special solutions for nonlinear wave-particle interaction

2
Volume 51A, number 3 PHYSICS LETTERS 24 February 1975 SPECIAL SOLUTIONS FOR NONLINEAR WAVE-PARTICLE INTERACTION* Flora YING FUN CHU** Department of Electrical and Computer Engineering, University of Wisconsin, Madison, Wisconsin53706, USA Received 22 January 1975 A class of exact solutions for a coupled set of nonlinear equations describing the interaction between two propagat- ing waves and a system of particles is found. These solutions include traveling wave solutions of the non-linear coupled equations. The set of coupled equation which describes the nonlinear interaction between two propagating waves with group velocities o 1 and o2 and a system of elec- tromagnetic particles (induced scattering) is [ 1] Ilt + °lllx = -°-/112 (1) I2t + 0212x = °JlI2" In (1), I 1 and 12 are the spectral densities, i.e., wave energy density per unit frequency and a is the cou- pling coefficient given by [2] a e) = -fwo,, k 1 , k2)(¢o 1 -- ' where w is the probability of scattering of the wave with wave number k 1 and frequency ~1 by a particle with momentum p into a wave number k 2 and fre- quency co 2 and f is the particle distribution function expressed in terms of e, the energy. The coupling co- efficient ct is positive if w I > w 2 and af/Oe < O. Hasegawa [1] has found traveling wave solutions for (1). However, (1) can have other more general sob utions. To see this, (1) is transformed into the follow- ing set of normalized equations 91~ = --919 2, 92r =91 92 (2) * Supported by Nationa/Science Foundation under Grant No. GK-37552. Present address: Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139. under the independent variable transformation 1 °2 ~---x - ~ - t o1 -- 02 o 1 -- 02 1 °1 7"----X - - t 02 -- o1 02 -- o 1 and the dependent variable transformation (3) 91 =tff 1 , 92 =or 2. (4) Equations (2) give (In 91)~r = 91~ =--9192 (ln 92)~r = -92r = -91 92 which can be integrated to In 91 = In 92 + In T(r) - In Z~(~) or 91/9 2 = T~(r)IZ~(g) where T(0 and Z(~) are arbitrary differentiable func- tions. Substituting (5) into (2) gives Z~(}) ~ 91Z~(}) ~ 912 911 i =--9~ L(T), k T(r) ]r- T(r)" (6) Eq. (6) can be integrated to give solutions of 91 and 92 in terms of two aribtrary functions T(r) and Z(~) 129

Upload: flora-ying-fun-chu

Post on 21-Jun-2016

218 views

Category:

Documents


3 download

TRANSCRIPT

Page 1: Special solutions for nonlinear wave-particle interaction

Volume 51A, number 3 PHYSICS LETTERS 24 February 1975

SPECIAL SOLUTIONS F O R N O N L I N E A R WAVE-PARTICLE INTERACTION*

Flora YING FUN CHU** Department of Electrical and Computer Engineering,

University of Wisconsin, Madison, Wisconsin 53706, USA

Received 22 January 1975

A class of exact solutions for a coupled set of nonlinear equations describing the interaction between two propagat- ing waves and a system of particles is found. These solutions include traveling wave solutions of the non-linear coupled equations.

The set of coupled equation which describes the nonlinear interaction between two propagating waves with group velocities o 1 and o 2 and a system of elec- tromagnetic particles (induced scattering) is [ 1 ]

I l t + ° l l lx = -°-/112 (1) I2t + 0212x = °JlI2"

In (1), I 1 and 12 are the spectral densities, i.e., wave energy density per unit frequency and a is the cou- pling coefficient given by [2]

a e) = - f w o , , k 1 , k 2 ) ( ¢ o 1 - - '

where w is the probability of scattering of the wave with wave number k 1 and frequency ~1 by a particle with momentum p into a wave number k 2 and fre- quency co 2 and f is the particle distribution function expressed in terms of e, the energy. The coupling co- efficient ct is positive if w I > w 2 and af/Oe < O.

