Traveling waves and defects in the complex Swift-Hohenberg equation.

Abstract

The complex Swift-Hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and, as such, admits solutions in the form of traveling waves. The properties of these waves are systematically analyzed and the dynamics associated with sources and sinks of such waves investigated numerically. A number of distinct dynamical regimes is identified and analyzed using appropriate phase equations describing the evolution of long-wavelength instabilities of both the homogeneous oscillating state and constant amplitude traveling waves.

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PHYSICAL REVIEW E 84, 056203 (2011)
Traveling waves and defects in the complex Swift-Hohenberg equation
Lendert Gelens1,*and Edgar Knobloch2,†
1Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium
2Department of Physics, University of California, Berkeley, California 94720, USA
(Received 21 June 2011; revised manuscript received 29 August 2011; published 7 November 2011)
The complex Swift-Hohenberg equation models pattern formation arising from an oscillatory instability with
a finite wave number at onset and, as such, admits solutions in the form of traveling waves. The properties
of these waves are systematically analyzed and the dynamics associated with sources and sinks of such waves
investigated numerically. A number of distinct dynamical regimes is identified and analyzed using appropriate
phase equations describing the evolution of long-wavelength instabilities of both the homogeneous oscillating
state and constant amplitude traveling waves.
DOI: 10.1103/PhysRevE.84.056203PACS number(s): 05.45.−a, 42.65.Sf
I. INTRODUCTION
Spatiallyextendedsystemsinmanyareasofphysicsexhibit
spontaneouspatternformationwhendrivenawayfromequilib-
rium [1–3]. The patterns that result can be classified according
to the linear instability of the spatially uniform system that
creates them. By increasing a control parameter beyond a
critical value this homogeneous equilibrium state may lose
stability with respect to perturbations with a characteristic
wave vector k0and/or with a characteristic frequency ω0at
the instability threshold. Near this threshold the system can
often be described by simpler, generic equations [3–5].
The complex Swift-Hohenberg equation (CSHE) is one
such universal equation that models pattern formation arising
from an oscillatory instability (ω0?= 0) with a finite wave
number(k0?= 0)atonset[6,7].TheSwift-Hohenbergequation
was originally obtained in the context of thermal convection
problems [8], but also finds applications in Couette flow [9],
magnetoconvection [10], liquid crystals [11], flame dynamics
[12], and nonlinear optics [13–20].
Inthiswork,wefocusontheapplicationoftheCSHEinthe
field of nonlinear optics. The derivation of the CSHE model
has been discussed in the context of optical parametric oscil-
lators (OPOs) [13–15,21,22], photorefractive oscillators [16],
and lasers [17–20]. When studying transverse dynamics in
large-aspect-ratiolasersystems,theMaxwell-Blochequations
provide a good description of pattern formation in single
longitudinal mode, two-level lasers. The behavior of the laser
system is then largely determined by the cavity detuning ν,
which is directly related to the difference between the atomic
frequency and the closest cavity resonance frequency. In the
case of negative detuning (ν < 0), Coullet et al. [23] showed
that the laser dynamics close to threshold could be described
by a complex Ginzburg-Landau equation (CGLE). In the
case of positive detuning (ν > 0), Newell and Moloney [24]
derived coupled Newell-Whitehead-Segel–like equations to
describe two-dimensional dynamics close to threshold. In
one dimension the latter equations reduce to two coupled
complex Ginzburg-Landau equations (CCGLEs) describing
the interaction between two counterpropagating traveling
*lendert.gelens@vub.ac.be
†knobloch@berkeley.edu
waves [25]. For a detailed study of the CCGLEs we refer
to Ref. [26]. A more general description of class A and C
lasers for both signs of the detuning ν is given by the CSHE
as shown by Lega et al. [17,18]. The validity of the CSHE
as an approximation to the general Maxwell-Bloch equations
is evaluated in Ref. [27]. The CSHE description of two-level
lasershasbeengeneralizedtomodelsemiconductorlasers[19]
and its validity in this context has also been investigated
[28,29].
In this paper we study the properties of the supercritical
CSHE in one spatial dimension, viz.,
ut= ru + iζuxx− (1 + iβ)?∂2
where u is a complex field representing the amplitude of a
traveling wave in a frame moving with the group velocity.
In optics applications u typically represents the complex
amplitude of a transverse electric field, for example, inside
a laser cavity.
The equation is fully parametrized by four real parameters
r, ζ, β, and b [6,7,17,18]. We treat the parameter r as
a bifurcation parameter, focusing on the behavior of the
solutions as a function of the parameter β, for fixed values of
the parameters b and ζ. We remark that in many applications
(e.g., Refs. [13–20]) the parameter β vanishes, although in
others (e.g., Ref. [29]) β may differ from zero. On the real line
the remaining parameter, the wave number k0, can be scaled
out of the problem, but we prefer to retain it in the formulation
in order to emphasize the role played by the intrinsic length
scale in the problem. The traveling wave solutions of Eq. (1)
can be thought of as generalizations of the time-independent
phase-winding states studied in previous work on the CSHE
with real coefficients [30]; when ζ = β = b = 0 such states
cannot travel because the equation has variational structure.
Our goal in this work is to provide a systematic study
of the generic CSHE, focusing on defects associated with
the presence of traveling waves, that is, sources and sinks
of traveling waves, and on their stability properties. The
work can be thought of as an extension of related studies
of the CGLE [1,5,31–33] in which the intrinsic length scale
2π/k0is infinite. The formation of defects is an inevitable
consequence of random initial conditions that are commonly
used in simulations of both CSHE and CGLE. A typical
x+ k2
0
?2u − (1 + ib)|u|2u,
(1)
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LENDERT GELENS AND EDGAR KNOBLOCHPHYSICAL REVIEW E 84, 056203 (2011)
solution consists of domains in which the waves propagate
in one or other direction. These domains are separated by
(typicallymoving)defects:sourcesthatsendoutTWandsinks
that absorb them [34–37]. Over time the defects can annihilate
ornewdefects canbecreated.Inaddition,defectscan undergo
a variety of instabilities leading, for example, to breathing or
chaotic motion, or they may form bound states that propagate
through the system with a common speed. Sinks and sources
differ in general in their properties, with sinks often stable
and sources prone to instabilities [26,34,36]. In the field of
nonlinear optics, for example, in laser systems, sources, and
sinks have been studied in Refs. [38–40].
Our analysis of traveling waves and defects between
these waves provides a comparison with other much used
universal equations, such as the CGLE and the CCGLEs. We
also show that there exists an intimate connection between
the CSHE and the Cahn-Hilliard and Korteweg-de Vries
equations. 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 [41],
but it also describes coarsening dynamics observed in many
other areas of physics. The Korteweg-de Vries equation was
derived to model shallow-water waves and is one of the
prototypical soliton-bearing equations [42]. We show that
both equations arise naturally as evolution equations for wave
number perturbations of solutions of Eq. (1) in the form
of nonlinear wave trains and relate their solutions to direct
numerical simulations of perturbed wave trains.
The paper is organized as follows. In Sec. II we describe
the properties of simple traveling wave (TW) solutions of
the CSHE and show that such waves are subject to long-
wavelength instabilities. In Sec. III we study the properties
of defects between such waves and describe the similarities
and differences between the behavior of the CSHE and a pair
of CCGLEs describing the dynamics near onset. In Sec. IV we
present the results of numerical simulations of the CSHE in
a variety of different parameter regimes. In Sec. V we derive
phase equations describing the evolution of long-wavelength
instabilities of both the homogeneous oscillating state and
the TW states identified in Sec. II. For the former the phase
equation takes the form of a generalized convective Cahn-
Hilliard equation that describes both coarsening behavior and
spatiotemporal chaos. For the latter the phase equation takes
the form, in appropriate regimes, of a perturbed Korteweg-de
Vries equation describing localized compression or dilation
of the TW. The paper concludes with a brief conclusion in
Sec. VII.
II. TRAVELING WAVE SOLUTIONS
Equation (1) has a nonzero homogeneous solution of the
form
u(x,t) = u0expi?t,
(2)
where
|u0|2= r − k4
0,? = −br + (b − β)k4
0.
