Long-wavelength instabilities of three-dimensional patterns
T. K. Callahan
1,2*
and E. Knobloch
1†
1
Department of Physics, University of California, Berkeley, California 94720
2
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Received 24 May 1999; revised manuscript received 24 April 2000; published 28 August 2001
Long-wavelength instabilities of steady patterns, spatially periodic in three dimensions, are studied. All
potentially stable patterns with the symmetries of the simple-, face-centered- and body-centered-cubic lattices
are considered. The results generalize the well-known Eckhaus, zigzag, and skew-varicose instabilities to
three-dimensional patterns and are applied to two-species reaction-diffusion equations modeling the Turing
instability.
DOI: 10.1103/PhysRevE.64.036214 PACS numbers: 89.75.Kd, 47.20.Ky, 47.54.+r
I. INTRODUCTION
Formation of structure via spontaneous symmetry-
breaking bifurcations is a topic of much current interest 1.
Despite this, little work has been done on pattern formation
in three dimensions, i.e., in systems that are translation in-
variant in three dimensions. The recent experimental discov-
ery of the Turing instability 2,3 provides one motivation for
extending the existing theory for two-dimensional patterns to
three dimensions. The Turing instability arises in reaction-
diffusion systems and the characteristic wavelength of the
pattern that is produced is intrinsic, i.e., it depends only upon
the reaction rates, concentrations, and diffusivities of the
chemicals involved, and not upon any externally imposed
length scale. Thus if the dimensions of the experimental ap-
paratus are much larger than the intrinsic length scale, the
instability can develop free from the influence of boundaries
and produce truly three-dimensional patterns. Other systems
exhibiting pattern formation in three dimensions include
block copolymer melts 4 and parametric oscillators in op-
tics 5. In the former a polymer consisting of long blocks of
different monomers starts in a spatially uniform state. As
time progresses, the different monomer types self-segregate
into distinct domains, frequently with spatial periodicity.
Both systems produce spatial structures that are similar to
those predicted by the general theory for three-dimensional
patterns on spatially periodic cubic lattices 6,7. This analy-
sis focuses on the vicinity of a steady state instability in
generic systems with translation invariance in three dimen-
sions, and determines the types of spatially periodic patterns
with the symmetry of the different types of cubic lattices and
their stability properties with respect to perturbations on
these lattices, but other types of perturbations have not been
considered. In particular the stability properties of the pre-
dicted stable states with respect to long-wavelength pertur-
bations remain unknown. In two dimensions such perturba-
tions are known to be important in so far as they are involved
in the various instabilities such as the Eckhaus, zigzag, and
skew-varicose instabilities that restrict the possible wave-
length of the pattern. These instabilities, originally identified
in the stability theory for convective rolls, are generic in the
sense that they destabilize roll-like states in all continuum
systems with Euclidean symmetry in two dimensions.
Calculations of this type provide useful information even
about systems that do not strictly satisfy all the hypotheses
used to construct the theory. The Turing instability is a case
in point. Actual experiments on Turing structures involve
concentration gradients of the feed chemicals, which neces-
sarily break both the homogeneity and the isotropy of the
system. This is the case, for example, in the experiments
reported in Refs. 2 and 3 in which the feed gradient was
imposed in the plane of the observed patterns and not per-
pendicular to it as in subsequent experiments. Despite this
the hexagons that were predicted for a homogeneous system
were still found. Thus a study of the corresponding problem
in three dimensions should likewise produce useful results.
In fact, because the characteristic length scale of the insta-
bility is so much smaller than all the external dimensions, the
authors of Refs. 2 and 3 conclude that the observed struc-
tures must in fact be three-dimensional and that the top-view
hexagonal pattern is actually a two-dimensional projection of
a body-centered-cubic bcc structure. More recently, two-
dimensional black-eye patterns in reaction-diffusion systems
have also been explained in terms of sections of a three-
dimensional bcc structure 8. The block copolymer melts
investigated in Ref. 4 do not suffer from these limitations
of the theory.
It is important, therefore, that the methods used to study
pattern formation and stability in two dimensions be ex-
tended to the three-dimensional case. For patterns on a spa-
tially periodic three-dimensional lattice the equivariant bifur-
cation theory approach has led to an almost complete
description of the possible stationary patterns on the simple-
cubic sc, face-centered-cubic fcc, and bcc lattices and
their stability properties with respect to all perturbations on
these lattices 6,7. Near onset these patterns are described
by real functions of the form
x =
i =1
N
z
i
e
ik
i
• x
+c.c.+n.l.t., 1.1
where | k
i
| =k
c
, i =1, . . . , N and N =3, 4, and 6, respec-
tively. The shorthand n.l.t. represents terms that are nonlinear
*Email address: timcall@math.lsa.umich.edu
†
Email address: knobloch@physics.berkeley.edu
PHYSICAL REVIEW E, VOLUME 64, 036214
1063-651X/2001/643/03621425/$20.00 ©2001 The American Physical Society 64 036214-1