Environmental Engineering Reference
In-Depth Information
Assuming the perturbation to be zero at large distances from the wave, that is
v
D
0
and
d
2
v
/
dx
2
D
0at
x
!1
, this equation reduces to
v
g
r
0
d
2
2
v
dx
2
v
v
D
u
,
2
with the solution
3
u
cosh
2
x
2
r
0
!
.
s
u
v
g
v
D
(5.111)
The wave described by (5.111) is concentrated in a limited spatial region and does
not diverge in time. The wave becomes narrower with an increase of its ampli-
tude, that is, the wave is concentrated in a restricted region, and its spatial width is
inversely proportional to the square root of the wave amplitude.
In other words, the Korteweg-de Vries equation has stationary solutions describ-
ing a nonspreading solitary wave, or soliton. The amplitude
a
and extension 1/
α
of solitons are such that the value
a
/
2
does not depend on the wave amplitude.
If the initial perturbation is relatively small, evolution of the wave packet leads to
formation of one solitary wave. If the amplitude of an initial perturbation is rela-
tively large, this perturbation in the course of evolution of the system will split into
several solitons. Thus, solitons are not only stable steady-state perturbations in the
system, but they can also play a role in the evolution of some perturbations in a
nonlinear dispersive medium.
α
5.4.4
Langmuir Solitons
The occurrence and propagation of solitons is associated with an electric field that
exists in a plasma as the result of a wave process, and this field confines the per-
turbation to a restricted region. This can be demonstrated using the example of
plasma oscillations. Denoting by
E
(
x
,
t
) the electric field strength of the plasma
oscillations, we have
E
2
8
W
(
x
)
D
π
for their energy density, where the bar means a time average. Assuming the equal-
ity of electron and ion temperatures,
T
e
D
T
i
D
T
, the pressure of a quasineutral
plasma is
p
2
N
e
T
. The plasma pressure is established with a sound velocity that
is greater than the velocity of propagation of long-wave oscillations. Then, because
of the uniformity of plasma pressure at all points of the plasma, we have
D
2
N
(
x
)
T
C
W
(
x
)
D
2
N
0
T
,
(5.112)
where
N
0
is the number density of charged particles at large distances where the
plasma oscillations are absent, and the plasma temperature is assumed to be con-
stant in space.
