Fundamentals of Plasma Physics

432 Chapter 15. Wave-wave nonlinearities

Usingω 2 −ω 3 =−ω 1 ,δ 2 −δ 3 =θ−δ 1 ,ω 1 −ω 3 =−ω 2 ,δ 1 −δ 3 =θ−δ 2 and
discarding non-resonant terms, these become

A ̇ 1 sin(ω 1 t+δ 1 )+A 1 δ ̇ 1 cos(ω 1 t+δ 1 ) = λA^2 A^3

4 mω 1

cos(θ−ω 1 t−δ 1 )

A ̇ 2 sin(ω 2 t+δ 2 )+A 2 δ ̇ 2 cos(ω 2 t+δ 2 ) = λA^1 A^3

4 mω 2

cos(θ−ω 2 t−δ 2 )

A ̇ 3 sin(ω 3 t+δ 3 )+A 3 δ ̇ 3 cos(ω 3 t+δ 3 ) = λA^1 A^2

4 mω 3

cos(θ+ω 3 t+δ 3 )

(15.13)

or
[ ̇
A 1 sin(ω 1 t+δ 1 )
+A 1 δ ̇ 1 cos(ω 1 t+δ 1 )

]

=

λA 2 A 3

4 mω 1

[

cosθcos(ω 1 t+δ 1 )

+sinθsin(ω 1 t+δ 1 )

]

[

A ̇ 2 sin(ω 2 t+δ 2 )

+A 2 δ ̇ 2 cos(ω 2 t+δ 2 )

]

=

λA 1 A 3

4 mω 2

[

cosθcos(ω 2 t+δ 2 )

+sinθsin(ω 2 t+δ 2 )

]

[ ̇

A 3 sin(ω 3 t+δ 3 )

+A 3 δ ̇ 3 cos(ω 3 t+δ 3 )

]

=

λA 1 A 2

4 mω 3

[

cosθcos(ω 3 t+δ 3 )

−sinθsin(ω 3 t+δ 3 )

]

.

(15.14)

Matching the time-dependent sine and cosine terms on both sides gives

A ̇ 1 = λA^2 A^3

4 mω 1

sinθ (15.15a)

A ̇ 2 = λA^1 A^3

4 mω 2

sinθ (15.15b)

A ̇ 3 = −λA^1 A^2

4 mω 3

sinθ (15.15c)

and

A 1 δ ̇ 1 =

λA 2 A 3

4 mω 1

cosθ

A 2 δ ̇ 2 =

λA 1 A 3

4 mω 2

cosθ

A 3 δ ̇ 3 =

λA 1 A 2

4 mω 3

cosθ. (15.16)

These equations can be used to establish various conservation relations. Multiplying
Eqs.(15.15) byω^2 jAj, and summing gives

1

2

d

dt

(

ω^21 A^21 +ω^22 A^22 +ω^23 A^23

)

=

(ω 1 +ω 2 −ω 3 )λA 1 A 2 A 3

4 m

sinθ=0

so
ω^21 A^21 +ω^22 A^22 +ω^23 A^23 =const. (15.17)
This shows that the sum of the energies of the three modes is constant, but allows for energy
transfer back and forth between the three modes.