Coarsening and frozen faceted structures in the supercritical complex Swift-Hohenberg equation

Abstract

The supercritical complex Swift-Hohenberg equation models pattern formation
in lasers, optical parametric oscillators and photorefractive oscillators.
Simulations of this equation in one spatial dimension reveal that much of
the observed dynamics can be understood in terms of the properties of exact
solutions of phase-winding type. With real coefficients these states take
the form of time-independent spatial oscillations with a constant phase
difference between the real and imaginary parts of the order parameter and
may be unstable to a longwave instability. Depending on parameters the
evolution of this instability may or may not conserve phase. In the former
case the system undergoes slow coarsening described by a Cahn-Hilliard
equation; in the latter it undergoes repeated phase-slips leading either
to a stable phase-winding state or to a faceted state consisting of an array
of frozen defects connecting phase-winding states with equal and opposite
phase. The transitions between these regimes are studied and their location
in parameter space is determined.

Full text

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Eur. Phys. J. D 59, 23–36 (2010)DOI: 10.1140/epjd/e2010-00132-6
Coarsening and frozen faceted structures in the supercritical
complex Swift-Hohenberg equation
L. Gelens and E. Knobloch

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Eur. Phys. J. D 59, 23–36 (2010)
DOI: 10.1140/epjd/e2010-00132-6
Regular Article
THE EUROPEAN
PHYSICAL JOURNAL D
Coarsening and frozen faceted structures in the supercritical
complex Swift-Hohenberg equation
L. Gelens1,aand E. Knobloch2
1Department of Applied Physics and Photonics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
2Department of Physics, University of California, Berkeley, CA 94720, USA
Received 30 November 2009 / Received in final form 7 April 2010
Published online 12 May 2010 – c ? EDP Sciences, Societ` a Italiana di Fisica, Springer-Verlag 2010
Abstract. The supercritical complex Swift-Hohenberg equation models pattern formation in lasers, optical
parametric oscillators and photorefractive oscillators. Simulations of this equation in one spatial dimension
reveal that much of the observed dynamics can be understood in terms of the properties of exact solutions of
phase-winding type. With real coefficients these states take the form of time-independent spatial oscillations
with a constant phase difference between the real and imaginary parts of the order parameter and may be
unstable to a longwave instability. Depending on parameters the evolution of this instability may or may
not conserve phase. In the former case the system undergoes slow coarsening described by a Cahn-Hilliard
equation; in the latter it undergoes repeated phase-slips leading either to a stable phase-winding state or
to a faceted state consisting of an array of frozen defects connecting phase-winding states with equal and
opposite phase. The transitions between these regimes are studied and their location in parameter space
is determined.
1 Introduction
The complex Swift-Hohenberg equation (CSHE) has a
number of applications in nonlinear optics. For example,
the CSHE describes, under appropriate conditions, class A
and C lasers [1–3]. The CSHE also describes nondegener-
ate optical parametric oscillators (OPOs) [4–6], photore-
fractive oscillators [7], semiconductor lasers [8] and pas-
sively mode-locked lasers [9]. More generally, the CSHE
models pattern formation arising from an oscillatory insta-
bility with a finite wavenumber at onset [10,11]. In most of
these examples a trivial spatially homogeneous state loses
stability with increasing forcing but the instability satu-
rates at small amplitude. Systems of this type are referred
to as supercritical. In general, the resulting supercritical
CSHE has complex coefficients and hence time-dependent
solutions. In the present paper, we restrict attention to an
important but special case of this equation, namely the
case of real coefficients and one spatial dimension. This
case admits stationary solutions and for large times these
are the only persistent states of the system. Understand-
ing of this special case is a prerequisite for gaining a deeper
insight into the behavior of the supercritical CSHE with
complex coefficients.
2 Formulation of the problem
The supercritical CSHE with real coefficients in one spa-
tial dimension is given by
ut= ru − (∂2
ae-mail: lgelens@vub.ac.be
x+ k2
0)2u − |u|2u,
(1)
where u is a complex field and the subscripts x and t indi-
cate partial derivatives with respect to position and time.
The equation is fully parametrized by the real parame-
ter r; in the following we find it convenient to retain the
wavenumber k0in the formulation despite the fact that it
can be scaled out. We study equation (1) on a periodic
domain with period L ? 2π/k0 as a function of both r
and k0 using a combination of analytical and numerical
tools [12]. The equation has variational structure with a
Lyapunov function (free energy) FSH given by
?L
Thus
dFSH
dtL
and FSH decreases with time until a stationary state is
reached corresponding to a local minimum of FSH. Both
the evolution and the final state are best visualized in
terms of either the real and imaginary parts of the order
parameter u(x,t) or in terms of its amplitude and phase:
FSH=1
L
0
[−r|u|2+1
2|u|4+ |(∂2
x+ k2
0)u|2]dx.
(2)
= −2
?L
0
|ut|2dx
(3)
u(x,t) = uR(x,t) + iuI(x,t) ≡ R(x,t)eiφ(x,t).
In terms of the latter equation (1) takes the form
(4)
Rt= (r − k4
+12Rxφxφxx− Rφ4
Rφt= −4k2
−4Rxφxxx− 4Rxxxφx+ 6Rφ2
0)R − R3− 2k2
0Rxx+ 2k2
x+3Rφ2
0Rφxx+ 4Rxφ3
0Rφ2
x+ 6Rxxφ2
x(5)
xx+ 4Rφxφxxx−Rxxxx,
x− 6Rxxφxx
xφxx− Rφxxxx.
0Rxφx− 2k2
(6)

