Problem 1
(a) Linearization is trivial (just like that of the standard SwiftHohenberg equation) and we get
so we immediately get the growth rate
of an infinitesimal solution
δ
u
=
e
iqx
e
σ
t
.
For
u
b
= 0 the real part of the growth rate is the same as
for the standard SHE:
so expanding it about the bifurcation point
r
c
= 0,
q
c
= 1 (i.e., it’s a typeI instability) we get
Since the bifurcation occurs at
r
c
= 0 we can just take
ε
=
r
, which immediately determines the
characteristic time scale
τ
= 1 and correlation
length
ξ
c
= 2 by comparing with the canonical
expression for a typeI instability. For nonzero
s
, the instability is oscillatory, i.e., typeI
o
with
critical frequency
ω
c
= Im
σ
q

q = 1
=
s
.
(b) Substituting the growth rate into the solution of the linearized equation we get
which by a change of variables
reduces to the corresponding expression for a
standard SHE. This change is equivalent to a translation of the growing pattern with speed
s
! It is
trivial to check that this change of variables reduces the modified SHE to the standard form.
,
3
)
1
(
2
2
2
u
u
u
u
s
u
r
u
b
x
x
t
δ
δ
δ
δ
δ
−
∂
+
−
∂
+
=
∂
isq
q
u
r
b
q
+
−
−
−
=
2
2
2
)
1
(
)
3
(
σ
,
)
1
(
Re
2
2
q
r
q
q
−
−
=
=
σ
γ
.
)
1
(
2
)
1
(
)
1
(
2
2
2
2
L
+
−
−
=
+
−
−
=
q
r
q
q
r
q
γ
]
)
)
1
(
exp[(
)]
(
exp[
2
2
t
q
r
st
x
iq
e
e
u
t
iqx
−
−
+
=
=
σ
δ
st
x
y
x
+
=
→
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