Periodic travelling waves in cyclic populations: Field studies and reaction-diffusion models

Abstract

Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reaction-diffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reaction-diffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical.

Full text

Propagating fronts in the complex Ginzburg-Landau equation generate fixed-width bands
of plane waves
Matthew J. Smith*
Microsoft Research, 7 J.J. Thompson Avenue, Cambridge CB3 0FB, United Kingdom
Jonathan A. Sherratt†
Department of Mathematics and Maxwell Institute for Mathematical Sciences,
Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
共Received 30 April 2009; revised manuscript received 1 September 2009; published 20 October 2009兲
Fronts propagating into an unstable background state are an important class of solutions to the cubic
complex Ginzburg-Landau equation. Applications of such solutions include the Taylor-Couette system in the
presence of through flow and chemical systems such as the Belousov-Zhabotinskii reaction. Plane waves are
the typical behavior behind such fronts. However, when the relevant plane-wave solution is unstable, it occurs
only as a spatiotemporal transient before breaking up into turbulence. Previous studies have suggested that the
band of plane waves immediately behind the front will grow continually through time. We show that this is in
fact a transient phenomenon and that in the longer term there is a fixed-width band of plane waves. Moreover,
we show that the phenomenon occurs for a wide range of parameter values on both sides of the Benjamin-
Feir-Newell and absolute instability curves. We present a method for accurately calculating the parameter
dependence of the width of the plane-wave band facilitating future experimental verification in real systems.
DOI: 10.1103/PhysRevE.80.046209 PACS number共s兲: 05.45.⫺a, 02.70.⫺c, 47.54.⫺r
I. INTRODUCTION
The cubic complex Ginzburg-Landau equation 共CGLE兲is
one of the most studied nonlinear equations in physics 关1兴.It
is a generic model for weakly nonlinear spatially extended
oscillatory media arising as the amplitude equation near a
supercritical Hopf bifurcation 关2兴. An important problem in
such systems is front propagation into an unstable state and
the CGLE has been fundamental to the study of this problem
in contexts including the Taylor-Couette system in the pres-
ence of through flow 关3,4兴and chemical systems such as the
Belousov-Zhabotinskii reaction 关5兴. Here we report on, and
explain, a previously unrecognized phenomenon associated
with such front propagation: a fixed-width band of plane
waves behind the front 共Fig. 1兲. This phenomenon occurs for
a wide range of parameters and we describe a method that
predicts the width of the band making this a natural target for
future experimental study.
The CGLE in one space dimension is given by
⳵
tA=A+共1+ib兲
⳵
x
2A−共1+ic兲兩A兩2A,共1兲
where the complex field Ais a function of space xand time
t, and band c⬎0 are real parameters. This equation has
a family of propagating front solutions connecting the un-
stable state A=0 to a plane wave. The latter is a fundamental
class of solutions of the CGLE with the general form
A=冑1−Q2eiQx−i
␻
t, where
␻
=共b+c兲Q2−cand −1 ⬍Q⬍1.
Straightforward substitution reveals a one-parameter family
of plane waves for all values of band c. The wave number
Q共v兲selected behind a propagating front is uniquely deter-
mined by the speed of the front v共which is ⱖ2冑1+b2关4兴兲
via v=共b−c兲Q+共b+c兲/Q关4,6兴. Our specific focus is on the
dynamics that results from the corresponding plane wave
A共v兲being unstable. In such cases the plane-wave solution is
either not observed or eventually undergoes a transition to
more complex dynamics such as a pattern of localized de-
fects or spatiotemporal chaos 共Fig. 1兲.
In a previous study 关7兴we described and explained the
phenomenon of fixed-width bands of plane waves in a sys-
tem of reaction-diffusion equations of so-called “Lambda-
Omega” type. These are simply CGLE 共1兲with the linear
dispersion term b=0. Superficially, our results appeared to be
qualitatively different from those of Nozaki and Bekki 关6兴
and of subsequent authors 共reviewed in 关4兴兲 for the full
CGLE. These previous studies demonstrated the existence of
a region of plane waves immediately behind the propagating
front, whose width grows through time at a constant rate.
However, when we performed longer term simulations of the
full CGLE 共1兲, we found this behavior to be a transient: the
size of the region of plane-wave solutions eventually reaches
a limit and remains at the limiting width for all subsequent
times. Figure 1illustrates three typical examples of the oc-
currence of plane-wave bands. In each case we apply a small
initial perturbation localized near the left-hand boundary to
the trivial state A=0. This induces a front to propagate across
the domain. Behind the front there is a region of plane
waves, whose width increases at early times, at a rate that
can be calculated via the theory of linear spreading speeds
关4,6兴. However, in each case the long term behavior is a
constant width plane-wave band.
In this paper we provide a detailed account of the occur-
rence and nature of the fixed-width band of plane waves in
the CGLE. We first highlight some numerical errors in the
study of Nozaki and Bekki 关6兴which led them to incorrect
conclusions about the nature of the dynamics behind the
*http://research.microsoft.com/~mattsmi;
matthew.smith@microsoft.com
†jas@ma.hw.ac.uk; http://www.ma.hw.ac.uk/~jas
PHYSICAL REVIEW E 80, 046209 共2009兲
1539-3755/2009/80共4兲/046209共6兲©2009 The American Physical Society046209-1

