86
9 Natural Measuring-Rods and Clocks
q
III
o (x, t) =
2a
π
arctan
sin
c o t
λ o
√
2
cosh
x
λ o
√
2
=
2a
π
arctan
sin(( o t)
cosh
x
λ o
√
2
,
(100a)
or, if we insert all the parameters,
q
III
o (x, t) =
2a
π
arctan
sin
π D
aρ
t
cosh
π D
aσ
x
.
(100b)
We leave the reader with the option of checking that the breather solution (100) really
does fulfil the sine-Gordon equation (95).
With q
III
o (x, t), we have received a line form that can only exist when in permanent
motion. What does this line form look like geometrically? Because of the hyperbolic
cosine in the denominator of (100), the dislocation line moves at any arbitrary time t
not only for x −→ +∞, but also for x −→ −∞ in the stabile position of equilibrium
q = 0 ; see Fig. 9.3.
For every fixed x the line oscillates with an angular velocity o ,
-
6
6
6
6
6
6
6
6
x
q
t 1
t 2
t 3
a
2
λ o
Fig. 9.3 Breather solution (100a), q III
o (x, t) =
2a
π arctan
sin
co t
λo
√
2
cosh
x
λo
√
2
, for t 1 = 0, t 2 =
√
2 πλo
2co , t 3 =
3
√
2 πλo
2co . (The arrows show the direction of motion of the oscillating dislocation line during its
passing through the zero position)