Fundamentals of Plasma Physics

424 Chapter 14. Wave-particle nonlinearities

andkb→−kb.The inverse Laplace transform gives

φ ̃upper 2 (t,x) = φaφb

4(ka−kb)^2

e

m

∫b+i∞

b−i∞

dp

2 πi

1

D(p,ka−kb)

∫

dv

∫b+i∞

b−i∞

dp′

2 π

×

ka

(p+i(ka−kb)v)^2 D(p′,−ka)

ikb

p−p′−ikbv

×

ept−(p−p

′)τ+i(ka−kb)x

D(p−p′,−kb)

∂f ̄ 0

∂v

. (14.142)

We first consider thep′integral and only retain the ballistic term due to the pole(p−p′)−
ikbv.Evaluating the residue associated with this pole givesp′=p−ikbvso

̃φupper 2 (t,x) = − φaφb

4(ka−kb)^2

e

m

∫b+i∞

b−i∞

dp

1

D(p,ka−kb)

∫

dv

i(ka−kb)

(p+i(ka−kb)v)^2

ikakb

D(p−ikbv,−ka)

ept−ikbvτ+i(ka−kb)x

D(ikbv,−kb)

∂f ̄ 0

∂v

.

(14.143)

The pole at(p+i(ka−kb)v)^2 is second-order and so the rule for a second-order residue
should be used, i.e.,
∮
dp

1

(p−p 0 )^2

g(p)=πi lim

p→p 0

d

dp

g(p). (14.144)

Thus, we obtain

φ 2 (t,x) = −

φaφbπi

4(ka−kb)^2

e

m

∫

dv lim

p→i(kb−ka)v

d

dp

(

i(ka−kb)

D(p,ka−kb)

ikakb

D(p−ikbv,−ka)

ept−ikbvτ+i(ka−kb)x

D(ikbv,−kb)

∂f ̄ 0

∂v

)

.

(14.145)

The strongest dependence onpis in the exponentialeptand so retaining only the contribu-
tion of this term to thed/dpoperator gives

̃φupper 2 (t,x) = − πiφaφb

4(ka−kb)^2

e

m

∫

dv lim

p→i(kb−ka)v

(

ti(ka−kb)

D(p,ka−kb)

ikakb

D(p−ikbv,−ka)

ept−ikbvτ+i(ka−kb)x

D(ikbv,−kb)

∂f ̄ 0

∂v

)

=

π

4

kakb

(ka−kb)^2

eφaφb

m

ei(ka−kb)x

∫

dv

i(ka−kb)t

D(i(kb−ka)v,ka−kb)D(−ikav,−ka)

ei(kb−ka)vt−ikbvτ

D(ikbv,−kb)

∂f ̄ 0

∂v

.

(14.146)