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24The European Physical Journal D
0100200 300
−0.2
−0.1
0
0.1
0.2
X
0 100200300
0.999
1
1.001
1.002
X
X
Time
100200300
(c)
0.5
1
1.5
2
2.5
3x 105
X
Time
100200300
(d)
0.5
1
1.5
2
2.5
3x 105
K = ∂xφ
K = ∂xφ
R
R
0
0
(a)
(b)
X
Time
100200300
(g)
0.5
1
1.5
2
2.5
3x 105
X
Time
100200 300
(h)
0.5
1
1.5
2
2.5
3x 105
0100200300
−3
−2
−1
0
1
2
3
X
0 100200 300
0.4
0.6
0.8
1
1.2
X
K = ∂xφ
K = ∂xφ
R
R
0
0
(e)
(f)
Fig. 1. Time evolution of the CSHE with r = 1, L = 300 and
N = 1024 discretization points, starting from random initial
conditions. Coarsening behavior is observed in panels (a)–(d)
for k0 = 0.2. Faceting behavior is observed in panels (e)–(h)
for k0 = 1. The spatio-temporal evolution of K ≡ ∂xφ(x,t) is
shown in panels (a) and (e), with the corresponding profiles at
t = 3 × 105in (c) and (g). The evolution of the amplitude R
is shown in panels (b) and (f), with the corresponding profiles
at t = 3 × 105in (d) and (h).
Figures 1a–1d show typical time evolution obtained using
direct numerical integration of equation (1) when r = 1
and k0= 0.2 starting from random initial conditions. Fig-
ure 1a depicts the spatio-temporal evolution of the phase
gradient K ≡ φx, while the evolution of the amplitude R
is depicted in Figure 1b. The profiles of both the phase
gradient K and the amplitude R at the final time step of
the numerical simulation are shown in Figures 1c and 1d.
Initially a modulational instability develops after which
several kink-antikink pairs or “bubbles” are created in the
system. These bubbles are unstable and in time smaller
bubbles repeatedly merge forming larger and larger struc-
tures, a process called coarsening [13]. This merging of
bubbles continues until only one bubble remains in the
system. Quite different behavior is observed when k0 is
larger, as indicated in Figures 1e–1h for k0 = 1, again
starting from random initial conditions. For this set of
parameters no coarsening dynamics is observed. Instead
the different bubbles present in the system are station-
ary and take the form of stable facets with wavenumbers
K = ±k0 connected by sharp interfaces with oscillatory
internal structure. The existence of these structures was
noted already by Raitt and Riecke [14,15] in the context
of the fourth order Ginzburg-Landau equation; in the fol-
lowing we refer to them as spatially localized states (LS).
In this paper, we provide a detailed understanding
of the dynamics in both the coarsening and the faceting
regimes of the CSHE. In particular we determine the pa-
rameter values associated with each regime, and elucidate
the nature of the transition between them and the role
played by phase slips in determining the final state. In
the following section, we summarize the basic properties
of the equation, following recent work on the subcritical
case [12]. In Section 4, we examine the stability proper-
ties of a special class of time-independent solutions called
phase-winding states that are frequently observed in sim-
ulations. These are solutions of the CSHE in which the
real and imaginary parts of the order parameter oscillate
(in space) with a constant, but nonzero, phase difference.
In Section 5, we study the evolution of the longwave insta-
bility of these states identified in Section 4 and show that
under appropriate conditions it follows Cahn-Hilliard dy-
namics, leading to slow coarsening in wavenumber space.
In Section 6, we show that in other regimes the longwave
instability generates phase-slips, resulting either in sta-
ble phase-winding states with a different wavenumber or
localized states that take the form of defects connecting
phase-winding states with equal and opposite phase lag
(Figs. 1e–1h) and show that bound states of such defects
can be stable over a wide range of parameter values. Fi-
nally, in Section 7, we examine the transition between the
coarsening and faceting regimes. Direct numerical simu-
lations of the CSHE complement the theory throughout.
The paper concludes with a brief discussion.
3 The spatially homogeneous solutions
Spatially homogeneous solutions of equation (1), hereafter
referred to as “flat”, take the form u = R0(t)eiφ0(t). Thus
R0t= (r − k4
φ0t= 0,
0)R0− R3
0,
(7)
(8)
with stationary solutions given by
R2
0= R2
s≡ 0,r − k4
φ0= φs,
0,
(9)
(10)
where φs is a constant (Rs ?= 0). The amplitude Rs of
these homogeneous states is shown in Figure 2.
3.1 Temporal stability
To determine the stability of these states we let u(x,t) =
(Rs+ δ(x,t))expiφs, where δ is a complex infinitesimal
perturbation satisfying
δt= rδ − (∂2
x+ k2
0)2δ − R2
s(2δ +¯δ).
(11)

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L. Gelens and E. Knobloch: Coarsening and frozen faceted structures in the supercritical CSHE25
0 1
r
2 3
0
0.4
0.8
1.2
1.6
Rs
T0
P
T
Fig. 2. Stationary homogeneous solutions Rsfor k0 = 1. These
states are always unstable to phase modes, although the am-
plitude mode becomes stable beyond the Turing bifurcation T
of the non-zero state (solid line).
Writing δ ≡ δR+ iδIwe obtain
δRt= rδR− (∂2
δIt= rδI− (∂2
Thus with (δR,δI) ∝ exp(ikx + σt) we find the pair of
growth rates
x+ k2
0)2δR− 3R2
x+ k2
sδR,
(12)
0)2δI− R2
sδI.
(13)
σR= −2R2
σI= (2k2
s+ (2k2
0− k2)k2,
(14)
0− k2)k2,
(15)
describing the stability of the flat states Rs?= 0 with re-
spect to amplitude and phase perturbations, respectively.
Temporal stability of the trivial state:
The state u = 0 is destabilized by periodic modulations
with wavenumber k = k0at r = 0 (point T0) in a Turing
bifurcation (also called modulational instability in the op-
tics literature). A band of unstable wavenumbers devel-
ops around k = k0for r > 0 and spreads to k = 0 when
r reaches r = k4
pitchfork bifurcation to nonzero flat states.
0(point P) corresponding to a reversible
Temporal stability of the nonzero flat states:
As can be seen from equations (14), (15), the nonzero flat
states are always unstable to phase perturbations with
wavenumber k in the range 0 < k2< 2k2
variance implies that the phase growth rate of these per-
turbations vanishes when k = 0. Thus the states beyond
the Turing point T (r > 3k4
phase-unstable, and for small wavenumbers (k ? k0) the
growth rate of the instability is positive but small. This
observation will be important in what follows.
Figure 2 summarizes these results. Solid/dashed lines
denote solutions that are stable/unstable with respect to
the amplitude mode.
0; translation in-
0/2) are amplitude-stable but
3.2 Spatial stability
In order to understand the presence of localized states
homoclinic or heteroclinic to the flat states we also need
to know their stability properties in space. For this purpose
we write R(x) = (Rs+δ(x))expiφs, where δ ≡ δR+iδI∝
eλx. The spatial eigenvalues λ satisfy the equations
λ4+ 2λ2k2
0+ 2R2
s= 0,
(16)
λ2(λ2+ 2k2
0) = 0.
(17)
The former gives the spatial eigenvalues of the amplitude
mode; these are as in the real Swift-Hohenberg equation.
The latter equation gives the spatial eigenvalues for the
phase mode. Evidently there is always a pair of zero spatial
eigenvalues, a consequence of the invariance of the CSHE
under phase shifts, together with spatial reversibility. In
addition, there is a pair of purely imaginary eigenvalues.
Neither of these depends on the state or the value of r.
These eigenvalues allow us to identify parameter regimes
in which localized states are possible and to determine
their asymptotic behavior as x → ±∞.
4 The phase-winding states
In the numerical simulations we encounter states in which
the phase is no longer constant in space: over large parts
of the domain the phase may vary linearly with the spa-
tial coordinate x. Stable states of this type, referred to as
phase-winding states, were observed in [16] and studied in
the context of the subcritical CSHE in [12]. These states
take the form u = Rsexp iφs, Rs?= 0, where
R2
φs= Ksx,
s= r − (K2
s− k2
0)2,
(18)
(19)
and Ksis a real constant. In the following we refer to the
quantity Ks ≡ ∂xφs as the wavenumber of the solution.
These states are characterized by the free energy
FSH,RW = −1
4[r − (K2
s− k2
0)2]2,
(20)
as obtained from equation (2). Thus states with Ks= k0
minimize the free energy at each fixed r. States with Ks?=
k0can, however, correspond to local energy minima and
hence can also be stable.
Figure 3 shows an example of a stable phase-winding
state when r = 0.5 and k0= 1, obtained using time inte-
gration of equation (1) with periodic boundary conditions
at x = 0,L and N = 512 mesh points. Observe that this
state is not symmetric under spatial reflection; generically
states of this type are expected to drift but here these
states are necessarily stationary. This is a consequence
of the variational nature of equation (1) when the coef-
ficients are real. In view of the constant phase difference
(in space) between the real and imaginary parts of the
order parameter u (Fig. 3, third panel) we refer to phase-
winding states of this type as rotating waves (RW): in the

