446 Chapter 15. Wave-wave nonlinearities
where the sign of the exponent is chosen to be consistent with the convention that a single
Fourier mode has the formexp(−iωt). The Fourier transform off(t)e±iω^3 twill therefore
be ∫
dt(
f(t)e±iω^3 t)
eiωt=f(ω±ω 3 ) (15.95)and the Fourier transform of∂f/∂twill be
∫
dt(
∂
∂tf(t))
eiωt=−iω∫
dtf(t)eiωt=−iωf(ω). (15.96)Application of Eqs.(15.95) and (15.96) to Eqs.(15.93) gives
(
−ω^2 −2iωΓ 2 +ω^22) ̃
E 2 (ω) = −λZ 3
2[
ψ ̃(ω+ω 3 )+ψ ̃(ω−ω 3 )]
(
−ω^2 −2iωΓ 1 +ω^21)
ψ ̃(ω) = −λZ^3
2[
E ̃ 2 (ω+ω 3 )+E ̃ 2 (ω−ω 3 )]
.(15.97)
By defining a generic linear dispersion relation
εj(ω)=−ω^2 −2iωΓj +ω^2 j, (15.98)these equations can be written as
ε 2 (ω)E ̃ 2 (ω) = −λZ 3
2[
ψ ̃(ω+ω 3 )+ψ ̃(ω−ω 3 )]
(15.99)
ε 1 (ω)ψ ̃(ω) = −λZ 3
2[
E ̃ 2 (ω+ω 3 )+E ̃ 2 (ω−ω 3 )]
. (15.100)
The frequencyωin Eq.(15.100) can be replaced byω±ω 3 so that
ψ ̃(ω±ω 3 )=− λZ^3
2 ε 1 (ω±ω 3 )[
E ̃ 2 (ω)+E ̃ 2 (ω± 2 ω 3 )]
which is then substituted into Eq.(15.99) to obtain
ε 2 (ω)E ̃ 2 (ω)=(
λZ 3
2) 2 [ ̃
E 2 (ω)+E ̃ 2 (ω+2ω 3 )
ε 1 (ω+ω 3 )+
E ̃ 2 (ω)+E ̃ 2 (ω− 2 ω 3 )
ε 1 (ω−ω 3 )]
.
(15.101)
The termsE ̃ 2 (ω± 2 ω 3 )can be discarded since they are non-resonant and therefore of
insignificant amplitude. What remains is
ε 2 (ω)=(
λZ 3
2) 2 [
1
ε 1 (ω+ω 3 )+
1
ε 1 (ω−ω 3 )]
(15.102)
which is called thenon-linear dispersion relation(Nishikawa 1968b, Nishikawa 1968a).
The non-linear dispersion relation is investigated by first writing
ω=x+iy (15.103)where it is assumed that
x≃ω 2 (15.104)