Fundamentals of Plasma Physics

420 Chapter 14. Wave-particle nonlinearities

is the usual self-consistent linear dielectric response function. Thus, the driven first-order
velocity distribution function can be written as

f ̃ 1 =−ie

m

k

p+ikv

φ ̃ext(p,k)

D(p,k)

∂f 0

∂v

. (14.122)

If an inverse transform were to be performed on f ̃ 1 , then three issues would have to be
taken into account. These are (i) a ballistic term∼exp(ikvt)associated with the pole
due top+ikvin the denominator, (ii) a Landau-damped plasma oscillation due to the
root ofD(p,k)in the denominator, and (iii) the direct inverse transform of the numerator
̃φext(p,k)which by itself would reproduce the original imposed potential, but which will

be modified due to the other factors.

Fourier-Laplace transform of the non-linear equation As discussed in Eq.(14.1)
nonlinear quantities provide sum and difference frequencies. For example, anoscillation at
frequencyωwould result from the nonlinear product of a term oscillating atω−ω′and a
term oscillating atω′since(ω−ω′)+ω′=ω.This can be written in a more formal and
more general way using convolution integrals for both Fourier and Laplace transforms.
Since Laplace transforms will be used for the temporal dependence and since theright
hand side of Eq.(14.114) involves a product term, it is necessary to considerthe Laplace
transform of a product,

L(g(t)h(t)) =

∫∞

0

g(t)h(t)e−ptdt

=

∫∞

0

[

1

2 πi

∫b+i∞

b−i∞

̃g(p′)ep

′t

dp′

]

h(t)e−ptdt

=

1

2 πi

∫b+i∞

b−i∞

̃g(p′)

(∫∞

0

h(t)e(p

′−p)t

dt

)

dp′

=

1

2 πi

∫b+i∞

b−i∞

̃g(p′) ̃h(p−p′)dp′. (14.123)

This means that the Laplace transform with argumentpresults from all possible products
of ̃g(p′)with ̃h(p−p′).Ifexp(α 1 t)is the fastest growing term ing(t)andexp(α 2 t)is the
fastest growing term inh(t)thenb>α 1 is required in order for the Laplace transform of
g(t)to be defined. Furthermore, in order for the Laplace transform ofh(t)to be defined, it
is necessary to have Rep−Rep′>α 2 orRep>Rep′+α 2 which impliesRep>b+α 2.
These requirements can be summarized asRep−α 2 >b>α 1.
Using

g ̃(k) =

∫∞

−∞

g(x)e−ikxdx (14.124a)

g(x) =

1

2 π

∫∞

−∞

̃g(k)e+ikxdk, (14.124b)