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26The European Physical Journal D
01020304050
X
607080 90 100
0.7071
0.7071
010 20 304050
X
60708090 100
−1.0053
Re(u), Im(u)
−1.0053
Ks = ∂xφs
0 10203040 50
X
6070 8090 100
−1
1
0
1
0
−0.2 0
−1
−0.8
Re(u)
Im(u)
−0.4−0.6 0.2 0.40.60.8
Rs
Fig. 3. RW: a phase-winding state in terms of its amplitude
Rs and the modulation wavenumber Ks ≡ ∂xφs = Kn, n = 16
(top two panels), the real (solid) and imaginary (dashed) parts
of the field u (third panel), and the field u in the complex
plane (bottom panel). Parameters: r = 0.5, k0 = 1, L = 100,
N = 512.
(Re(u),Im(u)) plane the RW correspond to closed orbits
around the origin (Fig. 3, bottom panel). In contrast, we
refer to the spatially periodic states in which Re(u) and
Im(u) oscillate in phase as standing waves (SW). In the
complex u plane the SW correspond to oscillations along
straight lines through the origin.
The solution shown in Figure 3 shows that time evo-
lution of equation (1) may result in stable states with
a wavenumber that differs from k0. In view of the im-
posed periodic boundary conditions the net phase jump
Δφ across a domain of length L associated with a solution
must be 2πn, where n is an integer. When this phase jump
is uniformly distributed it produces a constant phase gra-
dient specified by Kn≡ 2πn/L. In the event that Kn= k0
the corresponding phase-winding solution bifurcates from
u = 0 at r = 0, i.e., simultaneously with the SW branch.
However, although these two periodic states have, in this
case, the same wavenumber k0they are distinct.
4.1 Temporal stability
Writing R = Rs+δReikx+σtand φ = φs+δφeikx+σt, where
Rs and φs are given by equations (18), (19), one finds
that the temporal growth rate σ satisfies the quadratic
equation
σ2+ σ[2k4+ 11K2
sk2− 3k2
0k2+ 2R2
s]
− k2[2k2
0− k2− 6K2
s][k4− 2k2(k2
− 16k2K2
0− 3K2
s) + 2R2
s]
s[k2
0− k2− K2
s]2= 0.
(21)
When Ks = 0 this equation reduces to equations (14),
(15). In the following we refer to the two roots of this
equation as σ1,2. Figure 4 shows the larger root σ1 as a
0 0.2
k2
0.4 1 0.6 0.8
0
0.2
s1
-0.2
-0.4
-0.6
Fig. 4. Temporal growth rate σ1of the phase-winding solution
for r = 0.7, k0 = 1 and Ks = 1.3 as given by equation (21); σ2
is not shown since it is always more stable.
function of k2when Ks= 1.3, indicating that the phase-
winding states are unstable to perturbations with a suffi-
ciently small wavenumber.
4.2 Spatial stability
Writing R = Rs+ δReλxand φ = φs+ δφeλx, where Rs
and φs are given by equations (18), (19), one finds that
the spatial eigenvalues λ satisfy
λ2[(λ4+ 2(k2
0− 3K2
s)λ2+ 2R2
s][λ2+ 2k2
0− 6K2
0− K2
s]
+ 16λ2K2
s[λ2+ k2
s]2= 0.
(22)
This equation reduces to equations (16), (17) when
Ks= 0. When Ks?= 0 the pair of zero eigenvalues remains
but the remaining sixth order characteristic equation no
longer factors.
5 Coarsening dynamics: the Cahn-Hilliard
equation
In this section, we study the evolution of the longwave
instability of both the flat and the phase-winding states.
The analysis is motivated by the temporal stability results
which indicate the presence of a slowly growing long wave-
length phase mode (Eqs. (15) and (21)). We show that the
evolution of this mode is described by a Cahn-Hilliard-
type equation for the perturbation wavenumber. In the
region of validity of this equation, there is no locking of os-
cillatory tails and coarsening of the growing perturbations
is predicted. This prediction compares well with direct nu-
merical simulations of the CSHE described in Section 5.4.
Extensions of the theory to other regimes do predict lock-
ing of adjacent structures and hence evolution to a frozen
asymptotic state as described in Section 6.
5.1 Derivation of the Cahn-Hilliard equation
We consider the evolution of long wavelength phase mod-
ulation of the flat state R0 = Rs ≡
?r − k4
0, φ0 = φs,

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L. Gelens and E. Knobloch: Coarsening and frozen faceted structures in the supercritical CSHE27
where φsis a constant. We first rewrite equations (5), (6)
as coupled equations for the amplitude R(x,t) and mod-
ulation wavenumber K(x,t) ≡ ∂xφ(x,t):
Rt= rR − R3− 2k2
+12KKxRx+ 3K2
?
−2Rx
x
0Rxx− (k2
xR + 4KKxxR − Rxxxx
RK + 2Rxxx
R
?
Writing R = Rs(1 + u) and K = v, where u = O(?2),
v = O(?) and ? ? 1 is a small parameter measuring the
wavenumber of the perturbation, one obtains the following
equation for u:
0− K2)2R + 6K2Rxx
(23)
Kt= −22k2
0
Rx
K + 3Rxx
R
Kx+ 2Rx
RKxx
RK3
− 2k2
0Kxx− Kxxxx+ 2(K3)xx.
(24)
ut= −2R2
+3v2
su − 3R2
x+ 4vvxx− v4− 2k2
su2+ 2k2
0v2+ 2k2
0uv2
0uxx+ H.O.T.
(25)
We now write u = αv2+ w, where α = O(1) and w is of
order O(?4), and take ∂t ≡ O(?4), obtaining α = k2
and
1
2R4
s
0/R2
s
w =
[−(R2
s+k4
0)v4+(3R2
s−4k4
0)v2
x+4(R2
s−k4
0)vvxx].
(26)
Equation (24) now yields the result
vt= −[2k2
0/3R2
0v + vxx+ κ0v3]xx
(27)
where κ0≡ (8k4
Cahn-Hilliard equation [13]. In the CSHE context this
equation first appears in the work of Malomed et al. [17].
The Cahn-Hilliard equation was originally derived to
describe the dynamics of phase separation in systems with
a conserved quantity in the context of binary alloys [13].
However, the model equation arises in many other areas
of physics as well [18], including spinodal decomposition
in thin films [19], pattern formation on surfaces, disloca-
tions of microstructures, crack propagation and electromi-
gration, where it is used to describe progressive coarsen-
ing [18]. The equation also describes wavelength selection
in fixed flux thermal convection [20]. In the present con-
text the form of the equation is a consequence of phase
conservation across the domain, together with the sym-
metry of equations (23), (24) with respect the spatial re-
flection: x → −x, R → R, K → −K.
Equation (27) describes the evolution of the longwave
instability triggered by the negative diffusion coefficient
γ ≡ −2k2
zero mean, so that Δφ = 0 across the domain, as required
of any perturbation of a constant phase state.
s) − 2, correct to O(?5). This is the
0. The equation applies for perturbations with
5.2 Stationary solutions of the Cahn-Hilliard equation
The Cahn-Hilliard equation (27), has the Lyapunov func-
tional
FCH[v] =
dx
?
?1
2v2
x− V (v)
?
(28)
-4
-0.2-0.15-0.1-0.05 0
v
0.05 0.1 0.15 0.2
V(v)
-3
-2
-1
0
1
x 10-4
Fig. 5. Potential V (v) as a function of v for k0 = 0.1 and
κ0 = −2.
where (see Fig. 5)
V (v) ≡ k2
0v2+1
4κ0v4,
(29)
defined such that
vt= ∂xx[δFCH(v)/δv(x)].
(30)
Thus
dFCH
dt
= −
?L
0
[vxx+ V?(v)]2dx
(31)
and FCH[v] decreases until it reaches a stationary state of
equation (27). These stationary states satisfy
vxx+ 2k2
0v + κ0v3+ λ1+ λ2x = 0,
(32)
where λ1,λ2are integration constants. Symmetry with re-
spect to reflection in v requires that λ1 = 0, while the
requirement that v = 0 is a solution implies that λ2= 0.
Thus
1
2v2
we refer to the integration constant E as the energy.
Since V has two identical maxima (κ0≈ −2), there is
a pair of symmetry-related spatially homogeneous steady
states given by
vs≈ ±k0
with energy Es= −k4
is a family of spatially periodic nonlinear solutions of zero
mean whose period and amplitude depend on E. These
solutions are, however, known to be unstable [21,22]. As
E → Es from below these solutions degenerate into a
pair of heteroclinic states connecting a pair of symmetry-
related equilibria. These states are called kinks if the phase
increases across the associated defect, and antikinks if it
decreases. A kink-antikink pair is the long time zero-area
attractor of the equation [21–23]. Thus a system consist-
ing of several kink-antikink pairs or “bubbles” is unsta-
ble and in time exhibits coarsening dynamics, in which
smaller bubbles repeatedly merge together forming larger
x+ V (v) = E;(33)
(34)
0/κ0> 0. When 0 < E < Es, there