(3)
The stability of this solution is determined by writing
u(x,t) = u0[1 + δ(x,t)]expi?t and linearizing in the small
perturbation δ(x,t). The perturbation then evolves according
to the equation
δt=?iζ − 2k2
0(1 + iβ)?δxx− (1 + iβ)δxxxx
−(1 + ib)|u0|2(δ +¯δ).
(4)
This equation is solved using the ansatz δ(x,t) = δ+exp(σt +
ikx) +¯δ−exp(σt − ikx) leading to the dispersion relation
?σ− 2k2
0k2+ k4+ |u0|2?2+?k2?ζ − 2k2
0β?+ βk4+ b|u0|2?2
−(1 + b2)|u0|4= 0.
The two roots σ±of this equation are shown in Fig. 1 as a
function of the perturbation wave number k. One of the roots,
σ−, is always negative (stable) while the other, σ+, may be
positive for small k (long-wavelength instability). This root is
given explicitly by
(5)
σ+=?2k2
0− bζ + 2bβk2
?
0
?k2
?ζ − 2k2
−
1 + bβ +1 + b2
2|u0|2
0β?2?
k4+ ··· ,
(6)
implying that the homogeneous oscillation u0expi?t is
unstable with respect to long-wavelength perturbations when
1 + bβ > bζ/2k2
0.
(7)
This condition is similar to the condition for the presence
of the Benjamin-Feir instability of the Stokes solution of the
CGLE, except that here the instability is suppressed (bζ > 0)
orenhanced(bζ < 0)dependingonthesignoftheproductbζ.
In particular, in the former case instability may be absent for
small k0but present for larger k0, indicating that the intrinsic
scale of the problem has important consequences already for
the simplest solution of the CSHE.
In addition to homogeneous oscillations the CSHE also
admits solutions in the form of TWs. These take the form
u(x,t) = u0exp(i?t − iKx),
(8)
where
|u0|2= r −?k2
0− K2?2,
(9)
? = −br + (b − β)?k2
0− K2?2− ζK2.
The stability of the TW solution is determined by writing
u(x,t) = u0exp(i?t − iKx)[1 + δ(x,t)], and linearizing in
the small perturbation δ(x,t). The perturbation then evolves
according to the equation
δt=?iζ − 2k2
0(1 + iβ)?(−2iKδx+ δxx) − (1 + iβ)?4iK3δx
−6K2δxx− 4iKδxxx+ δxxxx
?− (1 + ib)|u0|2(δ +¯δ).
(10)
This equation is solved using the ansatz δ(x,t) =
δ+exp(σt + ikx) +¯δ−exp(¯ σt − ikx),where
σ
is now
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PHYSICAL REVIEW E 84, 056203 (2011)
complex. Although an exact expression for the real and
imaginarypartsofσ isreadilyobtained(seeFig.2)wecontent
ourselves with expressions for the (potentially) unstable
eigenvalue σ+in the long-wavelength limit k ? 1:
Reσ+=?−bζ + 2(1 + bβ)?k2
|u0|2(1 + b2)?k2
Imσ+= 2Kk?ζ + 2(b − β)?k2
These expressions reduce to earlier expressions in the case
K = 0. In contrast, when K = k0we obtain
Reσ+=?−bζ − 4(1 + bβ)k2
indicating that the TW and the homogeneous oscillations
have, in general, different stability properties. In particular,
it is possible for the TW with K = k0to be unstable while
the homogeneous oscillations with K = 0 are stable, and
0− 3K2??k2
+8K2k2
0− K2?2+ O(k4),
0− K2??+ O(k3).
(11)
(12)
0
?k2+ O(k4),
(13)
vice versa. Moreover, when K(k0− K) ?= 0 the instability
is enhanced by the second term in Eq. (11) and this
term always dominates close to onset (|u0| ? 1) leading to
modulational instability of all states with K(k0− K) ?= 0
at small amplitude. A separate calculation, also near onset
(|u0| ? 1), shows that the TW are initially stable with respect
to amplitude perturbations; as usual, this is a consequence
of the supercriticality of the primary bifurcation to the TW
states.
III. DEFECTS
When a numerical simulation is initialized with small
amplitude noise (e.g., of magnitude ≈10−4) superposed on
the nonzero homogeneous solution (2)–(3) or the TW solution
(8)–(9) the solutions generally evolve into different TWs that
travel in opposite directions in different locations. When this
is the case in a periodic domain an even number of defects
is created within the domain. In one dimension there are two
0.1 0.2 0.30.40.5
-2.05
-2.04
-2.03
-2.02
-2.01
-2.00
0
k
σ-
0.1 0.20.30.4 0.5
-0.020
-0.015
-0.010
-0.005
k
σ+
(a)
(b)
σ-
0.10.20.30.4 0.5
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
k
σ+
bζ=0
bζ>0
0.1 0.2 0.30.40.5
-2.03
-2.02
-2.01
-2.00
0
k
(c)bζ<0
σ-
0.1 0.20.3 0.4 0.5
-2.06
-2.05
-2.04
-2.03
-2.02
-2.01
-2.00
0
k
0.10.2 0.30.4 0.5
-0.004
-0.002
0.002
0.004
0
0
0
0
k
σ+
FIG. 1. The growth rates σ± of infinitesimal perturbations of the nontrivial homogeneous solution (2)–(3) as functions of the wave
number k. Only σ+leads to an instability. Parameters: r = 1, b = −0.5, k0= 0.2, β = 1, and ζ = 0, −0.1,0.1 in (a)–(c), respectively. The
long-wavelength instability is suppressed for bζ > 0 (b) or enhanced for bζ < 0 (c).
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LENDERT GELENS AND EDGAR KNOBLOCHPHYSICAL REVIEW E 84, 056203 (2011)
0.10.2 0.30.4 0.5
-2.06
-2.04
-2.02
-2.00
0
k
σ-
(a)
0.1 0.20.3 0.40.5
-0.0035
-0.0030
-0.0025
-0.0020
-0.0015
-0.0010
-0.00050
k
σ+
(b)
0.10.2 0.30.4 0.5
-2.06
-2.04
-2.02
-2.00
0
k
σ-
0.10.20.30.4 0.5
-0.0010
-0.0005
0.0005
0
0
k
σ+
FIG. 2. Stability analysis of the TW solution (8)–(9). Parameters: r = 1, b = −1, K = k0= 0.2, ζ = 0, and (a) β = 0.9 and (b) β = 1.1.
When K = k0and ζ = 0 there exists a long-wavelength instability for 1 + bβ < 0.
types of defects, sources and sinks. The source, as the name
suggests, acts as a source of waves, while a sink arises at
locations where oppositely traveling waves collide. Figure 3
shows a typical example of such defects in the CSHE. Much
is known about the properties of these defects, mostly gleaned
fromstudiesofdefectsintheCCGLEs[26,33].Reference[37]
providesageneralclassificationofdefectsinoscillatorymedia.
Related structures, called domain walls, connecting homoge-
neousstateswithconstantbutoppositephasehavebeenwidely
studiedinoptics,inbothone[43]andtwo[44–46]dimensions.
A. Onset of absolute instability
A source serves to select the wave number of the waves
emitted from it. This wave number is determined from the
nonlinear dispersion relation by the frequency of the waves
emitted from the core. When the core is broad (i.e., the
amplitude of the solution is small over a broad region) the
frequency selected is the frequency of infinitesimal waves.
These obey the dispersion relation
iσ = r − iζk2− (1 + iβ)?k2
0− k2?2,
(14)
FIG. 3. Temporal simulation of the CSHE with complex coefficients starting from a TW with wave number K0= 0.063 and superposed
small amplitude noise. (a)–(d) Space-time plots of the time evolution of Reu, Imu, the local wave number K ≡ φx, and the amplitude R,
respectively. Panels (a) and (b) show the last 25 time steps of the much longer simulation shown in panels (c) and (d) that show the details
of the slow coarsening process ultimately resulting in a single source and a single sink. White/black regions correspond to high/low values of
K or R. (e)–(h) The profiles at the last time step of (a)–(d). Parameters: r = 1, b = −0.5 and k0= 0.2, L = 300, N = 1024 (parameters not
mentioned are zero).
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where σ ≡ λ + iω and k is the perturbation wave number.
These waves set in at r = 0 with k = k0, and for r > 0
waves with (k2
and frequency ω = −ζk2− β(k2
we can define the phase speed cp= ω/k and the group
speed cg= ∂ω/∂k. The phase speed cp is in general large
compared to cg(this is certainly the case when k is small).