Hasegawa [1] has found traveling wave solutions for (1). However, (1) can have other more general sob utions. To see this, (1) is transformed into the follow- ing set of normalized equations

91~ = - - 9 1 9 2, 92r =91 92 (2)

* Supported by Nationa/Science Foundation under Grant No. GK-37552. Present address: Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139.

under the independent variable transformation

1 °2 ~ - - - x - ~ - t

o 1 - - 02 o 1 - - 02

1 °1 7 " - - - - X - - t

02 -- o 1 02 -- o 1

and the dependent variable transformation

(3)

91 =tff 1 , 92 =or 2. (4)

Equations (2) give

(In 91)~r = 91~ =--9192

(ln 92)~r = -92r = -91 92

which can be integrated to

In 91 = In 92 + In T ( r ) - In Z~(~)

or

91/9 2 = T~(r)IZ~(g)

where T( 0 and Z(~) are arbitrary differentiable func- tions. Substituting (5) into (2) gives

Z~(}) ~ 91Z~(}) ~ 912

911 i =--9~ L ( T ) , k T(r) ] r - T(r )" (6)

Eq. (6) can be integrated to give solutions of 91 and 92 in terms of two aribtrary functions T(r) and Z(~)

129

Page 2: Special solutions for nonlinear wave-particle interaction

Volume 51A, number 3 PHYSICS LETTERS 24 February 1975

T(T) Zt(~) 91 = Z(~) - T0") 92 = Z(~) - T(¢)" (7)

Since Z and T are arbitrary, (7) generates an infinite set of exact solutions for (2). For example, if T r = T O = constant and Z~ = Z 0 = constant, (7) gives

To z o 91 =ZOO_ To,r + 70, 92 - Z O O _ Tcr +TO'

T O constant, (8) If T and Z are chosen such that

T = T O exp(~r) , Z = Z 0 exp(g ~) (9)

where K, w, TO, Z 0 are constants,

91 = ½ ~ [ - 1 + tanh ½(K~ - o~ + ln(-Zo/To))]

92 = ½~ [1 + tanh ½(K~ - coy + ln(-Zo/To))]. (lO)

Eq. (10) are the traveling wave solutions of (1) dis- cussed by Hasegawa [1] where 91 is a rarefaction wave and 9 2 is a shock wave when co < 0 and r > 0. These waves move at a velocity Kv 2 + ~Vl/~ + w in the laboratory frame. However, solutions in the form of (7) are interesting because they can also generate solutions for the interaction of 91 and ~ when these spectral densities are no longer traveling wave solutions of (2) or (1). For example, choosing

where T1, T 0 ,Z 1 ,Z0, o~ and K are constants, will gen- erate

91 = ½~[--1 + tanh(½(K~ -- cot + l n f l ) ) ]

9 2 = ½/¢ [1 + tanh(½ (K~ - cot - In f2))]

where

f l = In (2x/Z0(Z 1 -- T1)/T20

X cosh(L:~+log~/Zo/Z 1 - T 1 ) ) (12)

f2 =In (2~/T0(T 1 -Z1)/Z2

× cosh(½g~+logx/To/T 1 - Z 1 ) )

i f Z 1 - T 1 4 :0 and Zo/IZoI = -To/ITo I. The solutions (12) are similar to the traveling wave solutions (10) except now, the phases ln(f l) and ln(f2) are slowly varying functions of space and time.

References

[ 1 ] A. Hasegawa, Propagation of wave intensity shocks in nonlinear interaction of waves and particles, Phys. Lett. 47A (1974) 165.

[2] V.N. Tsytovich, Nonlinear effects in plasma (Plenum Press, New York, 1970), 207.

T=Toexp(mr)+T 1 Z=Zoexp(K~)+Z 1 (11)

130