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28The European Physical Journal D
and larger structures. This process is driven by the mu-
tual interaction between the kink and antikink pairs and
is present whenever the spatial eigenvalues of vs ≈ ±k0
are real, i.e., provided there is no pinning. At large times
the width of the broadest bubble increases exponentially
slowly as a result of the exponentially small interaction
between kinks and antikinks when these are widely sepa-
rated [24,25].
5.3 Derivation of the Cahn-Hilliard equation
for phase-winding states
In this section we consider the growth of unstable pertur-
bations of phase-winding states given by R = Rs, K = Ks,
where R2
before we write R = Rs(1 + u), K = Ks(1 + v) but this
time take both u and v to be O(?). At the same time we
suppose that spatial derivatives are O(?) and time deriva-
tives of at least O(?3). Substitution into equations (23)
and (24) yields coupled equations for u and v. As before
we can solve for u in terms of v:
s= r − (k2
0− K2
s)2and Ks?= 0 is a constant. As
u = β1v + β2v2+ β3v3+ β4vxx+ O(?4),
(35)
where
β1=2K2
s(k2
0− K2
R2
s
0− 3K2
R2
s
s)
(36)
β2=K2
s(k2
s)
−5
2β2
1
(37)
β3= −β1(β2
1+ 3β2) +K2
s
R2
s
[(k2
0− K2
s)(β1+ 2β2)
−2(β1+ 1)K2
β4=−(k2
s](38)
0− 3K2
s)β1+ 2K2
R2
s
s
.
(39)
It follows that v satisfies the equation [26]
?
where the coefficients are given by
vt=
γv − δvxx−1
2κ1v2−1
3κ2v3
?
xx
,
(40)
γ = −2k2
δ = 1 + 4β1+ 4(k2
κ1= 4[k2
κ2= −4k2
−2K2
Note that since equation (1) is variational, the phase dy-
namics is also variational.
Equation (40) resembles the Cahn-Hilliard equa-
tion (27) except for the presence of an asymmetrical po-
tential V . When γ = O(?2), ? ? 1, the dynamics is
described by
?
0(1 + 2β1) + 2K2
s(3 + 2β1)
s)β4
s(3 + 3β1− β2
1− 2β2+ 3β1β2− 6β3)
s(3 + 6β1− 6β2
(41)
(42)
0− K2
1+ 2β2) − K2
1− β3
0(β1− β2
0(2β2
1+ 2β2)](43)
(44)
1+ 2β3
1+ 12β2− 6β1β2+ 6β3).
vt=
γv − δvxx−1
2κ1v2
?
xx
,
(45)
provided v = O(?2). Such an equation is thus valid for
small wavenumber changes near the stability boundary
γ = 0 (Fig. 6). A similar equation arises in the classical
theory of the Eckhaus instability based on the complex
Ginzburg-Landau equation with real coefficients [27,28],
where γ < 0 in the Eckhaus-unstable regime (white re-
gions in Figure 6 near Ks= ±k0) and the nonlinear term
drives v, and hence u, to such large amplitudes that R
reaches zero and a defect forms. In this case phase is no
longer conserved: the conservation law
plied by equations (24) and (45) requires that R > 0
throughout the evolution of R and K. Equations of the
form (45) arise in other systems with variational dynam-
ics, including the rupture of thin liquid films [29] and com-
pressible fixed heat flux convection [30], and require that
both γ and v are taken so small that the stabilizing fourth
derivative term is competitive. In all these applications the
resulting dynamics leads to the formation of a singularity.
In the present case we find that if the effective value
of ?, as defined by the magnitude of γ, is not too small,
the quartic term in the potential does enter into the dy-
namics and may, under appropriate conditions, prevent
the formation of defects. In these circumstances the Cahn-
Hilliard equation (40) describes the behavior of the CSHE
for all time, and one again observes coarsening. In the
CSHE this is the case for small wavenumbers Ks, i.e.,
near band center. In contrast, the Ginzburg-Landau ap-
proximation applies near Ks = ±k0, and here the band
center is Eckhaus-stable (Fig. 6 and [14]). However, away
from the band center in the CSHE, i.e., for Kssufficiently
far from zero, one does observe defect formation instead
of coarsening, much as in the Eckhaus-unstable regime in
the Ginzburg-Landau equation. We discuss this behavior
next.
?L
0v dx = 0 im-
5.4 Numerical verification of coarsening dynamics
In this section we confirm that the CSHE equation does
indeed follow the coarsening dynamics predicted from the
Cahn-Hilliard equations with Ks= 0 and Ks?= 0 provided
that Ksis appropriately small.
In Figure 7 we show the results of a numerical sim-
ulation starting from a phase-winding state with a net
phase jump Δφ = 2πn across the domain, where n is a
small integer. Such a state has wavenumber Ks= Kn≡
2πn/L ? 1. In the region of validity of equation (40) this
phase jump should be conserved by the evolution, i.e.,
the area A ≡
lution. When γ < 0 (white regions in Fig. 6) the phase-
winding state is modulationally unstable. Figure 7 shows
the spatiotemporal evolution of the real and imaginary
parts of the complex field u and of the wavenumber K
when k0= 0.2 (Fig. 6a) and r = 2, Ks= 2π/100 = 0.063
(n = 1). The simulations confirm the conservation of the
area A and demonstrate that the phase-winding state de-
velops a modulational instability after which the system
coarsens in time until only one stationary bubble remains.
From area conservation the width d−of the K < 0 part
?L
0K(x,t)dx = 2πn throughout the evo-

