76
8 The sine-Gordon Equation of a Dislocation
∂
2
∂
x
λ o
2 q(x, t) −
∂
2
∂
t
τ o
2 q(x, t) = sin
q(x, t)
.
(92)
Using the dimensionless coefficient of measure x for the determination of space and
the dimensionless coefficient t for the determination of time,
x =
x
λ o
,
t =
t
τ o
,
⎫
⎪ ⎬
⎪ ⎭
(93)
the following is developed,
∂
2
∂x 2 q
λ o x, τ o t
−
∂
2
∂t 2 q
λ o x, τ o t
= sin
q(λ o x, τ o t)
.
(94)
In physics, it is normal to remove uncomfortable parameters in equations in such a
fashion that the units of measure, in which the quantities found in the equations are
defined, are suitably assimilated. Only the product out of the coefficient of measure
and the unit of measure can be determined. It is of no difference if we either use 1
cm, or 10
8 Å, if we use either 3600 s, or if we use 1 h, 1 cm = 10
8 Å, 3600 s = 1 h etc.
For a further simplification of Eq. (94), we agree to measure all distance in multiples
of λ o ; in other words, we measure in gm (the distance λ o then has the coefficient of
measure 1) and all time intervals in multiples of τ o ; in other words, we measure in
gs (the time τ o then has the coefficient of measure 1). Using this agreement, we get
a ‘simplified’ version of the sine-Gordon equation for a function or, as one says in
physics, for a (dimensionless) field q = q(x, t) in dependence of the dimensionless
variables x and t according to
∂
2 q
∂x 2 −
∂
2 q
∂t 2 = sin q,
q = q(x, t).
⎫
⎬
⎭
sine-Gordon equation
(95)
Equation (95) is also the sine-Gordon equation of a dislocation. It is purely a matter
of taste whether we use (88) or (95) for our calculations. Equation (95) is less tedious
in writing terms. One does however have to take into consideration that in (95) all
units of length are measured in gm and all units of time in gs, respectively, (see
Eqs. (91) and (92)) and that q is according to (89) the ratio of the real deflection q
of a dislocation to the lattice parameter a , multiplied by 2π . Once one has found a
solution q = q(x, t) of Eq. (95), then the corresponding solution q(x, t) of Eq. (88)
is
q(x, t) =
a
2π
q
x
λ o
,
t
τ o
=
a
2π
q
x
λ o
,
c o t
λ o
(96)
8 The sine-Gordon Equation of a Dislocation
∂
2
∂
x
λ o
2 q(x, t) −
∂
2
∂
t
τ o
2 q(x, t) = sin
q(x, t)
.
(92)
Using the dimensionless coefficient of measure x for the determination of space and
the dimensionless coefficient t for the determination of time,
x =
x
λ o
,
t =
t
τ o
,
⎫
⎪ ⎬
⎪ ⎭
(93)
the following is developed,
∂
2
∂x 2 q
λ o x, τ o t
−
∂
2
∂t 2 q
λ o x, τ o t
= sin
q(λ o x, τ o t)
.
(94)
In physics, it is normal to remove uncomfortable parameters in equations in such a
fashion that the units of measure, in which the quantities found in the equations are
defined, are suitably assimilated. Only the product out of the coefficient of measure
and the unit of measure can be determined. It is of no difference if we either use 1
cm, or 10
8 Å, if we use either 3600 s, or if we use 1 h, 1 cm = 10
8 Å, 3600 s = 1 h etc.
For a further simplification of Eq. (94), we agree to measure all distance in multiples
of λ o ; in other words, we measure in gm (the distance λ o then has the coefficient of
measure 1) and all time intervals in multiples of τ o ; in other words, we measure in
gs (the time τ o then has the coefficient of measure 1). Using this agreement, we get
a ‘simplified’ version of the sine-Gordon equation for a function or, as one says in
physics, for a (dimensionless) field q = q(x, t) in dependence of the dimensionless
variables x and t according to
∂
2 q
∂x 2 −
∂
2 q
∂t 2 = sin q,
q = q(x, t).
⎫
⎬
⎭
sine-Gordon equation
(95)
Equation (95) is also the sine-Gordon equation of a dislocation. It is purely a matter
of taste whether we use (88) or (95) for our calculations. Equation (95) is less tedious
in writing terms. One does however have to take into consideration that in (95) all
units of length are measured in gm and all units of time in gs, respectively, (see
Eqs. (91) and (92)) and that q is according to (89) the ratio of the real deflection q
of a dislocation to the lattice parameter a , multiplied by 2π . Once one has found a
solution q = q(x, t) of Eq. (95), then the corresponding solution q(x, t) of Eq. (88)
is
q(x, t) =
a
2π
q
x
λ o
,
t
τ o
=
a
2π
q
x
λ o
,
c o t
λ o
(96)