430 Chapter 15. Wave-wave nonlinearities
hole in the plasma density. The pump wave can dig itself a channel in the plasma and then
this channel can guide, focus, and even trap the pump wave.
Solitons
Plasma waves are dispersive in general so, after propagating some distance, an initially
sharp pulse will become less sharp because the various components of the Fourier spec-
trum constituting the pulse propagate at different phase velocities. However, it is found in
certain situations that pulses with an amplitude exceeding some critical value will prop-
agate indefinitely without broadening even though the medium is nominally dispersive.
This high amplitude non-dispersive pulse is called a soliton and its existence results from
nonlinearities competing against dispersion in a way such that the two effects cancel each
other.
15.2 Manley-Rowe relations
Before investigating wave-wave nonlinearities, it is instructive toexamine a closely
related, but simpler system consisting of three non-linearly coupled harmonic oscillators
(Manley and Rowe 1956). This simple system consists of a particle of massmmoving in
a three dimensional space with dynamics governed by the Hamiltonian
H=1
2 m(
P 12 +P 22 +P 32
)
+
1
2
(
κ 1 Q^21 +κ 2 Q^22 +κ 3 Q^23)
+λQ 1 Q 2 Q 3. (15.1)In theλ→ 0 limit this Hamiltonian describes three independent harmonic oscillatorshav-
ing respective frequenciesω 1 =
√
κ 1 /m,ω 2 =√
κ 2 /mandω 3 =√
κ 3 /m.In the more
general case of finiteλ, Hamilton’s equations for theP 1 ,Q 1 conjugate coordinates are
P ̇ 1 = −∂H
∂Q 1
=−κ 1 Q 1 −λQ 2 Q 3 , (15.2a)Q ̇ 1 = ∂H
∂P 1
=P 1 /m (15.2b)with similar equations for theP 2 ,Q 2 andP 3 ,Q 3 conjugates. Using relationships such as
Q ̈ 1 =P ̇ 1 /m, three coupled oscillator equations result, namely
Q ̈ 1 +ω^21 Q 1 = −λ
mQ 2 Q 3
Q ̈ 2 +ω^22 Q 2 = −λ
mQ 1 Q 3
Q ̈ 3 +ω^23 Q 3 = −λ
mQ 1 Q 2. (15.3)
For smallλ, each oscillator may be assumed to oscillate nearly independently so that during
each cycle of a given oscillator, the oscillator experiences only slight amplitude and phase
changes due to the nonlinear coupling with the other two oscillators. Thus,approximate
solutions for the oscillators can be written as
Q 1 (t) = A 1 (t)cos[ω 1 t+δ 1 (t)]
Q 2 (t) = A 2 (t)cos[ω 2 t+δ 2 (t)]
Q 3 (t) = A 3 (t)cos[ω 3 t+δ 3 (t)] (15.4)