Fundamentals of Plasma Physics

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.