This is also clear from the solution shown in Fig. 3 where
the drift velocity of the defects v ≈ −10−4[see panels (c)
and (d)] is small compared to the phase speed cpof the TW
[|cp| = ω/|K| ≈ 2.5, ω ≈ 0.5, K ≈ ±k0= 0.2, see panels
(a) and (b)]. Because of this large difference in speed the
oscillatory instability is convective at onset. Sources can be
present only when the instability becomes absolute so that
the traveling perturbations grow faster than they propagate.
The computation of the threshold for absolute instability
follows standard procedure [26,34,47]. Using Eq. (14), where
k ≡ kR+ ikI, and seeking the location in the complex k plane
where cg= 0, one finds
iζ = 2(1 + iβ)?k2
Equation (15) together with Eq. (14) represent four equations
which determine kR, kI, the threshold value r = ra, and the
corresponding frequency ω = ωa. It is this frequency that
in turn determines the wave number of the nonlinear waves
emitted by the source. This wave number will in general differ
from the wave number kRselected at r = ra. We obtain
ζ2
4(1 + β2),
together with the equations
k2
I= ρ,
where
ρ ≡1
21 + β2
Thus,
k2
2
?ρ − k2
and the profile of the source at x = 0 is exponential at small
amplitude and given by |u| ∼ exp(±kIx). Note that kR is
nonzero,incontrasttotheCCGLEcasestudiedinRef.[47].In
particular, when ζ = 0 we have |kR| = |kI| = k0; when ζ ?= 0
and k2
The above analysis suggests that when r > ra stationary
sources are present and that these select the frequency ωa
when r ≈ ra; this frequency in turn selects the wave number
K of the emitted waves in the far field according to
ωa= −bra+ (b − β)?k2
With increasing r these symmetric sources may lose stability
in different ways. This loss of stability may arise via a
parity-breakingbifurcationleadingtoadriftingdefectemitting
waves with different wave numbers fore and aft, or via a
Hopf bifurcation leading to a pulsating source. A reflection
symmetry-breaking Hopf bifurcation is also possible [48];
this leads to a one-dimensional spiral wave, where the source
0− k2)2< r have a positive growth rate λ
0− k2)2. Using these results
0− k2?.
(15)
ra=
ωa= −ζk2
0+
ζ2β
4(1 + β2),
(16)
0− k2
R+ k2
kRkI= −ρ
2β,
(17)
?
??k2
βζ
?
.
(18)
R=1
k2
?k2
0− ρ +
0− ρ?2+ ρ2/β2?,
(19)
I=1
2
0+
??k2
0− ρ?2+ ρ2/β2?,
0= ρ > 0 we have |kR| = |kI| =√|ρ/2β|.
0− k2?2− ζK2.
(20)
drifts first in one direction and then in the opposite direction,
resulting in a zigzag space-time trajectory.
On the other hand, when r < rano stationary sources are
expected although stationary sinks may remain. Sinks are,
in general, much more robust since they form as a result
of a collision of waves, and these cannot be perturbed away
[26,47].Theirstructureisdeterminedlocallybytheamplitude,
frequencies, and wave numbers of the incoming wave trains,
and these are, in turn, determined by the sources. If the
incoming waves have different frequencies and wave numbers
the sink may drift. If the incoming waves are Benjamin-
Feir unstable, the sink may undergo irregular back-and-forth
oscillations [26,47]. Thus, both sources and sinks can undergo
complex dynamical behavior. However, we expect that in the
regime 0 < r < ra, present provided ζ ?= 0, time-dependent
sources may remain, just as in the CCGLE case, coexisting
with either stationary or nonstationary sinks, depending on the
properties of the waves emitted by the sources.
An example of the destabilization of the source as r is
reduced below ra≈ 0.125 is shown in Fig. 4. Figures 4(a)
and 4(b) are computed with k0= 0.2 starting from an initial
condition consisting of two TW with wave numbers ±k0each
occupying a region of width L/2. The simulations show, in
agreement with theory, that for r < ra, a time-dependent solu-
tionispresent,whileforr > raatime-independentstructureis
the long-time attractor of the system. This solution resembles
states described by the convective Cahn-Hilliard equation,
as discussed further in Sec. VIA. In contrast, for k0= 1,
for which the Cahn-Hilliard equation no longer applies, no
time-dependent sources are found for r < ra, and the system
insteadevolvestoasingleTW(notshown).However,asshown
in Fig. 4(c), when r > rastable sources are recovered.
B. Onset of convective instability and the CCGLE description
In order to compare our results with those of [26] on the
CCGLEs we derive here the CCGLEs from the CSHE. Such
a derivation requires a judicious scaling of the parameters
and the dependent variable, as appropriate for a computation
nearthethresholdforconvectiveinstability.Wewriter = r0+
?2r2,ζ = ?ˆζ,andintroducetheslowtimeT = ?2t andtheslow
spatial scale X = ?x. Here ? ? 1 is a small parameter whose
magnitude may be defined by setting r2= 1. As a result the
calculation that follows is restricted to a small neighborhood
of r = r0, where r0is determined below. In this neighborhood
we expect the solution u to be small and hence write
u(x,t,?) = ?u0(x,X,t,T) + ?3u2(x,X,t,T) + ···. No term of
the form ?2u1(x,X,t,T) appears in the expansion owing to the
symmetry of CSHE with respect to u → −u. The remaining
parameters k0, b, and β are all treated as O(1) quantities.
These scalings capture the expected balance between forcing
via the term (r − r0)u, advection at the group velocity, weak
dispersion and weak nonlinearity, all of which enter at O(?3)
in the theory that follows. As mentioned below, this is not the
case for other possible scalings of r − r0, ζ, and the dependent
variable u, and such regimes are described by evolution
equations that differ from the classical CCGLE description.
We seek to describe the evolution of the leading order
solutionu0(x,X,t,T)overlongtimes[T = O(1),equivalently
t ∼ ?−2] and large scales [X = O(1), equivalently x ∼ ?−1].
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FIG. 4. TemporalsimulationoftheCSHEwithcomplexcoefficients.(a)–(c)Space-timeplotsofthetimeevolutionofthelocalwavenumber
K. (d)–(f) The profiles K(x) at the last time step of (a)–(c). White/black regions correspond to high/low values of K. Parameters: b = −0.5,
β = −1, ζ = −1, L = 300, N = 1024, ra= 0.125. (a) k0= 0.2, r = 0.1 < ra. (b) k0= 0.2, r = 0.3 > ra. (c) k0= 1, r = 0.3 > ra.
At O(?) we obtain
u0t= r0u0− (1 + iβ)?k2
This equation is solved by
0+ ∂2
x
?2u0.
(21)
u0=AL(X,T)exp(iω0t +ikx) + AR(X,T)exp(iω0t −ikx),
(22)
representing a superposition of left- and right-traveling waves
with slowly varying amplitudes AL(X,T) and AR(X,T),
respectively.SubstitutionofthisexpressionintoEq.(21)yields
the dispersion relation
iω0= r0− (1 + iβ)?k2
0− k2?2.
(23)
Thus,
r0=?k2
0− k2?2,ω0= β?k2
0− k2?2.
(24)
Hence, the onset wave number is given by k = k0, that is, the
gravest mode has wave number equal to the intrinsic wave
number k0, and we have r0= ω0= 0. This is as expected
since we have taken r to be real, thereby eliminating the
leading order frequency. The value r0= 0 corresponds to
the threshold for convective instability, and indeed r0< ra
provided ζ ?= 0. Note also that ω0= 0, while at the absolute
instability threshold ωa?= 0.
The CSHE is identically satisfied at O(?2), while at O(?3)
we obtain
u2t− r0u2+ (1 + iβ)?k2
+4u0xxXX
0+ ∂2
x
?2u2
= −u0T+ r2u0+ 2iˆζu0xX− (1 + iβ)?2?k2
0+ ∂2
x
?u0XX
(25)
?− (1 + ib)|u0|2u0.
