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