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9 Wave Equations
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way under mild conditions on the initial wave function. By this limit process the
Vlasov equation of classical mechanics (see Chapter 1) is obtained:
n
w t + ξ · grad x w −grad x V(x) · grad ξ w = 0, x and ξ ∈ R , (9.21)
where the positive measure w denotes a weak limit point of the sequence W as
ε → 0. We remark that the equation (9.21) can be interpreted in the following
way: the measure w is constant along the characteristics of the linear hyper-
bolic equation (9.21), which are precisely the Newtonian trajectories of classical
mechanics:
dx dξ
= ξ , = −grad x V(x) . (9.22)
dt dt
This dynamical system is a formulation of Newton’s second law, which states
that force (here given by the field −grad x V(x)) is equal to mass (has been scaled
2
to 1) times acceleration d x 2 . In this way the fundamental law of classical mechan-
dt
ics is recovered from the quantistic Schr¨ odinger equation. This limiting process,
here carried out formally, has been justified rigorously in the references [13], [4].
Many important wave phenomena are inherently nonlinear and cannot be
described by linear wave equations. A typical example is the motion of ocean
waves before and after breaking.
Here we mention the cubically nonlinear Schrödinger equation as an often
used model for nonlinear wave motion:
ε 2
2
n
iεu t = − Δu + V(x)u + a|u| u , x ∈ R , t ∈ R . (9.23)
2
If the real parameter a is positive, the equation is defocusing, i.e. all energy
contributions are non-negative, and if a is negative, the equation is focusing, i.e.
the energy contribution stemming from the nonlinearity (the so called inter-
action energy) is non-positive. The cubic nonlinearity models binary particle
interactions,higherorderinteractionsareneglectedhere.ThecubicSchrödinger
equation has various applications, ranging from tsunami modeling to a quan-
10
tistic phenomenon called Bose–Einstein condensation ,represented by atem-
perature phase transition in the nano-Kelvin range, occurring in boson gases,
leading to the formation of a ‘super-atom’. In the latter case V is typically a har-
monic (quadratic) potential representing the laser confinement of the conden-
sate, and the nonlinear Schrödinger equation is referred to as Gross–Pitaevski
equation [15], [1].
We remark that the mathematical features of focusing and defocusing non-
linear Schrödinger equations are totally different. Focusing nonlinearities often
lead to finite time blow-up of solutions, depending on the space dimension, the
polynomial order of the nonlinearity and the initial datum. In this case the posi-
tion density ρ = ρ(x, t) features a concentration (i.e. formation of a Dirac mass)
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http://www.colorado.edu/physics/2000/bec/