Page 8

L. Gelens and E. Knobloch: Coarsening and frozen faceted structures in the supercritical CSHE29
Ks
r
0 1 2 -1 -2
0
0.5
1
1.5
2.0
Ks
0 1 2 -1 -2
r
0
0.5
1
1.5
2.0(a)
(b)
Fig. 6. (Color online) The condition Rs > 0 with γ > 0 (gray)
and γ < 0 (white) in the (Ks,r) plane. No solutions Rs are
present in the black region. (a) k0 = 0.2. (b) k0 = 1.
of the bubble is given by
d−=
LK+
K+− K−
−
2πn
K+− K−,
(46)
where n is an integer determined by the initial condi-
tion (here n = 1) and K± are the wavenumbers con-
nected by the pair of heteroclinic connections that form
as t → ∞. We find that K± = ±Ks(1 + v±), where v±
correspond to the two maxima of the potential V2(v) ≡
λ1v−1
Of course, such a connection is only possible when the two
maxima are identical, a requirement that selects the quan-
tity λ1. With this choice of λ1 the potential V2 remains
asymmetrical but resembles Figure 5. The numerical sim-
ulations indicate, in addition, that the resulting values
of v±are such that K±≈ ±k0, i.e., the evolution of the
system replaces the phase-winding state with wavenumber
Kswith the preferred wavenumber k0. Thus the asymmet-
rical potential is a consequence of the fact that Ks?= k0
and the associated phase gradient prevents relaxation of
the final bubble to the symmetric state d+= d−= L/2,
i.e., it pins the front separating the ±k0phases to a loca-
tion determined by equation (46) [12].
To understand the numerical results in more detail we
rewrite equation (40) in terms of the wavenumber K =
Ks(1 + v) obtaining
?
Figure 8a shows the quantity ρ ≡ κ2−1
the importance of the quadratic term in equation (47) as
a function of k0. Thus for small to moderate values of
k0 the quadratic term is indeed small, and coarsening is
expected. Figure 8b shows, moreover, that in this regime
the selected wavenumbers satisfy
?
For larger values of Ks, or equivalently of γ, the quadratic
term in the potential is no longer negligible, and the evo-
lution begins to resemble behavior familiar from earlier
studies of the nonlinear evolution of the Eckhaus insta-
bility [27,28]. In this regime defects form and phase is no
2γv2+1
6κ1v3+1
12κ2v4associated with equation (40).
Kt=(γ + κ1− κ2)K − δKxx+
ρ
KsK2−
κ2
3K2
s
K3
?
xx
(47)
.
2κ1that measures
K±= ±Ks
3(γ + κ1− κ2)/κ2≈ ±k0.
(48)
X
Time
20406080 100
(c)
2
4
6
8
10
x 104
020406080 100
(d)
−0.2
−0.1
0
0.1
0.2
X
X
Time
20406080 100
(e)
2
4
6
8
10
x 104
020406080 100
(f)
−2
−1
0
1
2
X
X
Time
20406080 100
2
4
6
8
10
x 104
020406080 100
−2
−1
0
1
2
X
Re(u)
Im(u)
K = ∂xφ
(a)(b)
Fig. 7. Time evolution of the CSHE with k0 = 0.2 and
r = 2 starting from an unstable phase-winding state with ini-
tial wavenumber Kn = 2πn/L, where L = 100 and n = 1. For
these parameter values equation (1) evolves via coarsening, as
revealed by the spatio-temporal evolution of the wavenumber
(a) K, (c) Re(u) and (e) Im(u). The resulting (almost) station-
ary profiles are shown in panels (b), (d), (f). The correspond-
ing coefficients in the phase equation (40) are: γ = −0.0563,
δ = 1.00, κ1 = −0.0473 and κ2 = −0.0237.
0 0.20.40.60.81.0
k0
0
1
2
3
0
0.005
-0.005
-0.01
-0.015
-0.02
r
K+,-
(a)
(b)
K+
-K-
Fig. 8. (a) The quantity ρ ≡ κ2 −1
K± of the maxima of the resulting potential V2 when these are
equal, both as functions of k0. Parameters: r = 2 and Ks = 0.1.
2κ1 and (b) the location
longer conserved. A typical example of this type of evo-
lution is shown in Figure 9. We again take k0= 0.2 and
r = 2, but now consider a phase-winding state with Kn=
2π(18/100) ≈ 1.131 (n = 18). As shown in Figure 6a for
these parameters γ < 0 indicating that this phase-winding
state is also modulationally unstable. However, coarsening
does not take place and one observes instead successive

Page 9

30The European Physical Journal D
Table 1. Outcome of the modulational instability of phase-winding states when k0 = 0.2, r = 2, L = 300, N = 1024. The top
row specifies the wavenumber of the unstable initial state while nfinal specifies the phase jump across the domain in the final
state. The letters C (coarsening) and E (Eckhaus) indicate the evolution type; EC indicates evolution via phase slips followed
by coarsening.
n = KsL/2π
behavior
nfinal
1
C
1
2
C
2
3
C
3
4
C
4
5
C
5
47
E
44
48
E
31
49
E
26
50
E
24
51
E
17
52
E
16
53
E
15
54
E
7
55
EC
5
56
EC
0
57
EC
–2
X
204060 80 100
5
10
15
20
X
20406080 100
Time
500
1000
1500
2000
2500
3000
5
10
15
20
Time500
1000
1500
2000
2500
3000
00
0
10
Time
20
18
14
10
6
2
Df / 2 p
(a)
(b)
(c)
Fig. 9. Time evolution of the CSHE with k0 = 0.2 and r =
2 starting from an unstable phase-winding state with initial
wavenumber Kn = 2πn/L, where L = 100 and n = 18. Panels
show the spatio-temporal evolution of (a) Re(u) and (b) Im(u),
while (c) shows the evolution of the total phase jump over the
domain.
formation of defects. The associated change in phase is
tracked in Figure 9c; the final state that results is a phase-
winding state with a stable wavenumber Kn, n = 5. Thus
the modulational instability acts to shift the wavenumber
of the state from an unstable wavenumber to one that is
stable (shaded region in Figure 6a).
5.5 Transition from coarsening to Eckhaus dynamics
In an attempt to locate the transition from coarsening be-
havior to evolution via phase slips we have performed a
series of numerical computations spanning the unstable
regions (γ < 0) in Figure 6. We describe here the results
for k0= 0.2, r = 2 on a domain of length L = 300 with
N = 1024 mesh points. In each case we initialize the cal-
culation with an unstable phase-winding state, specified
by n, and allow numerical error to trigger the instability.
As a result we do not control the initial condition and
so find the most “likely” outcome of the instability. We
emphasize that multiple outcomes are in general possi-
ble, depending on initial conditions, and that even small
Time
20
0
40
0
Time
20
0
40
−2
−1
0
1
2
-0.5
0.5
0100200300
X
0100200300
(d)
X
X
100200300
0
X
100200300
(c)
0
Re(u)
K = ∂xφ
(a)
2e3
4e3
6e3
8e3
1e4
K = ∂xφ
0
-0.4
0.4
(b)
2e3
4e3
6e3
8e3
1e4
−2
−1
0
1
2
Re(u)
Fig. 10. Time evolution of the CSHE with k0 = 0.2 and r =
2 starting from an unstable phase-winding state with initial
wavenumber Kn = 2πn/L, where L = 300 and n = 55. Panels
show the spatio-temporal evolution of (a) K ≡ ∂xφ and (c)
Re(u), with the corresponding profiles at t = 50 and t = 104
shown in (b) and (d).
changes in the resolution (for example) lead to different
results. This is because the Eckhaus instability in effect
amplifies small amplitude noise which develops into a dis-
tribution of phase slips that depends on the noise details.
Thus in the Eckhaus-unstable regime the outcome of the
instability requires a probabilistic description.
The results are summarized in Table 1. The top
row specifies the wavenumber Kn of the unstable ini-
tial state while the third row lists nfinal ≡ Δφfinal/2π
that measures the total phase jump across the domain
in the final state of the simulation. The letter C indi-
cates evolution via coarsening. Since this process is phase-
conserving, nfinal= ninitial. This type of behavior domi-
nates throughout the unstable band near Ks= 0. In the
remaining unstable regions in Figure 6a the evolution is
via phase slips and we label this type of evolution with the
letter E. The final states reached consist of new phase-
winding states with wavenumber in the Eckhaus-stable
regime. The table reveals that near the inner boundary
of these regions the Eckhaus instability generates only a
small number of defects, ninitial− nfinal. However, this
number increases approximately linearly with ninitial, un-
til the total phase jump across the domain is almost zero.