Since both components of the general solution (22) are in the
kernel of the linear operator on the left side of Eq. (25) there
are two solvability conditions for this equation:
AL,T− c0AL,X= r2AL+ 4k2
0(1 + iβ)AL,XX
−(1 + ib)(|AL|2+ |AR|2)AL,
0(1 + iβ)AR,XX
−(1 + ib)(|AR|2+ |AL|2)AR,
(26)
AR,T+ c0AR,X= r2AR+ 4k2
(27)
where c0= −2ˆζk0is the group velocity. After rescaling space
and time, these equations form a special case of the equations
studied in Ref. [26], viz.,
AL,T− s0AL,X= r2AL+ (1 + ic1)AL,XX
−(1−ic3)|AL|2AL−g2(1−ic2)|AR|2AL,
(28)
AR,T+ s0AR,X= r2AR+ (1+ ic1)AR,XX
−(1−ic3)|AR|2AR−g2(1−ic2)|AL|2AR,
(29)
with s0= −ˆζ, c1= β, c2= c3= −b, g2= 1. In contrast,
when ζ = O(1), r = O(?2) one obtains instead the nonlo-
cal CCGLEs derived in Ref. [49], while with ζ = O(1),
r = O(?) one obtains the hyperbolic equations derived in
Refs. [50,51]. The relation between the properties of the local
and nonlocal CCGLEs is explored in Refs. [49,52,53]. When
ˆζ = 0 the above equations reduce to a special case of those
studied in Ref. [54], with γ = 1 in their notation. As shown in
Ref. [54] (see also Ref. [26]), this parameter value falls near
the transition from spatiotemporal intermittency to a steady
state with domains of (AL,AR) = (AL,0) and (AL,AR) =
(0,AR) separated by domain walls. When ˆζ = O(1) or
equivalently ζ = O(?) the critical value of the parameter r2
at which the source width diverges identified in Refs. [26,34],
viz., r2= s2
expression (16a).
0/4(1 + c2
1), agrees with the more general
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We mention that the CCGLEs (28) and (29) contain two
extra parameters as compared with Eq. (1), assuming that the
parameter k0is fixed. Consequently, the behavior exhibited by
Eq. (1) is expected to fall between that of a single CGLE and
that exhibited by general CCGLEs. In particular, some aspects
of the mutual suppression of left-TWs by right-TWs and vice
versa may be expected in the dynamics but mutual excitation
should be absent.
IV. NUMERICAL EXPLORATION
From studies of the supercritical CSHE with real coeffi-
cients [30]1we know that the intrinsic wave number plays
an important role in the observed dynamics. In particular,
for small k0 the system exhibits coarsening behavior with
the number of defects gradually decreasing over the course
of time and a corresponding increase in the size of the
defect-freedomainsinbetween.Thisevolutionisdescribedby
the Cahn-Hilliard equation for the phase gradient that can be
derived from the CSHE on the assumption that this gradient is
small. On the other hand, for larger k0the system evolves via
phase slips and any coarsening that may take place initially
is arrested by the formation of bound states of (stationary)
defects[30].Weexploreherethecorrespondingbehaviorwhen
thecoefficientsintheCSHEarecomplexandthedefectsmove.
A. Dynamics in the CSHE with a large intrinsic
length scale (k0small)
1. b ?= 0, β = ζ = 0
Westartournumericalexplorationwiththeregimeinwhich
coarsening is observed in the CSHE with real coefficients
assuming that b ?= 0, β = ζ = 0, and scan the dynamical
behaviorasthecoefficientb becomesmoreandmorenegative.
Figure 3 shows a numerical simulation of the time evolution
in the CSHE with periodic boundary conditions and r = 1,
b = −0.5, and k0= 0.2 with the remaining parameters zero.
Here and in the remainder of this section small amplitude
random noise superposed on the nonzero homogeneous
solution (2)–(3) is taken as the initial condition. The figure
shows that the final state of the system consists of one source
and one sink separating two counterpropagating TWs [see
Figs. 3(e) and 3(f)]. This situation corresponds to a pair of
connections or fronts between a positive and a negative phase
gradientK ≡ φx,therebyformingapairoflocalizedstructures
(LSs); this structure manifests itself in the amplitude as well
[seeFigs.3(g)and3(h)].Thesourceandsinkaredistinguished
by the asymptotic phase speed far from the LS: If this speed is
outward the defect corresponds to a source; if it is inward
the defect is a sink [see Figs. 3(a) and 3(b)]. The source
is an active coherent structure that sends out waves in both
directions, while the sink is sandwiched between TWs whose
phase velocity points inward. It is clear from Fig. 3 that
initially several LSs form and that these annihilate pairwise
leaving only a single sink and a single source which form a
slowly drifting bound state (drift speed v ≈ −10−4). Similar
coarsening behavior is observed for less negative values of
1Equation (37) in Ref. [30] should read β2=
K2
s(k2
0−3K2
R2s
s)
−1
2β2
1.
0
-0.8 -0.7-0.6
b
-0.5 -0.4
-10-4
v
0 100 200300
-0.2
0.2
0
-0.2
0.2
0
-0.2
0.2
0
b = -0.2
b = -0.45
b = -0.8
10-4
2 x10-4
3 x10-4
X
K
K
K
FIG. 5. The velocity of a front in the derivative of the phase
(ashocklikestructureinthelocalwavenumberK ≡ φx)asafunction
of the parameter b obtained by numerical continuation of the solution
in Figs. 3(g) and 3(h) for a fixed spatial average of K (¯ K = 0.063).
Other parameters are as in Fig. 3. The right panels show the resulting
K profiles for three different values of b.
b as well, and is described by a Cahn-Hilliard type equation
derived in Sec. V.
The drift velocity v of the bound state in Fig. 3 is nonzero
because of the presence of an asymmetry between left- and
right-TWs.Thisasymmetryisquantifiedbythespatialaverage
of the wave number K, here
constant.TheresultsofnumericalcontinuationshowninFig.5
reveal thatthedriftspeeddepends ontheparameter b and does
soinanonmonotonicfashion,largelybecauseofchangesinthe
width of this state. For the solution in Figs. 3(g) and 3(h) (b =
−0.5) the drift velocity v ≈ −10−4and is small compared
to the phase speed cp of the waves (|cp| = ω/|K| ≈ 2.5,
ω ≈ 0.5, K ≈ ±k0= 0.2). As b decreases from b = 0, the
(leftward) drift velocity first increases but starts to decrease
again when b ≈ −0.5 and crosses zero when b ≈ −0.67
and thereafter the bound state drifts toward the right. The
solution profiles shown in the right panels in Fig. 5 show that
this reversal in the direction of drift is a consequence of an
overshoot in the profile that develops with decreasing b.
The change in dynamical behavior when b decreases is
exemplified by Fig. 6. When b = −1, one observes an initial
back-and-forth movement of the phase gradient fronts (and
the associated LSs), but after this transient the remaining
LSs (a source and a sink) form a stationary bound state
[see Figs. 6(a) and 6(b)]. With further decrease in b a regime
of spatiotemporal intermittency is encountered [see Figs. 6(c)
and 6(d)]. In this regime, defect chaos coexists with TWs.
Patches of TWs are separated by various LSs, some of
which are created and destroyed in rapid succession, while
others persist for a longer period and display the same type
of back-and-forth movement, as seen in the transient in
Figs. 6(a) and 6(b). Similar spatiotemporal intermittency has
been studied in the context of the CGLE [5,26,31,54]. When
b is decreased even further, the system exhibits more and
more spatiotemporally disordered regimes, whose presence
we attribute to modulational instability of the basic TW state.
The coarsening behavior described here involves sources
and sinks of counterpropagating TWs, in contrast to earlier
studies of coarsening behavior between domain walls in
one [43] and two [44–46] dimensions. However, curvature
effects that give rise to different types of growth laws in two
dimensions [44–46] are absent.
¯ K = 0.063, which remains
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FIG. 6. Temporal simulation of the CSHE with complex coef-
ficients. Space-time plots of the time evolution of the local wave
number K ≡ φx[(a), (c), (e)] and the amplitude R [(b), (d), (f)] for
increasing values of |b|. White/black regions correspond to high/low
values of K or R. Parameters: r = 1, b = −1 [(a), (b)], b = −1.5
[(c), (d)], b = −2 [(e), (f)] and k0 = 0.2, L = 300, N = 1024
(parameters not mentioned are zero).
2. b ?= 0, β ?= 0, ζ = 0
Figures7–9showtheresultsofnumericalsimulationswhen
bremainsfixed(b = −1)andtheparameterβ isvariedinstead.
