434 Chapter 15. Wave-wave nonlinearities
Figure 15.2: Photon at frequencyω 3 decaying into a photon at frequencyω 1 and a photon
at frequencyω 2.
An additional conservation equation can be obtained by subtracting the last of
Eqs.(15.16) from the sum of the first two to obtain
θ ̇ = δ ̇ 1 +δ ̇ 2 −δ ̇ 3
=
(
λA 2 A 3
4 mA 1 ω 1
+
λA 1 A 3
4 mA 2 ω 2
−
λA 1 A 2
4 mA 3 ω 3
)
cosθ
=
(
A ̇ 1
A 1
+
A ̇ 2
A 2
+
A ̇ 3
A 3
)
cosθ
sinθ
(15.23)
and then integrating to find
A 1 A 2 A 3 cosθ=const. (15.24)
We now consider some solutions to the system of equations given by Eqs.(15.15b)-
(15.15c). Suppose thatA 3 >>A 2 ,A 1 initially. In this case Eq.(15.15c) givesA 3 ≃const.
Solving Eq.(15.15b) forA 1 and substituting the result in Eq.(15.15a) gives
1
sinθ
d
dt
(
1
sinθ
dA 2
dt
)
=
λ^2 A^23
16 m^2 ω 1 ω 2
A 2 (15.25)
which has exponentially growing solutions ifω 1 ω 2 > 0 .By defining
τ=
∫t
0
dt′sinθ(t′) (15.26)