Page 10

L. Gelens and E. Knobloch: Coarsening and frozen faceted structures in the supercritical CSHE31
The corresponding wavenumbers, as measured by nfinal
are once again in the unstable regime near Ks = 0 and
one therefore anticipates a second phase of the evolution
resembling coarsening. We indicate this type of behavior
using the symbol EC.
Figure 10 shows an example typical of the EC regime.
Numerous phase slips take place that reduce the effective
n from its initial value ninitial= 55 to nfinal= 5. At this
point the solution is still far from a phase-winding state
with a linear phase gradient but from this point on no ad-
ditional phase slips take place and evolution via coarsening
takes over, as expected from a Cahn-Hilliard equation ini-
tialized with a large number of “bubbles”. The final state
of the system is therefore a single stable bubble whose area
is determined by the net phase jump across the domain at
the end of the Eckhaus phase.
6 Faceting of phase-winding states
6.1 Origin of frozen facets
In this section we describe the corresponding results for
the case k0 = 1 (Fig. 6b). Figures 11 and 12 show the
time evolution of the CSHE for r = 2 starting with unsta-
ble phase-winding states with wavenumbers Kn≡ 2πn/L
with L = 100 and n = 2,22, respectively. As antici-
pated the solutions evolve by generating defects, thereby
changing their wavenumber into a stable one. However,
the manner with which this is accomplished depends on
both parameters and initial conditions. In Figure 11, cor-
responding to n = 2, the system evolves into a state with
wavenumbers K = ±k0connected by (a pair of) hetero-
clinic connections or fronts. Owing to the complex spa-
tial eigenvalues λ of these states the fronts lock to each
other via their overlapping oscillatory tails [14,31] form-
ing a bound state of two fronts, i.e., a bound state of a
kink and an antikink, and this structure is reflected in the
amplitude R as well. The formation of such a bound state
suppresses further evolution of the system and no coars-
ening is observed, even when multiple bound states are
present. In contrast, Figure 12, corresponding to n = 22,
reveals behavior that conforms to standard Eckhaus evo-
lution, resulting in the formation of a stable phase-winding
state with n = 19. Here the phase gradient remains ho-
mogeneous in space and no facets form.
6.2 Bifurcation structure of the frozen facets
In this section, we examine the properties of faceted states
of the type shown in Figure 11. These states are the typical
final states obtained in numerical simulations starting ei-
ther from an unstable RW as initial condition or starting
with random noise. These structures are spatially local-
ized in both amplitude and phase, and can be symmetric
or asymmetric under reflection in x. States of this type ex-
hibit pronounced localization in amplitude together with
almost linear phase variation, typically with slopes ±k0on
X
Time
20406080 100
5
10
15
20
25
30
X
Time
20 406080 100
5
10
15
20
25
30
020406080 100
(e)
−2
−1
0
1
2
X
020 406080 100
(f)
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
X
X
Time
20406080 100
5
10
15
20
25
30
X
Time
20 406080 100
5
10
15
20
25
30
K = ∂xφ
R
5101520 25 30
14
10
6
2
Df / 2 p
(a)
(b)
(c)(d)
(g)
Time
Fig. 11. Time evolution of the CSHE with k0 = 1 and r =
2 starting from an unstable phase-winding state with initial
wavenumber Kn = 2πn/L, where L = 100 and n = 2. Panels
show the spatio-temporal evolution of (a) K and (b) R, while
panels (c), (d) show the solution profiles at the last time step of
(a), (b). Panels (e), (f) show the evolution of Re(u) and Im(u),
respectively, while (g) shows the evolution of the total phase
jump Δφ across the domain.
X
Time
20406080 100
(c)
20
40
60
80
100
120
140
X
Time
2040 6080 100
(d)
20
40
60
80
100
120
140
X
Time
20406080 100
20
40
60
80
100
120
140
X
Time
20406080 100
20
40
60
80
100
120
140
(a)(b)
Fig. 12. Time evolution of the CSHE with k0 = 1 and r =
2 starting from an unstable phase-winding state with initial
wavenumber Kn = 2πn/L, where L = 100 and n = 22. Panels
show the spatio-temporal evolution of (a) K and (b) R, (c)
Re(u) and (d) Im(u). The final state has a uniform amplitude
and wavenumber corresponding to n = 19.

Page 11

32The European Physical Journal D
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
r
Energy
SW
RW, n=16
MW
BS
(2)
(1)
(4-6)
(7)
(3)
A
Fig. 13. Bifurcation diagram showing the classical periodic
states (SW, black), together with the new phase-winding (RW,
black) and faceted states (MW/BS, gray). The integers indi-
cate locations used for solution profiles in subsequent figures.
Solid (dashed) lines depict stable (unstable) solutions. Since
the stable branches of BS and MW have very similar energies
the former have been displaced slightly downward. Parameters:
k0 = 1, L = 100, N = 512.
0
5x 10−8
102030 4050
X
60 708090 100
0
1
R
0
1
102030 40 50
X
60 70 8090100
−5
0
01020 304050
X
60708090100
−1
0
0
−0.20
Re(u)
Im(u)
−0.6−10.20.61
−0.2
0.2
0.5
Re(u), Im(u)
K = ∂xφ
Fig. 14. SW: Solution profile at location (2) in Figure 13. Top
two panels: amplitude R and modulation wavenumber K ≡
∂xφ. Third panel: real (solid) and imaginary (dashed) parts of
the field u. Bottom panel: the field u in the complex plane.
Parameters: r = 0.5, k0 = 1, L = 100 and N = 512.
either side of the resulting localized structure. The result-
ing phase variation thus resembles a faceted surface, with
the relatively abrupt changes in the phase gradient asso-
ciated with the amplitude defect. In general, the widths
of the amplitude and wavenumber defects are comparable.
In the presence of periodic boundary conditions LS of this
type necessarily come in pairs. In view of the variational
structure of equation (1) asymmetric LS remain station-
ary and hence can easily be continued in parameter space
using standard continuation techniques.
Figure 13 shows the bifurcation structure of the differ-
ent types of LS using the L2norm (hereafter, the energy)
as a function of the control parameter r when k0 = 1.
Stable (unstable) solutions are shown in solid (dashed)
lines. The branch of in-phase oscillations with wavenum-
0 1020304050
X
60708090100
0
R
010 203040 50
X
6070 80 90100
−10
0.5
0
10
01020304050
X
60708090100
0
−0.10
Re(u)
Im(u)
0.2 −0.20.4
0.2
-0.2
0
-0.5
0.2
0.4
Re(u), Im(u)
K = ∂xφ
010 20 30 4050
X
60708090 100
0
1
R
01020 3040 50
X
607080 90100
−5
0
5
01020304050
X
60 708090100
−1
0
1
0
1
0
Re(u)
Im(u)
0.2−0.20.40.60.8−0.4−0.6−0.8
−1
0.5
Re(u), Im(u)
K = ∂xφ
Fig. 15. MW: Solution profiles at locations (3), (4) in Fig-
ure 13, r = 0.04,0.5, respectively. Top two panels: amplitude
R and modulation wavenumber K ≡ ∂xφ. Third panel: real
(solid) and imaginary (dashed) parts of the field u. Bottom
panel: the field u in the complex plane. Parameters: k0 = 1,
L = 100 and N = 512.
ber k0, labeled as SW (Standing Waves), bifurcates from
u = 0 at r = 0. An example of an SW (profile (2)) can be
seen in Figure 14. As r increases beyond r = 0, the SW
branch bifurcates at point A resulting in a branch of Mod-
ulated Rotating Waves (MW), consisting of an equidistant
pair of identical defects of opposite chirality. Once again,
the phase outside of these defects varies almost linearly,
with alternating gradients ±k0and no overall phase jump
Δφ = 0. In this type of solution the amplitude defects
are symmetric under reflection since each defect lies ex-
actly midway between its neighbors on either side. Near
A the phase gradient is markedly nonuniform in space
(see Fig. 15, profile (3)). However, with increasing r the
nonuniformity in both amplitude and the phase gradi-
ent gradually decreases, and the solution resembles more
and more a front connecting equal and opposite wavenum-
bers ±k0(see Fig. 15, profile (4)). In space these bifurca-
tions correspond to the spatial analogue of the direction-
reversing Hopf bifurcation from a group orbit of periodic
states discussed in [32].