Figure 7 shows typical behavior that is observed when β < 0.
For all β < 0 coarsening dynamics are found, and coarsening
proceeds until only one source-sink pair is left in the system.
Both the source and the sink are then stable and either remain
stationary or slowly drift in time.
FIG. 7. Temporal simulation of the CSHE with complex coeffi-
cients. Space-time plots of the time evolution of (a) the local wave
number K ≡ φxand (b) the amplitude R. White/black correspond to
high/low values of K or R. Parameters: r = 1, b = −1, β = −2, and
k0= 0.2, L = 300, N = 1024 (parameters not mentioned are zero).
When 0 < β ? 1 spatiotemporal intermittency is found.
Figure 8, computed for β = 1 shows typical space-time plots
in this regime. The figure reveals a uniformly drifting complex
oscillation in the real and imaginary parts of u with the
property that R ≈ 1 throughout the duration of the time series
[see Fig. 8(h)]. Thus, the oscillation shown is associated with
phase variation and not amplitude variation. However, for
larger β the behavior changes qualitatively and for β ? 1
drifting TW-like states are no longer found in numerical simu-
lations. With random initial conditions, the system evolves to
a state which oscillates with wave number comparable to k0in
both the phase gradient K = φxand the amplitude R, while
slowly drifting. A state of this type is shown in Figs. 9(a)
and 9(b), which reveal that the spatial oscillations are likely
quasiperiodic, with a longer wavelength modulation super-
posed on the 2π/k0wavelength. This modulation becomes
more and more pronounced as β increases. We can see, for
example, in Figs. 9(c) and 9(d), that an increase in β leads
to an increase of the local wave number K and the creation
and destruction of defects resulting in spatiotemporal chaos.
Such perturbations of the basic state [panels (a) and (b)] arise
spontaneously, indicating the presence of a linear instability.
The dynamics evolve on faster time scales and no coherent
structurescanbeobserved.Similarbehaviorisalsoobservedin
the CGLE [5,26]. In the case of the CGLE, this spatiotemporal
chaos has been described using the Kuramoto-Sivashinsky
equation [31]. We show in Sec. V that this is so also in the
present case.
B. Dynamics in the CSHE with a larger intrinsic
length scale [k0= O(1)]
InpreviousworkontheCSHEwithrealcoefficients[30,55]
we have shown that coarsening terminates when the intrinsic
wavenumberk0islarger,andcharacterizedthetransitionfrom
coarseningdynamicstothefrozenstatethatresults.Thefrozen
stateconsistsofboundpairsofdefectslockedtogetherviatheir
oscillatory tails, and the crossover transition can be accurately
attributed to the first appearance of these tails.
In this section, we numerically explore the dynamical
behavior that arises in the CSHE with complex coefficients
when k0= 1, a value large enough for oscillatory tails to exist.
As before, we first consider the case β = 0 and then allow
both b and β to have nonzero values.
1. b ?= 0, β = ζ = 0
The results for β = ζ = 0 and small negative values of b
are shown in Fig. 10. Starting from random initial conditions,
the system relaxes after a brief transient behavior to a bound
state of two breathing LSs which drifts to the left at a constant
speed.Incontrasttothecaseofsmallk0,morethanonedrifting
source and sink pair can be present in the system as long-time
coarsening no longer takes place. The constituents of these
boundstatesaretypicallyunsteady,exhibitingsmallamplitude
oscillations that may be periodic or chaotic. Moreover, Fig. 10
also reveals that the wave number within the bound state is no
longer constant and instead varies linearly with the position x
within the structure. This type of behavior is characteristic of
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FIG. 8. Temporal simulation of the CSHE with complex coefficients. Space-time plots of the time evolution of (a) Reu, (b) Imu, (c) the
local wave number K ≡ φx, and (d) the amplitude R ≡ |u|. (e)–(h) The profiles at the last time step of (a)–(d). White/black regions correspond
to high/low values of K or R. Parameters: r = 1, b = −1, β = 1, and k0= 0.2, L = 300, N = 1024 (parameters not mentioned are zero).
FIG. 9. Temporal simulation of the CSHE with complex coefficients. Space-time plots of the time evolution of (a), (c) the local wave
number K ≡ φxand (b), (d) the amplitude R. (e)–(h) The profiles at the last time step of (a)–(d). White/black regions correspond to high/low
values of K or R. Parameters: r = 1, b = −1, β = 2 (a), (b), (e), (f), β = 3 (c), (d), (g), (h), and k0= 0.2, L = 300, N = 1024 (parameters
not mentioned are zero).
FIG. 10. Temporal simulation of the CSHE with complex coefficients. Space-time plots of the time evolution of (a), (c) the local wave
number K ≡ φxand (b), (d) the amplitude R. (e)–(h) The profiles at the last time step of (a)–(d). White/black regions correspond to high/low
values of K or R. Parameters: r = 1, b = −0.5 (a), (b), (e), (f), b = −1 (c), (d), (g), (h), and k0= 1, L = 300, N = 1024 (parameters not
mentioned are zero).
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LENDERT GELENS AND EDGAR KNOBLOCHPHYSICAL REVIEW E 84, 056203 (2011)
FIG. 11. Temporal simulation of the CSHE with complex coef-
ficients. Space-time plots of the time evolution of the local wave
number K ≡ φx[(a), (c), (e)] and the amplitude R [(b), (d), (f)], for
decreasingvaluesofb.Panels(c)and(d)representazoomoftheblack
boxin(a)and(b).White/black regionscorrespondtohigh/lowvalues
of K or R. Sinks (Si) are stationary and stable for long time intervals,
while sources (So) oscillate wildly. Nucleation points creating or
destroying a source/sink pair are denoted by N1, N2. Parameters:
r = 1, b = −3 (a)–(d) and b = −5 (e) and (f), and k0= 1, L = 300,
N = 1024 (parameters not mentioned are zero).
systems described by a Burger’s type equation, as discussed
further in Sec. VI.
Figure 11 shows spatiotemporal chaotic regimes that are
found when b is more negative. Instead of drifting LSs, one
observes a stable stationary sink solution Si, while the source
So undergoes apparently chaotic oscillations. This behavior
is reminiscent of the source-induced bimodal chaos that is
observed in CCGLEs [26], although in the latter case the sink
slowly drifts. However, the bimodal chaos is only observed
in CCGLEs and is not present in a single CGLE. While
the sink remains stationary and stable over long periods of
time, one can see in Figs. 11(a) and 11(b) that at certain
points in time the sink Si is abruptly destroyed and recreated
elsewhere in space (b = −3 in this case). Panels (c) and
(d) show a magnification of such an event [indicated by
the black box in Figs. 11(a) and 11(b)]. The magnification
shows that a new stable sink/chaotic source pair nucleates at
point N1 first and does so via the original chaotic source;
soon after this occurs the original sink and chaotic source
annihilate one another in point N2. Thus, after this process is
completed only one stable sink/chaotic source pair is again
present in the system, but at a different location than before.
This behavior repeats intermittently in time. To the best of
our knowledge, such behavior has not been observed in other
systems.
Further decrease in b leads to multiple stable sink/chaotic
source pairs within the domain as illustrated in Figs. 11(e)
and 11(f) for b = −5. For this parameter value nucleation
of new sink/source pairs (and subsequent annihilation) is
apparently absent and the observed dynamic structure persists
indefinitely, much as in the CCGLEs [26].
Qualitatively the same behavior is observed for positive
values of b as well. For small values of b, single or multiple
drifting source/sink pairs are observed, while increasing b
eventually leads to the creation of one stable sink/chaotic
sourcepair(e.g.,forb = 3).Foryethighervaluesofb,multiple
coexisting stable sink/chaotic source pairs are again found
(e.g., for b = 5).
2. b ?= 0, β ?= 0, ζ = 0
Figure 12 shows the dynamics that are observed when
β ?= 0. Since a complete characterization of the spatiotem-
poral dynamics in the (b,β) parameter space is beyond the
scope of this work we limit ourselves to analyzing the
dynamical behavior along the b = −1 slice through this
space, scanning β from −5 to 5. The following behavior is
observed.2
(i) For β < −3.4, the whole space is filled with coexisting
stationary source/sink pairs. For the chosen domain width
L = 300, either three or four pairs are present in the system
[see, e.g., Fig. 12(a); β = −5].
