Waves, Instabilities, and Structures in Excited and Ionized Gases - Fundamentals of Ionized Gases - page 324

one can write the particle velocity in the form v D v g C v 0 ,where v g is the group

velocity and v 0 is the particle velocity in the frame of reference where the wave is at
rest. Because v 0 v g ,wehave

@ v 0
@

C v g @ v 0
@

F
m D

0

t x

in the linear approximation, where the last term is a linear operator with respect
to v 0 . In the harmonic approximation we have v 0 t ). We seek the

form of the operator F / m that leads to dispersion relation (5.107) in this approach.
This gives

exp( ikx i ω

C v g @ v 0
@ x 3

0

3

v 0 @ v

@

r 0 @ C D

0.

t x @

The last term takes into consideration a weak dispersion of long-wave oscillations.
To account for nonlinearity of these waves, we analyze the second term of this
equation. In the linear approach we replace the particle velocity v by the group
velocity v g . Returning this term to its initial form, that is, taking into account weak
nonlinear effects, we obtain

3

@ v
@

C v @ v
@ x C v g r 0 @

v

D 0 . (5.109)

x 3

t @

This equation is called the Korteweg-de Vries equation [67] and was originally ob-
tained in the analysis of wave propagation processes in shallow water.
The Korteweg-de Vries equation accounts for nonlinearity and weak dispersion
simultaneously, and therefore serves as a convenient model equation for the anal-
ysis of nonlinear dissipative processes. As applied to plasmas, it describes propa-
gation of long-wavelength waves in a plasma for which the dispersion relation is
given by (5.107).

5.4.3
Solitons

The Korteweg-de Vries equation has solutions that describe a class of solitary
waves. These waves are called solitons and conserve their form in time. It fol-
lows from dispersion relation (5.107) that short waves propagate more slowly than
long waves, but nonlinear effects compensate for the spreading of the wave. We
now show that this property holds true for waves that are described by the Ko-
rteweg-de Vries equation. We consider a wave of velocity u .Thespatialandtem-
poral dependence for the particle velocity has the form v D f ( x ut ). This gives

@ v / @ t D u @ v / @ x , and the Korteweg-de Vries equation is transformed to the form

u ) d

dx C v g r 0 d 3 v

dx 3

( v D 0 . (5.110)