Page 12

L. Gelens and E. Knobloch: Coarsening and frozen faceted structures in the supercritical CSHE 33
0 10203040 50
X
60708090100
0
5
1
R
0102030 4050
X
60708090100
−5
0
0 10203040 50
X
607080 90100
−1
0
1
0
1
0
Re(u)
Im(u)
0.2 −0.2 0.40.60.8 −0.4−0.6 −0.8
−1
0.5
Re(u), Im(u)
K = ∂xφ
0102030 40 50
X
60 7080 90 100
0
5
1
R
0102030 4050
X
607080 90100
−5
0
01020304050
X
60708090100
−1
0
1
0
1
0
Re(u)
Im(u)
0.2 −0.2 0.40.60.8 −0.4−0.6−0.8
−1
0.5
Re(u), Im(u)
K = ∂xφ
010 20304050
X
60708090100
0
5
1
2
R
0 10 203040 50
X
6070 80 90100
−5
0
0 1020 304050
X
60708090100
−1
0
1
−1
0
1
0
Re(u)
Im(u)
0.40.8−0.4 −0.8
Re(u), Im(u)
K = ∂xφ
Fig. 16. BS: Solution profiles at locations (5)–(7) in Figure 13,
r = 0.5,0.5,0.8, respectively. Top two panels: amplitude R and
modulation wavenumber K ≡ ∂xφ. Third panel: real (solid)
and imaginary (dashed) parts of the field u. Bottom panel: the
field u in the complex plane. Parameters: k0 = 1, L = 100 and
N = 512.
For comparison, Figure 13 also shows a branch of
phase-winding states (RW) from equations (18), (19), with
constant phase gradient Kn ≡ 2πn/L, n = 16. Figure 3
shows this state at location (1). In this case this state
was the outcome of a numerical simulation of the CSHE,
and indeed the wavenumber selected, K16, minimizes the
energy FSH,RW in a domain with finite spatial period
L = 100. In general, however, numerical simulations pro-
duce one or more either equally or unequally spaced LS
(MW, BS in Fig. 13). The figure shows that the latter
fall on disconnected solution branches created in saddle-
node bifurcations (only the stable part of these branches
is shown). A typical example of the latter is shown in Fig-
ure 16, profiles (5)–(7). We refer to solutions of this type
as bound states (BS). The BS form via locking between
the oscillatory tails of individual LS forming, at each r,
a discrete family of BS with different separations. Each
of these states lies on a distinct solution branch with like
behavior [14]. Similar bifurcation structure is also present
in the subcritical case [12].
7 Transitions between coarsening, frozen
faceted structures and Eckhaus dynamics
7.1 Transition from coarsening to frozen faceted
structures
As discussed in the previous two sections, the behavior
of faceted structures in the CSHE depends on the char-
acteristic length scale 2π/k0. When k0is small the facets
coarsen indefinitely as described by a Cahn-Hilliard equa-
tion, while for larger values of k0 the coarsening ceases
leading to a frozen faceted structure. In this case the fronts
connecting equal and opposite phase gradients k0 form
bound states permitting the coexistence of multiple stable
bubbles. In this section we examine the transition between
these two regimes.
In order to predict the value of k0at which the tran-
sition from coarsening to faceting behavior takes place we
examine the spatial eigenvalues of the phase-winding state
with Ks = k0 as described by equation (22). Since the
decay of spatial perturbations around this state is con-
trolled by the slowest eigenvalue, i.e., the eigenvalue with
the smallest nonzero real part, coarsening will take place
whenever this eigenvalue is purely real and oscillatory tails
are absent. On the other hand, whenever this particular
eigenvalue has a nonzero imaginary part oscillatory tails
will be present allowing adjacent kink and antikinks to
lock to one another [31]. In this case one expects to ob-
serve faceting. It follows, therefore, that the transition be-
tween coarsening and faceting dynamics is given by the
point in parameter space where the eigenvalue with the
smallest real part acquires a non-zero imaginary part.
In Figure 17 we show the real and imaginary parts
of the spatial eigenvalues of a phase-winding state with
wavenumber Ksin the special case in which Ks= k0. This
choice is motivated by our numerical simulations which
show that the phase gradients involved in the formation
of a localized structure are always ±k0. For the param-
eters r = 1 and Ks = k0 ∈ [0,1] the eight eigenvalues
(as obtained from Eq. (22)) are organized as follows: there
is a quartet of complex eigenvalues (λ1,2,3,4= ±λr±iλi),
two purely real eigenvalues (λ5 = −λ6) and a double
zero eigenvalue (λ7,8 = 0). As already mentioned zero

Page 13

34The European Physical Journal D
0
Re(l)
0.4
0.8
-0.4
-0.8
0 0.2 0.4 0.6 0.8 1
Ks = k0
l1,2
l3,4
l5
l7,8
(a)
Kcross
l6
0
Im(l)
0 0.2 0.4 0.6 0.8 1
Ks = k0
-2
-1
1
2
l1,2
l3,4
l5,6,7,8
(b)
Fig. 17. The (a) real and (b) imaginary parts of the spa-
tial eigenvalues of the phase-winding state as a function of the
phase gradient Ks, 0 ≤ Ks ≤ 1, and Ks set equal to k0 as ob-
served in all numerical simulations resulting in localized states.
Thus the plot also represents the variation of the real and imag-
inary parts of the relevant spatial eigenvalues as a function of
the basic wavenumber k0 when r = 1.
eigenvalues are a consequence of the invariance of the
CSHE under phase shifts, together with spatial reversibil-
ity. From Figure 17 it is clear that there is a cross-over
between the magnitude of the real parts of the complex
quartet and the purely real eigenvalues. This cross-over
occurs at Kcross ≈ 0.43 and splits the graph in two re-
gions: a coarsening region for Ks = k0 < Kcross and a
faceting region where the LS can pin to the oscillatory
tails of the fronts (Ks= k0> Kcross). This prediction of
the structure of the spatial eigenvalues is verified in Fig-
ure 18, where we show two temporal simulations of the
CSHE, one for Ks= k0= 0.35 < Kcross(Fig. 18a) and
one for Ks = k0 = 0.55 > Kcross (Fig. 18c). In both
cases we use the same initial condition consisting of two
bubbles, one of which is much smaller in width than the
other, and identical values of r and L. The results reveal
an unambiguous qualitative change in the dynamical be-
havior of the system, from coarsening to faceting. More-
over, the front profiles of the bubbles (Fig. 18b, 18d), con-
firm the absence/presence of oscillatory tails when coars-
ening/faceting takes place.
X
Time
50 100150200250 300
1
2
3
4
5
x 106
50 100150
X
200250 300
0
0
0.4
-0.4
K = ∂xφ
0
-0.2
0.2
X
Time
50100150200250300
1
2
3
4
5
x 106
0 50100150
X
200250 300
0
0
0.4
-0.2
-0.4
-0.6
-0.2
-0.6
K = ∂xφ
Fig. 18. Spatio-temporal simulation of the CSHE from an
initial condition consisting of two bubbles of different widths.
Parameters: r = 1, L = 300, N = 1024 and k0 = 0.35, 0.55
in panels (a)/(b) and (c)/(d), respectively. Panels (a) and (c)
show the space-time contour plot of K ≡ ∂xφ. High/low K is
color-coded by white/black. Panels (b) and (d) show profiles
taken from panels (a) and (c) close to onset.
7.2 Transition from frozen faceted states
to Eckhaus dynamics
In Figure 18 the evolution was started from an initial state
with two bubbles to examine the transition from contin-
ued coarsening to locking. However, when k0= 1, r = 2
and the simulations are initialized with an unstable phase-
winding state the evolution always leads to phase slips.
Depending on the wavenumber Knof the unstable state
the phase slips may lead to a frozen faceted structure (EF)
or to a stable phase-winding state with wavenumber in the
Eckhaus-stable region in Figure 6b (E). Table 2 lists the
results for k0 = 1, r = 2, L = 300 and N = 1024 and
different initial wavenumbers Kn (N = 2048 is used for
n ≥ 66). In addition to the initial and final values of n and
the labels EF and E distinguishing the different types of
behavior the table also lists the number nfacetsof frozen
facets in the final state. We mention that for these pa-
rameter values γ reaches maximum (maximum stability)