(ii) In the region −3.4 < β < −2.5, the same source-sink
pairsstillcoexist,butarenolongerstationaryandexhibitslight
“breathing” (time-periodic) behavior [see, e.g., Fig. 12(b);
β = −3].
(iii) When β increases within the interval −2.5 < β <
1.2, the number of source/sink pairs in the system tends
to diminish. Furthermore, on top of the “breathing” be-
havior, the sources and sinks start to drift [see, e.g.,
Figs. 12(c) and 12(d); β = −2.5 and β = 1]. We remark that
around β ≈ −0.1 the drift speed of the structures becomes
very slow.
(iv) For 1.2 < β < 4, both source and sink are again
stabilized and remain stationary [see, e.g., Fig. 12(e); β = 3].
(v) In the interval 4 < β < 4.2, time-periodic “breathing”
behavior is again observed, but with time-periodic oscillations
that are much more pronounced than in the previous cases.
Moreover,onlythesourceshowstime-periodicbehavior,while
the sink remains stationary and stable [see, e.g., Fig. 12(f);
β = 4.1].
(vi) When increasing β in the interval 4.2 < β < 4.8, the
time-periodicity ofthe sourceislostand isreplaced bychaotic
2We remark that the boundaries mentioned in this classification
of dynamical behavior have been numerically determined with an
accuracy of ?β = 0.1.
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FIG. 12. Temporal simulation of the CSHE with complex coefficients in terms of space-time plots showing the time evolution of the
local wave number K ≡ φx and the amplitude R. White/black regions correspond to high/low values of K or R. Parameters: r = 1, b =
−1, β = −5, (a) −3, (b) −2.5, (c) 1, (d) 3, (e) 4.1, (f) 4.5, (g) and 5, (h) k0= 1, L = 300, N = 1024 (parameters not mentioned are
zero).
behavior; the sink remains stable and more or less stationary
[see, e.g., Fig. 12(g); β = 4.5].
(vii) Finally, for β > 4.8, the source is again only mildly
chaotic but additional time-dependent substructures appear in
the middle of each TW domain that oscillate in phase with the
source [see, e.g., Fig. 12(h); β = 5].
Stable breathing sinks similar to Fig. 12(f) have recently
been computed in doubly diffusive convection [56].
In the next section, we derive a nonlinear phase equation
modeling the evolution of long-wavelength perturbations of
the homogeneous amplitude solutions. This phase equation
will prove useful for interpreting some of the numerically
observed dynamics.
V. NONLINEAR PHASE EQUATIONS
Large length scale perturbations of the phase of a wave
evolve on a slow time scale relative to perturbations of
the amplitude. This fact allows us to decouple the amplitude
perturbations from those of the phase, through a procedure
analogous to center manifold reduction, a reduction procedure
employed in studies of finite-dimensional dynamical systems.
The equation that results is an equation for the slow evolution
of φ, the wave phase, or, equivalently, for the wave number
K ≡ ∂xφ. We consider two cases, the first being the evolution
of spatial perturbations of the homogeneous oscillations in
time (|K| ? 1) and the second studying perturbations of a
TW [K = O(1)].
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WebeginbyrewritingtheCSHEintermsoftheamplitudeR andthephasegradientK ≡ ∂xφ,whereu(x,t) ≡R(x,t)expiφ(x,t).
Thus,
Rt= rR − R3− k4
+β?−4K3Rx+ 6KxRxx+ 4RxKxx+ 4KRxk2
Kt= −2bRRx−
?
0R +?−2k2
0Rxx+ 6K2Rxx+ 12KRxKx− RK4+ 2RK2k2
0+ 4KRxxx− 6RK2Kx+ 2Rk2
0+ 3RK2
x+ 4KKxxR − Rxxxx
0Kx+ RKxxx
?
?− ζ[2RxK + RKx], (30)
0Kx+ 4k2
R
?
x
?
4Rxxx
R
K + 6Rxx
R
Kx+ 4Rx
RKxx+ Kxxx− 4Rx
RK3− 6K2Kx+ 2k2
0KRx
?
x
+β
−2k2
0
Rxx
R
+ 6K2Rxx
R
+ 12KKxRx
R
− K4+ 2k2
0K2+ 3K2
x+ 4KKxx−Rxxxx
R
+ ζ
?Rxx
R
− K2
?
x
.
(31)
In this formulation the temporal oscillation frequency ? of the
wave train is absent.
Equations (30) and (31) are exact. In the following we
consider two distinct cases. In the first we take a finite
amplitude spatially homogeneous oscillation, corresponding
to R = R0(a constant) and K = 0. We consider wave number
perturbations with K = O(?), where ? ? 1; that is, we
consider the evolution of the system from initial conditions
that vary slowly in space, on an O(?−1) scale only. In order to
balancetheevolutionontheselargescalesweneedtoselectthe
correct size of the perturbation R = R0+ u. Since the basic
oscillation is invariant under reflection x → −x the resulting
evolution equation must also respect this symmetry. In the
second case, we take R = R0 (a constant) and K = O(1),
representing a homogeneous wave train. The balances in this
case are different, and since the basic wave train is no longer
invariant under x → −x the resulting evolution equation for
wavenumberperturbationswillalsobedifferent.Inparticular,
the wave number perturbations are now expected to propagate
and at the same time to steepen.
In order to capture transitions between different regimes
we do not explicitly scale the variables with ? but employ an
iterative method that can be checked for consistency at each
step.UnlessotherwisestatedallparametersaretreatedasO(1)
quantities, in contrast to the CCGLE derivation.
A. The |K| ? 1 case
We write the perturbation of a homogeneous oscillation
in the form R = R0+ u, K = v, where R2
take v to be O(?), thereby defining the modulation scale ?−1.
We suppose that the amplitude perturbation u varies on the
same scale, and hence take all spatial derivatives to be O(?).
Temporal derivatives are taken to be slow, relative to ?−1, and
will, in fact, be O(?4), as can be checked a posteriori. The
primary balance in Eq. (30) is then provided by
0≡ r − k4
0, and
ut= −2R2
0u + 2k2
0R0(βvx+ v2) − ζR0vx+ h.o.t.,
(32)
where h.o.t. denotes higher order terms. Since vxand v2are
both O(?2) we conclude that we must take u = O(?2); that is,
welinktheperturbationamplitudeutothe(inverse)scaleofthe
modulation as specified by v. Thus, R0u ≈ (k2
0β −1
2ζ)vx+
k2
results in a modified convective Cahn-Hilliard-type equation:
vt= [γv − avxx− κ0v3]xx+?1
This equation is equivariant with respect to the operation
x → −x, v → −v, a symmetry inherited from the fact that
v ≡ φx, with the phase φ a scalar. When the coefficients γ and
D are both O(1) the wave number v evolves on an O(?−2)
time scale and is of Burger’s type at leading order, that is,
at O(?3). The shock formation that results is regularized by
the diffusion coefficient γ provided γ > 0. However, when
γ < 0 (Benjamin-Feir instability) the diffusion term leads to
singularitiesinfinitetimeandtocontroltheseitisnecessaryto
carry the derivation of the phase equation to O(?5), resulting
in the presence of several additional terms, all of which are
included in Eq. (33). These terms become important on an
O(?−4) time scale, implying that the above expansion is
no longer uniform in time. Of the new terms, those in the
first bracket constitute the classical Cahn-Hilliard equation
describingcoarseningoverlongtimescales.Theappearanceof
this equation in the context of phase evolution goes back to the
workofMalomedetal.[35](seealso[30,55,57,58]).However,
the terms in the second bracket lead to nonlinear steepening
resulting in instabilities that arrest and disrupt the slow
coarseningprocessasdescribedforc = d = e = 0inRef.[59]
and in the general case in Sec. VIA. For other choices of
the coefficients Eq. (33) resembles the Kuramoto-Sivashinsky
equation, and on large domains exhibits spatiotemporal chaos
[31]. A discussion of this regime and a comparison of its
properties with those of the CSHE is deferred to Sec. VI.
The coefficients in Eq. (33) are given by
0v2. Substituting this result in Eq. (31) for the phase gradient
2Dv2+ cv2
x+ dv4+ evvxx
?
x
+h.o.t.(33)
γ = −2k2
κ0= −2
D = 4k2
0(1 + bβ) + bζ,
1 −k4
R2
0
0(β − b) − 2ζ,
?2k2
?2k2
(34)
?
