Fundamentals of Plasma Physics

422 Chapter 14. Wave-particle nonlinearities

Equation (14.129) may be solved forf ̃ 2 (p,k)to give

f ̃ 2 (p,k)=−ie

m

k

(p+ikv)

∂f 0

∂v

φ ̃ 2 (p,k)+^1

(p+ikv)

∂

∂v

χ(p,k,v). (14.131)

However, the Fourier-Laplace transform of the second-order Poisson’s equation gives

−k^2 φ ̃ 2 (p,k)=

e

ε 0

∫

f ̃ 2 (p,k)dv (14.132)

and so substituting forf ̃ 2 (p,k)gives

−k^2 ̃φ 2 (p,k)=

e

ε 0

∫ [

−i

e

m

k

(p+ikv)

∂f 0

∂v

φ ̃ 2 (p,k)+^1

(p+ikv)

∂χ

∂v

]

dv (14.133)

or

̃φ 2 (p,k) = − e

k^2 ε 0 D(p,k)

∫

1

(p+ikv)

∂χ

∂v

dv

=

e

k^2 ε 0 D(p,k)

∫

ikχ

(p+ikv)^2

dv. (14.134)

Substitution forχand using the velocity-normalized distribution functionf ̄ 0 =f 0 /n 0
discussed in Eq.(14.4) gives

̃φ 2 (p,k) = ω

2

p

k^2 D(p,k)

e

m

∫

dv

∫∞

−∞

dk′

2 π

∫b+i∞

b−i∞

dp′

2 π

{

ik

(p+ikv)^2

k′ ̃φext(p′,k′)

D(p′,k′)

i ̄k′

p ̄′+i ̄k′v

φ ̃ext( ̄p′, ̄k′)

D( ̄p′, ̄k′)

∂f ̄ 0

∂v

}

.

(14.135)

Double-impulse source function and its transform
In order to proceed further, the form of the external source must be specified. We
assume that the external source consists of two sets of periodic grids whichare pulsed
sequentially. The first set of grids has wavenumberkaand is pulsed att= 0whereas the
second set of grids has wavenumberkband is pulsed after a delayτ.Thus, the external
source has the form

φext(x,t)=φacos(kax)δ(ωpt)+φbcos(kbx)δ(ωp(t−τ)). (14.136)
The Fourier-Laplace transform of this source function gives

φ ̃ext(p,k) =

∫∞

−∞

∫∞

0

φext(x,t)e−ikx−ptdxdt

=

π

ωp

∑

±

{

φaδ(k±ka) +φbδ(k±kb)e−pτ

}

. (14.137)

Since we are interested in the non-linear interaction between theaandbpulses, only
the contribution from theapulse in the first ̃φextfactor in Eq.(14.135) and the contribution