sl = s2 = s3,

7~(t) = C t a n h {tanh -~ ()'~o/C) - Csl ( a ~ 3 ) ~ t}

7~o >

if

~12 --12 ~ 1 ~ 2 ~20 = ~ 1 ~ 3 ~ 3 0 ,

(40)

and 71('0 ,

=

~'1o + C2tan(slC(oc2~x3) ¢ t ) 1 71o t a n (siC (~2~3) ~ t)

if

2 1 2 -t 2 71o < a l O ~ 72o = o¢t~3 73o, (41)

where C = (y~o - oqa2-XT~o)}. W h e n sl = 1 a n d 71o > 0, y t ( t ) given b y (41) is explosive. Finally, for the special case where si --- s, oq = o¢ for i --- 1, 2, 3 o r

Iffll Iz21 z Iz312 = 1if21 Izll 2 Iz312 = 1ff3l Izll 2 IzW,

(42)

450

P.K.C. WANG

and ~io = 70 > 0, i = 1, 2, 3, performing the integration in (347 gives ~/,(t)

= 7o/(1 -

s~,Vot),

i = 1, 2, 3.

(43)

This solution corresponds to the ray solution as given in ref. 11. It is explosive when s = 1 and 7o > 0. Similar to the two-wave interaction discussed earlier, the direction of the waveenergy transfer for the ith wave along a phase-invariant solution x j t ) = 7~(t) z~ depends only on s~, since

d ]x~(O[2/dt = 2 Re { g ' x * (t) x2(t) Xa(t)} = 2sg I~l

j=

7j(t) IzjI -

(44)

Now, we apply the foregoing results to the nonlinear interaction between two transverse waves (o)~, k~), (e)2, k2) (with same kind of circular polarization) and a longitudinal wave (0)3, k3) in a magnetized collisionless plasma consisting of particles with mass m and charge q. It is assumed that (,)~ and k~ are all real, and all the waves p r o p a g a t e along the lines of a constant magnetic field Bo (parallel to the x-axis in a rectangular coordinate system (x, y, z)). F o r this case, it has been shown by Stenflo 9) that under the matching conditions k3 = kl

-

and

k2

(')a = (ol - ('~2,

(45)

the evolution of the complex wave amplitudes x~(t) with time is describable by (27) with #~ given by

,

(46)

where

C =

eorn2(o~o~2

(o~3 - k3vx) 2

k3 +_ o)¢

f - klk2

e)l - klvx +_ (%

--

('~2 - k2vx +_ (%

dv

v~ c~Fo/(?v.~ dr}, (~ol - klvx +_ c%) ((o 2 - k2v x +_ o¢) (~)3 - k3vx)

(47)

PHASE-INVARIANT SOLUTIONS OF NONLINEAR SYSTEMS

q2

e (~o, k) = 1

f (o9 - kv~) Fo - kv~ ~Fo/c~v~ dv ..... ,

eomO) 2 J

q2

451 (48)

~0 -- ](.Vx q- (De

i OFo/~Vx dr,

(49)

e~ (,o, k) = 1 + eomk O ~--~7 kv----~

and where Fo = Fo (Vx, v~ + v 2) is the equilibrium velocity distribution function; (,~¢ is the gyrofrequency. The positive (resp. negative) sign in front of ~o¢ in (47) and (48) corresponds to the right-hand (resp. left-hand) polarization of the transverse waves. Consider the situation where (~ > 0 and all the waves have positive energy, i.e.

> O,

i = 1, 2;

--

> O.

(50)

Then, in view of (46), the phase angles associated w i t h / h can be taken as ¢1 = 0, ¢2 = q53 = rr. T o satisfy the phase conditions in (28), we m a y set nl = 0 and n2 = n 3 = l s o t h a t

01 - 0 2 -

03 = 0

(51)

and S 1 :=

1,

s2 = s3 = - 1 .

