450 Chapter 15. Wave-wave nonlinearities
By defining the normalized variables
τ = ωpet/ 2
χ =
A
2
√
κTe/me
ξ = xωpe/c
η = 2Γ/ωpe (15.128)
Eq.(15.127) can be put in the standardized form
i
∂χ
∂τ
+iηχ+|χ|^2 χ+∇^2 ξχ=0; (15.129)
this is called a non-linear Schrödinger equation since ifη= 0,this equation resembles a
Schrödinger equation where|χ|^2 plays the role of a potential energy.
In order to exploit this analogy, we recall the relationship between the Schrödinger
equation and the classical conservation of energy relation for a particle in a potential well
V(x).According to classical mechanics, the sum of the kinetic and potential energies gives
the total energy, i.e.,
p^2
2 m
+V(x)=E. (15.130)
However, in quantum mechanics, the momentum and the energy are expressed as spatial
and temporal operators,p=−i∇andE= i∂/∂twhich act on a wave functionψso
that Eq.(15.130) becomes
−
^2
2 m
∇^2 ψ+Vψ=i
∂ψ
∂t
(15.131)
or, after re-arrangement,
i
∂ψ
∂t
−Vψ+
^2
2 m
∇^2 ψ=0. (15.132)
Equation (15.130) shows that a particle will be trapped in a potential well ifV(±∞)>
E > VminwhereVminis the minimum value ofV.From the quantum mechanical point
of view,|ψ|^2 is the probability of finding the particle at positionx.Thus, existence of
solutions to Eq.(15.132) localized to the vicinity ofVminis the quantum mechanical way of
stating that a particle can be trapped in a potential well. Comparison of Eqs.(15.129) and
(15.132) shows that−
∣
∣A ̄
∣
∣^2 plays the role ofVand so a local maximum of
∣
∣A ̄
∣
∣^2 should act
as an effective potential well. This makes physical sense because Langmuir waves reflect
from regions of high density and the amplitude-dependent ponderomotive force digsa hole
in the plasma. Thus, regions of high wave amplitude create a density depression and the
Langmuir wave reflects from the high density regions surrounding this density depression.
The Langmuir wave then becomes trapped in a depression of its own making. Formation of
this depression can be an unstable process because if a wave is initiallytrapped in a shallow
well, its energy|χ|^2 will concentrate at the bottom of this well, but this concentration of
|χ|^2 will make the well deeper and so concentrate the wave energy into a smaller region,
making|χ|^2 even larger, and so on.