Fundamentals of Plasma Physics

14.3 Echoes 421

a similar procedure for Fourier transforms gives

F(g(x)h(x)) =

∫∞

−∞

g(x)h(x)eikxdx

=

∫∞

−∞

[

1

2 π

∫∞

−∞

̃g(k′)e−ik

′x

dk′

]

h(x)eikxdx

=

1

2 π

∫∞

−∞

dk′g ̃(k′)

∫∞

−∞

h(x)ei(k−k

′)x

dx

=

1

2 π

∫∞

−∞

dk′g ̃(k′) ̃h(k−k′). (14.125)

Thus, the Fourier-Laplace transform of Eq.(14.114) gives

(p+ikv)f ̃ 2 (p,k) +ik

e

m

∂f 0

∂v

̃φ 2 (p,k)

= −

e

m

∂

∂v

[∫

∞

−∞

dk′

2 π

∫b+i∞

b−i∞

dp′

2 π

k′ ̃φ 1 (p′,k′)f ̃ 1 (p−p′,k−k′)

]

.(14.126)

Because the convolution integrals are notationally unwieldy, to clarifythe notation we
define the new dummy variables

̄k′ = k−k′

̄p′ = p−p′ (14.127)

so that Eq.(14.126) becomes

(p+ikv)f ̃ 2 (p,k) +ik

e

m

∂f 0

∂v

φ ̃ 2 (p,k)

= −

e

m

∂

∂v

[∫

∞

−∞

dk′

2 π

∫b+i∞

b−i∞

dp′

2 π

k′φ ̃ 1 (p′,k′)f ̃ 1 ( ̄p′,k ̄′)

]

. (14.128)

The factors in the convolution integral can be expressed in terms of the original driving
potential using Eqs.(14.120) and (14.122) to obtain

(p+ikv)f ̃ 2 (p,k) +ik

e

m

∂f 0

∂v

φ ̃ 2 (p,k)=∂

∂v

χ(p,k,v) (14.129)

where the non-linear convolution term is

χ(p,k,v) =

(e

m

) 2 ∫∞

−∞

dk′

2 π

∫b+i∞

b−i∞

dp′

2 π

{

k′

φ ̃ext(p′,k′)

D(p′,k′)

×

i ̄k′

p ̄′+i ̄k′v

φ ̃ext( ̄p′, ̄k′)

D( ̄p′,k ̄′)

∂f 0

∂v

}

. (14.130)