Page 14

L. Gelens and E. Knobloch: Coarsening and frozen faceted structures in the supercritical CSHE35
Table 2. Outcome of the modulational instability of phase-winding states when k0 = 1, r = 2, L = 300, N = 1024 (or N = 2048
for n ≥ 66). The top row specifies the wavenumber of the unstable initial state while nfinal specifies the phase jump across the
domain in the final state. The letters E (Eckhaus) and EF (Eckhaus-faceting) indicate the evolution type.
n = KsL/2π
behavior
nfinal
nfacets
n = KsL/2π
behavior
nfinal
nfacets
n = KsL/2π
behavior
nfinal
nfacets
n = KsL/2π
behavior
nfinal
nfacets
1234567
E
50
0
17
E
52
0
27
E
45
0
72
EF
42
1
89 10
E
53
0
20
E
51
0
30
E
41
0
EF
24
2
11
E
52
0
21
E
52
0
66
E
57
0
EF
8
4
12
E
52
0
22
E
47
0
67
E
53
0
EF
0
3
13
E
51
0
23
E
48
0
68
E
51
0
EF
11
3
14
E
52
0
24
E
47
0
69
E
49
0
EF
14
4
15
E
51
0
25
E
46
0
70
E
59
0
EF
9
5
16
E
52
0
26
E
46
0
71
E
45
0
EF
16
4
18
E
50
0
28
E
44
0
73
E
49
0
EF
44
1
19
E
51
0
29
E
44
0
74
EF
13
3
31
E
37
0
when Ks= 1.18, corresponding to n = 56.5. On the other
hand the Knminimizing the Swift-Hohenberg energy FSH
corresponds to n = 48. Thus the evolution of the insta-
bility only rarely selects the lowest energy phase-winding
state and the other states listed in the table correspond to
thermodynamically metastable states. The frozen faceted
states have even larger energies FSH but these are also
linearly stable, and hence also correspond to local minima
of the energy landscape.
8 Conclusions
We have described a new class of time-independent states
in the supercritical complex Swift-Hohenberg equation
with real coefficients which we have called phase-winding
states. These complex-valued solutions oscillate periodi-
cally in space, like the periodic states of the real Swift-
Hohenberg equation, but with a well-defined phase differ-
ence between the real and imaginary parts. Such states
are the spatial analogue of states known as rotating waves
in the time domain. The solutions fall into different fam-
ilies characterized by the overall phase jump across the
domain, which must be a nonzero integer multiple of 2π.
These states can be obtained analytically and are easily
found in numerical simulations.
Associated with the spatially extended phase-winding
states one finds a new class of localized states taking the
form of defects connecting phase-winding states with equal
and opposite phase lag. The resulting phase defect is re-
flected in the amplitude of the complex field as well. These
states are easily found in numerical simulations starting
with an unstable phase-winding state, and then followed
in parameter space using numerical continuation. Defects
of this type have been studied in the context of pattern-
forming systems with a near-degenerate neutral curve.
When the neutral curve admits two nearby minima the
patterns that result may consist of domains consisting
of wave trains with two distinct wavenumbers connected
by fronts. As shown in [14,15] an envelope description of
such domain structures leads to a fourth order Ginzburg-
Landau equation, an equation that can be written in the
CSHE form (1).
Our work extends existing work [14,15] on such domain
structures in two directions. (a) It shows that such states
can be readily followed in parameter space using numerical
continuation and their bifurcation properties determined,
and (b) it explains how their presence relates on the one
hand to phase slips generated by the Eckhaus instabil-
ity in some wavenumber regimes and on the other to the
coarsening dynamics observed in others.
Specifically, we have seen that these defect structures
are associated with the presence of fronts connecting equal
and opposite phase gradients. The fronts that are observed
for order-one wavenumber k0have oscillatory tails allow-
ing Pomeau locking between kinks and antikinks, stabiliz-
ing the localized states. In this region the phase variation
resembles a faceted surface with abrupt changes in the
phase gradient. On the other hand, when the characteris-
tic wavenumber k0is small, stable facets and defects are no
longer present. Instead numerical simulations reveal coars-
ening dynamics of the fronts. A theoretical analysis of the
stability of the zero phase gradient state with respect to
long wavelength perturbations showed that the observed
coarsening is described by the Cahn-Hilliard equation for
the perturbation phase gradient, thereby confirming the

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36The European Physical Journal D
coarsening results obtained from simulations of the CSHE.
For larger values of k0the cubic term in the effective po-
tential becomes dominant and simulations reveal the for-
mation of defects that shift the wavenumber of the phase-
winding state towards increased stability, behavior that
is familiar from existing studies of the Eckhaus insta-
bility. We have quantified the parameter regimes where
this type of evolution leads to stable phase-winding states
and where the final state consists of two or more time-
independent defects. Each defect connects phase-winding
states with wavenumber ±k0 and pairs of defects form
bound states locked to fixed separations determined by
the oscillatory tails of the fronts connecting the ±k0phase-
winding states. In some cases the Eckhaus instability of
the phase-winding state with large Ks selects unstable
wavenumbers near k0; if the spatial eigenvalues of this
state are real the initial Eckhaus phase is followed by
slow evolution via coarsening into a single bubble state. In
other cases the eigenvalues are complex and the Eckhaus
instability leads to a frozen defect state.
Throughout the paper we have chosen to vary the basic
length scale 2π/k0in the CSHE even though on the real
line this length scale can be scaled out of the equation.
Such a rescaling changes the bifurcation parameter r to
r/k4
of the scaled bifurcation parameter, potentially explaining
why the coarsening behavior described here is not more
widely known. On the other hand for k0= 1 faceting re-
sulting from phase slips is the generic behavior of the sys-
tem. One-dimensional fronts in the two-dimensional Swift-
Hohenberg equation for a real order parameter may also
undergo a faceting or zigzag instability [33,34]. This in-
stability results in a faceted front and is distinct from the
faceting in the spatial phase of the pattern described here.
0. Thus small values of k0correspond to large values
In future work we will confront the predictions ob-
tained here for the supercritical CSHE with direct numer-
ical simulations of the Maxwell-Bloch equations [16] or
the equations modeling nondegenerate optical parametric
oscillators [4–6]. We will also extend the present study to
the CSHE with complex coefficients.
This work was supported by the Belgian Science Policy Office
under grant No. IAP-VI10, the Research Foundation - Flan-
ders and the National Science Foundation under grant DMS-
0908102.
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