(52)

Thus, if we choose the initial phases 01 of the waves such that (51) is satisfied and ]zi] ~ 0 for i = 1, 2, 3, then the corresponding solution will be phase-invariant. As evident f r o m (52) and (32), this solution is nonexplosive. Also, note that (51) specifies a plane through the origin in the (01,02, 03)-space. Thus, any triplet of initial phases in this plane will produce a phase-invariant solution. F o r the case where the first wave has positive energy, and the second and third waves have negative energy, i.e. >0, 00)

( o l , kl )

<0, ]"~C ( O

(to2 ' k2 )

<0,

(53)

~(O (o93, ka )

we m a y set ~b~ = 0, n, = 0 and & = cos (ny0 = 1 for i = 1, 2, 3. Again, w e h a v e a phase-invariant solution if the initial phases 0i satisfy (51) and [z~[ ~ 0 for i = 1, 2, 3. Moreover, if the amplitude of the longitudinal wave is initially zero [second condition in (36)], then (Y3o = O) and 0 < 1/~21 [z112 7 120 < [ # 1 ] [Z2[ 2 ~Y20 2 an explosive phase-invariant longitudinal wave is generated. Its complex ampli-

452

P.K.C. WANG

tude is given by [using (33) and (35b)1: x3(t) = {(~3/=xl) (y2(t) - 72o)} ~ [z3l exp ( j (0~ - 02)) = ([/z3[/[/z~l)~ [zll 2 exp {j (01 - 0,)} 7~o tn [7=olzzl (1~11 I/~31)~ t, k'}, (54) where 2

k,2 = ~

f#21 [z~l 2 7~o

(55)

I~, f Iz2l 2 y ~~o "

It is evident that along solution (54), the longitudinal wave gains energy (negative) from the transverse waves for all t e [0, T'), where 3' is the explosion time given in (37). In a similar manner, we may seek phase-invariant solutions of other types of nonlinear three-wave interactions in plasmas or for other plasma models such as those based on the fluid equations13).

4. Concludipg remarks The main point of this paper is to demonstrate that for a certain class of nonlinear wave-wave interacting systems, it is possible to find solutions for which the nonlinear interactions only affect the magnitude of the wave amplitudes and not their phases. In many systems, such solutions can be found simply by choosing appropriate combinations of the initial phases for the waves. It may be interesting to verify experimentally the existence of the phase-invariance phenomenon in certain physical nonlinear wave-wave interactions in plasmas or nonlinear dielectric media. Finally, although in this paper, the results are developed for the case where the phases of the waves are invariant with time. They are also applicable to the case where t corresponds to a spatial coordinate variable, and (1) describes the spatial evolution of the wave amplitudesZ4).

Acknowledgements This work was completed during author's visit with the Departement de la Physique du Plasma et de la Fusion Contr616e, Centre d'Etudes Nucl6aires, Fontenayaux-Roses, France. Their hospitalities are greatly appreciated. The author also wishes to acknowledge the support provided by an A F O S R grant No. 74-2662.

PHASE-INVARIANT SOLUTIONS OF NONLINEAR SYSTEMS

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References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

V.N. Tsytovich, Nonlinear Effects in Plasma (Plenum, New York, 1970). R.Z. Sadeev and A.A. Galeev, Nonlinear Plasma Theory (Benjamin, New York, 1969). R.C. I)avidson, Methods in Nonlinear Plasma Theory (Academic, New York, 1972)." T.Sato, Phys. Fluids 14 (1971) 2426. J.A. Armstrong, N. Bloembergen, J. Ducuing and P. S. Pershan, Phys. Rev. 127 (1962) 1918. J.Fukai, S.Krishan and E.G.Harris, Phys. Fluids 13 (1970) 3031. H.Wilhelmsson, L. Stenflo and F.Engelmann, J. Math. Phys. 11 (1970) 1738. H.Wilhelmsson, J. Plasma Phys. 3 (1969) 215. L.Stenflo, J. Plasma Phys. 4 (1970) 585. P.K.C.Wang, J. Math. Phys. 16 (1975) 251. P.K.C.Wang, I1 Nuovo Cimento 28B (1975) 56. E.A.Coddington and N.Levinson, Theory of Ordinary Differential Equations (McGrawHill, New York, 1955). 13) C.N.Lashmore-Davis, Plasma Phys. 17 (1975) 281. 14) I.Fidone, Plasma Phys. 12 (1970) 31.