42
1 Mechanics of a One-Dimensional Crystal
where l is a parameter determining the crowdion width
l =
s
2
0
−V
2
ω
0
=
s
0
ω
0
1 −
V
s
0
2
. (1.6.13)
When V approaches th e limiting velocity s
0
, the crowdion width experiences a rela-
tivistic reduction. In the limit V = s
0
there a rises a step with δu = a.AtV = 0 a
fixed inflection of width l
0
= s
0
/ω
0
remains in the crystal.
Since a homogeneous translation onto the crystal period is a symmetry transforma-
tion, the physical state of a 1D crystal far from the point x = x
0
(|x − x
0
|x
0
)is
similar to that in the absence of a crowdion. The derivative du/ dx coinciding with a
linear chain deformation is evidence of the change in the state of a one-dimensional
system in the presence of a crowdion. It follows from (1 . 6.12) (in in itial dimensional
quantities)
du
dx
= −
a
πl
1
cosh
x − x
0
l
. (1.6.14)
The localized d eformation (1.6.14) moving in the crystal has th e form of a solitary
wave and is another example of a strain soliton in a 1 D crystal. Finally, coming back
to a dimensional displacement and to a complete dependence on the coordinates and
time in (1.6.12), we get the final expression for the displacements
u(x, t)=
2a
π
arctan
exp
−ω
0
x −Vt
s
2
0
−V
2
. (1.6.15)
The perturbation described by this formula moves with velocity V (necessarily less
than that of a sound s
0
). This makes it different from the shock-wave perturbation.
The velocity is determined by the total energy associated with this perturbation.
The last observation will be supported with a certain qualitative argument of general
character. It follows from a comparison of the properties of solitons of the two types
considered by us. Irrespective of the character of nonlinearity that generates a soliton,
the value of its limiting velocity is completely determined by th e dispersio n law of
harmonic vibrations that can exist in the system under study.
If a plot of the dispersion law of linear vibrations is convex upwards similar to
the plot of the dispersion law (1.2.13), it is characterized by some maximum phase
velocity s
0
. The velocity of a soliton (if it arises) will exceed s. If a plot of the
dispersion law shows convexity downwards, as in the case of a sine-Gordon equation,
it is characterized by the minimum p hase velocity s
0
, and the velocity of the existing
soliton should be less than s
0
. This conclusion will be proved in the following.