Fundamentals of Plasma Physics

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)