0
+1
6
γk2
R2
0
0
?
,
(35)
(36)
a = 1 +
0β − ζ?2
0
2R2
,
(37)
c = 3β −
b
4R2
0
0β − ζ?2−2k2
0
R2
0
?2k2
0β − ζ?,
(38)
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d = −β −bk4
?
0
R2
?
0
+4k2
,
(39)
e = 4β
1 −2k4
0
R2
0
0ζ
R2
0
.
(40)
Since a > 0 Eq. (33) is well posed even when γ < 0.
Assuming that v = O(?), ∂x= O(?), and ∂t= O(?4) the
terms in Eq. (33) are all of the same order provided we take
γ and D are both O(?2). In this case the steepening from the
Burgers term and the Benjamin-Feir instability all occur on
the same time scale as its saturation via higher-order diffusion
and nonlinearity. If this is the case the expansion becomes
uniform in time and Eq. (33) formally describes the whole
time evolution of any initial wave number distribution with
only O(?−1) spatial variation. This condition is valid for all b
and β when
k2
0∼ ζ = O(?2).
(41)
B. The K = O(1) case
The corresponding derivation for the case K ?= 0 is con-
siderably more involved largely as a consequence of the slow
drift identified in Eq. (12) when the modulation wave number
k is small. As a result we write ξ = x − ct, where c is the
(unknown) drift speed and write Eqs. (30) and (31) in the
comoving frame:
Rt= cRξ+ rR − R3− k4
+β?−4K3Rξ+ 6KξRξξ+ 4RξKξξ+ 4KRξk2
Kt= cKξ− 2bRRξ−
?
0R +?−2k2
?
Rξξ
R
0Rξξ+ 6K2Rξξ+ 12KRξKξ− RK4+ 2RK2k2
0+ 4KRξξξ− 6RK2Kξ+ 2Rk2
4Rξξξ
R
0+ 3RK2
0Kξ+ RKξξξ
ξ+ 4KKξξR − Rξξξξ
?− ζ[2RξK + RKξ], (42)
0Kξ+ 4k2
?
ξ
?
K + 6Rξξ
RKξ+ 4Rξ
RKξξ+ Kξξξ− 4Rξ
RK3− 6K2Kξ+ 2k2
0KRξ
R
?
ξ
?
+β
−2k2
0
+ 6K2Rξξ
R
+ 12KKξRξ
R
− K4+ 2k2
0K2+ 3K2
ξ+ 4KKξξ−Rξξξξ
R
+ ζ
?Rξξ
R
− K2
ξ
.
(43)
We next rewrite these equations in terms of amplitude and
wave number perturbations u, v defined by R = R0(1 + u),
K = K0(1 + v), where the subscript zero refers to the TW
solution (9) with R0= |u0| and K = K0. We solve the
resulting equations on the assumption that both u and v are of
order ? ? 1, and suppose that the ξ derivatives are of order
? as well; that is, we write (u,v) → ?(ˆ u,ˆ v), ξ →ˆξ/?, where
the quantities with a hat are formally of order one. In the
calculationthedriftateveryordermustbeincorporatedintothe
speed c. Thus, c = c0+ ?c1+ ?2c2+ ···, and the remaining
time derivative refers to growth arising from the real part of
the unstable eigenvalue only; as shown below, this derivative
enters the calculation at O(?3). Since v is the slow mode the
perturbation u is slaved to the (slow) evolution of v and we
write (omitting hats)
u = Av + ?Bv2+ ?Cvξ+ ?2Dv3+ ?2Evvξ+ ?2Fvξξ
+ ··· ,
and assume that both u and v are functions of the slow times
τ = ?2t and T = ?3t, that is, u(ξ,t) = u(ξ,τ,T), v(ξ,t) =
v(ξ,τ,T). We summarize below the major steps in the
derivation of the phase equation which we write in the
form
(44)
K0vt= ?2Qvξξ+ ?2S(v2)ξ+ ?3Uvξξξ+ ?3V(vvξ)ξ
+?3W(v3)ξ+ O(?4),
where ∂t= ?2∂τ+ ?3∂T and Q,...,W are coefficients de-
pending on the parameters k0, b, β, ζ, and the wave number
K0of the basic wave train and computed below. Note that
this equation is no longer equivariant with respect to x → −x,
(45)
v → −v. This is a consequence of the nonzero value of the
wave number K0.
At O(?) the above approach yields
?k2
A =2K2
00− K2
R2
0
0
?
,
(46)
while at O(?2) we find that
B =K2
0
R2
0
?k2
0− 3K2
0
?−2K4
0
R4
0
?k2
0− K2
0
?2,
(47)
C =
K0
2R2
0
?
−4βK2
?4K2
0+?−ζ + 2β?k2
0− K2
R2
0
0− K2
0
??
0− K2
×
0
?k2
0
?
+ 1
?
+2c0K0
R2
0
?k2
0
??
,
(48)
where c0is thus far unknown. Substituting these results into
the v equation we obtain an equation of the form
K0vt= ?Pvξ+ ?2c1vξ+ ?2Qvξξ+ ?2S(v2)ξ+ O(?3).
Since the drift is already incorporated in the drift speed c the
O(?) term on the right side cannot be balanced by the time
derivative and we must therefore demand that P = 0, that is,
that
?k2
an expression that yields the result (12) for the frequency of
small wave number perturbations. With this value of c0we see
that v evolves on an O(?−2) time scale in the comoving frame,
(49)
c0= 4(b − β)K0
0− K2
0
?+ 2ζK0,
(50)
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LENDERT GELENS AND EDGAR KNOBLOCH PHYSICAL REVIEW E 84, 056203 (2011)
that is, on the time scale τ = O(1). Moreover, we also obtain
expressions for the coefficients Q and S. Specifically,
Q
K0
= ζb− 2(1+ βb)?k2
0− 3K2
0
?−8K2
0
R2
0
(1+ b2)?k2
0− K2
0
?2,
(51)
a result that agrees with the real part of the unstable phase
mode eigenvalue given in Eq. (11), while
?ζ + 2(b − β)?k2
Finally, since the variable ξ is, by definition, the correct
comovingframe,noadditionaldriftmayoccuratO(?2)andwe
must take c1= 0. Thus, to O(?) the evolution of the instability
is described by Burger’s equation [60],
S = −K2
00− 3K2
0
??.
(52)
K0vτ= Qvξξ+ S(v2)ξ+ O(?),
provided only that Q/K0> 0 so that diffusion arrests the
steepening arising from the nonlinear term. We remark that
c0= −∂?/∂K0and S = (1/2)K2
thedispersionrelation(9b),inaccordwithgeneraltheory[61].
However, in the regime of greatest interest (the Benjamin-Feir
unstable regime), the diffusion coefficient Q is destabiliz-
ing (Q < 0) and the Burger’s problem is ill posed unless
higher order terms are included. We calculate these terms
next.
AtO(?3)theequationfortheevolutionoftheperturbationv
includes the terms c2vξ+ Uvξξξ+ V(vvξ)ξ+ W(v3)ξ, where
U = −2bR2
+β?−2k2
0
+(1 + b2)(k0− K0)2?ζ − 2βk2
0
(53)
0∂2?/∂K2
0, where ?(K) is
0F + 4K3
0A + 6K2
0/R4
0C − 4K0k2
0A + 4K2
??−8b(1 + b2)K2
−(b − β)R4
V = −2bR2
+12K3
=4K3
R4
0
+3K4
and
0C
0
?k2
?+ ζA
0+ 6βK2
=?4K2
00− K2
0
?3
0
?R2
0
?,
(54)
0(AC + E) + 4K3
0− 4k2
?8(1 + b2)K2
0
0(2B − A2+ 3A)
0K0(A − A2+ 2B)
?k2
0+ 3(1 + bβ)R4
0
00− K2
0
?3− 4(1 + b2)?k4
??,
0− 4k2
0K2
0
?R2
0
(55)
W = −2bR2
To evaluate the above expressions we computed the coeffi-
cients D, E, F in Eq. (44). Once again we may take c2= 0,
obtaining
?R2
R6
00
×?ζ − 2(2b + β)?k2
R6
00
×?ζ − 2βk2
0(AB + D) − 4βK4
0= 4(b − β)K4
0.
