Phys 7268
Midterm exam
Due:
3/10/11
Problem 1
Generalizations of SwiftHohenberg equation.
The form of the SwiftHohenberg equation is determined by
the symmetries of the system, such as translational and rotational symmetries.
For instance, breaking the parity
symmetry leads to an extra term
s∂
x
u
with
s
an arbitrary constant:
∂
t
u
=
ru
+
s∂
x
u

(1 +
∂
2
x
)
2
u

u
3
.
(a) Compute the growth rate and determine the type of instability (i.e., I
s
, III
o
, etc.)
of the trivial solution
u
= 0 which arises for different values of parameter
s
.
Furthermore, determine the coherence length
ξ
c
, the
characteristic timescale
τ
and, if the instability is oscillatory, the characteristic frequency
ω
c
.
(b) By substituting the growth rate into the linear solution determine the meaning of the new term.
Can you
show that by a suitable change of variables the above equation can be reduced to a standard SwiftHohenberg
equation?
(b) Determine the stability of the two nontrivial uniform solutions to the above equation. If it happens that these
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 Spring '11
 Staff
 Elementary algebra, Parametric equation, Dictyostelid, SwiftHohenberg equation

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