(56)
R6
0D = −2K4
00+?k2
?k2
?4(1 + 2b2)K2
0− K2
0
?2??R2
0− 2k2
0K2
0+ 2K4
0
?,
(57)
0E = −24bK5
0− K2
?3− 6βK3
?k2
0R4
??,
0− R2
0K3
0
?k2
0− K2
0
?
0− 3K2
0
(58)
?
(59)
0F = 2K2
0− K2
?R2
0
?3− b?k2
0
0− K2
0
0+ 6βK2
00+ R4
?.
The expressions for E and F include contributions from the
leading order estimate of the term uτ= Avτ+ O(?). In view
of Eq. (49) this term is O(?3) in unscaled variables.
Equation (45) generalizes Eq. (33) to the case K0?= 0,
that is, to long wave perturbations of TWs, and shows that
even in the comoving frame the wave number perturbation
v evolves on two time scales, t = O(?−2) and t = O(?−3).
The perturbation grows and steepens on the faster time scale
and only equilibrates on the slower time scale, once again
indicating that if Q < 0 the resulting evolution is nonuniform
in time.
When Q < 0, that is, Reσ+> 0 [see Eq. (11) with K =
K0], the uniform wave number state (v = 0) is unstable. Near
the threshold Q = 0 the time scale balance changes since
the growth rate of the instability is slower than assumed
in the above derivation. Specifically, if we take Q = ?2ˆ Q,
where ˆ Q = O(1), we must assume for consistency that v =
?w. Here w = O(1) and hence the unscaled wave number
perturbation v = O(?2), that is, smaller than assumed in
the derivation of Eq. (45) where the coefficients Q,S,...
are all formally of order one. In this case w evolves on
an O(?−3) time scale and we write w = w(ξ,T), where
T = ?3t. Repeating the above derivation we find that in the
near-threshold regime O(?2) wave number perturbations obey
the perturbed Korteweg-de Vries equation
K0wT− 2Swwξ− Uwξξξ
= ?[ˆ Qwξξ+ V(wwξ)ξ+ Zwξξξξ] + O(?2),
derivedinanumberofpreviousstudies[62–64].Equation(60)
is also known as the modified Kawahara equation [65,66] and
in other regimes as the generalized Kuramoto-Sivashinsky
equation [67]. Once again, in this equation the coefficients
S, U,ˆ Q, V, and Z are all formally of order one, with Z given
by
(60)
Z = −K2
0
2R6
0
?64(1 + 2b2)K4
−24bK2
+??ζ − 2βk2
In order that Eq. (60) be well posed we demand that Z < 0.
Whenthisisnotthecaseayethigherordercalculationbecomes
necessaryinordertoidentifytermsthatmayarresttheresulting
shortwave instability.
In writing Eq. (60) we have included O(?) terms. These
terms govern the evolution of w on yet longer time scale,
t = O(?−4), but are required in order to select among the
different solutions of the leading order Korteweg-de Vries
equation
0
?k2
0− K2
0
?4
0
?k2
0− K2
?2+ 4?8k2
0
0
?2?ζ − 2βk2
?R4
0+ 6βK2
0
?β?K2
?R2
0
00+ 3?ζ − 2βk2
0
00
−4(8 − 9β2)K4
0+ 2R6
?.
(61)
K0wT− 2Swwξ− Uwξξξ= 0.
The O(?) terms are essential since the CSHE is a dissipa-
tive system while the Korteweg-de Vries equation (62) is
conservative and completely integrable. As discussed below
these higher order terms select the amplitude and speed
(relative to the moving frame ξ) of both solitonlike solutions
and cnoidal solutions which is undetermined within (62),
(62)
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PHYSICAL REVIEW E 84, 056203 (2011)
a process that takes place on an O(?−4) time scale as
discussed in Refs. [63,65,68,69]. Following [63] we obtain the
solution
w = Nsech2[ρ(ξ − sT)] + O(?),
(63)
where
ρ =
?SN/6U,s = −2SN/3K0.
(64)
Theamplitudeofthissolutionevolvesontheslowertimescale
T?≡ ?T according to the (corrected) equation
SN2
K0U2[21ˆ QU − (10SZ − 12UV)N].
dN
dT?= −
8
945
(65)
Consequently, we expect, in this regime, the presence of
solitons of amplitude
N0=
21ˆ QU
10SZ − 12UV.
(66)
Within the perturbed Korteweg-de Vries equation these
solitary waves are amplitude-stable whenever K0S(12UV −
10SZ) > 0. An example of such a solitary wave solution
is shown in Fig. 13(a) for parameter values such that the
coefficients in the perturbed Korteweg-de Vries equation (60)
are all O(1), with Q < 0, Z < 0 and ? small, while Fig. 13(b)
shows the soliton in panel (a), appropriately rescaled, in terms
of the corresponding wave number perturbation in the CSHE.
Such a choice demands a fine-tuning of the CSHE parameters
as the coefficients S,Q... can be very sensitive to changes
in these parameters. Since the length scale in the system has
been defined by demanding that |ˆ Q| = |K0|, implying that
? =√|Q|/|K0|, we can only regard Eq. (60) as a perturbation
of the Korteweg-de Vries equation (62) if the coefficient Q is
small. Figures 14(a) and 14(b) show the dependence of Q on
ζ and K0, with the remaining parameters as in Fig. 13(a).
Thus, Q is negative and small only when (a) ζ ≈ −0.1
(K0= 0.505) and (b) K0≈ 0.5 (ζ = −0.1), motivating the
choice of parameters used in Fig. 13(a). Panels (c) and (d) of
Fig.14showthecorrespondingvariationinρ,whilepanels(e)
and (f) depict the variation in the amplitude N0of the soliton
solution. For this parameter set, only dark solitons (N0< 0)
exist close to Q ≈ 0, representing a localized dilation of the
wavelength of the original wave train. We mention that nu-
merical simulations of TW doubly diffusive convection reveal
steadily traveling compression pulses (N0> 0) in appropriate
parameterregimes[70].Westudysolitarywavesolutionsinthe
perturbed Korteweg-de Vries equation (60) and in the CSHE
in Sec. VIB.
Because of the absence of oscillatory tails the Korteweg-de
Vriessolitonsdonotformboundstates.However,boundstates
form for larger values of ? when additional terms enter into
the equation, and the solitary wave becomes asymmetric. This
is so both for the modified Kawahara equation [64] and for
the convective Cahn-Hilliard equation [71], suggesting that
the observation of traveling bound states in the CSHE such as
those in Figs. 10(a) and 10(b) is likely a common occurrence.
Similar bound states form in the generalized Kuramoto-
Sivashinsky equation, that is, Eq. (60), with ? = O(1) and
V = 0 [67].
The Korteweg-de Vries equation (62) also admits spatially
periodic, traveling solutions called cnoidal waves. These take
the form
?SN
−2S?− 3k2
wcn= w0+ Ncn2
?1
cnw0+ N − 2k2
3K0k2
cn
kcn
6U
?
ξ
cnN?
T
?
,kcn
?
, (67)
parametrized by w0, kcn, and the amplitude N. The elliptic
modulus kcn is determined by the spatial period L of the
domain. If the domain contains n wavelengths,
L = n?(kcn) = 2nK(kcn)kcn
?6U
SN
?6U
SN
1
cnsin2sds.
= 2nkcn
?π/2
0
?1 − k2
(68)
In order that wcnrepresent a perturbation of the basic wave
numberK0itmustsatisfythecondition?L
N of the cnoidal solution is selected by the perturbation terms
on the right side of Eq. (60). To determine N we multiply both
0wcn(ξ)dξ = 0.This
requirementdeterminesthequantityw0.Finally,theamplitude
FIG. 13. (a) A soliton solution in the perturbed Korteweg-de Vries equation (60) for b = −1, β = 0.9, ζ = −0.1, K0= 0.505, k0= 0.1,
and r = 1, corresponding to the coefficient values S = −0.7062, U = 1.199,ˆ Q = −K0= −0.505, V = −0.6916, Z = −0.5397, ? = 0.098.
(b) The solution (a) interpreted within the CSHE as a perturbation of the wave number K (here a localized wavelength dilation).
056203-15