propagating front. Next we give a brief overview of the
methods we used to predict the width of the plane-wave
band. We then report our predictions for the parameter de-
pendence of the width, which we confirm with simulations.
Finally, we discuss likely physical systems that could be used
to test our predictions.
II. CORRECTION TO THE STUDY
OF NOZAKI AND BEKKI [6]
The first study that we are aware of on propagating fronts
in the CGLE is that of Nozaki and Bekki 关6兴. These authors
discuss fronts connecting the unstable background state to
both stable and unstable plane waves. We began our own
work by attempting to reproduce the numerical simulations
in 关6兴. Despite 关6兴being very well cited, we found that the
simulations contained previously unrecognized major quali-
tative errors due to problems with numerical truncation. This
is most easily explained for Fig. 2 of 关6兴, in which the au-
thors show a propagating front, behind which there is a re-
gion of low amplitude plane waves, followed by plane waves
of higher amplitude; both plane waves are stable. Moreover,
the front shown in Fig. 2 of 关6兴undergoes a relatively abrupt
change in speed part way through the simulation. The initial
condition used in 关6兴is A
˜
=sech共0.05x
˜
兲; here we use tildes
to denote Nozaki and Bekki’s variables, which differ from
those in Eq. 共1兲via scalings. Our own simulations of this
case show a uniformly propagating front followed by a
single plane wave 关Fig. 2共a兲兴. It seems that Nozaki and Bekki
performed their computations in single precision. Thus,
in reality they used initial conditions of the form
A
˜
=sech共0.05x
˜
兲if sech共0.05x
˜
兲⬎10−5 and A
˜
=0 otherwise 共the
exact threshold would depend on details of their numerical
implementation兲. In Fig. 2共b兲we show the results of a simu-
lation done at high precision but using this truncated initial
condition, which reproduces Fig. 2of Nozaki and Bekki.
Figure 2共c兲shows the results for A
˜
共x
˜
,0兲=0 apart from a
perturbation localized to the left-hand boundary. In parts 共a兲
and 共c兲, a single stable wave train is selected behind the
invasion front. However, the truncated initial conditions lead
to a propagating front that is initially “pushed” before tran-
sitioning to a “pulled” front. These different propagating
front speeds lead to two different plane-wave solutions being
selected. The interface between these plane-wave bands
gradually moves with the lower amplitude wave train replac-
ing that of higher amplitude 关19兴.
There is a similar problem in Fig. 3 of 关6兴, which uses
parameter values giving an unstable plane wave behind the
front. In this case, the authors do not state their initial con-
ditions explicitly, but the tacit implication is that again
FIG. 1. Numerical simulations of pulled propagating fronts in CGLE 共1兲for different band cvalues. The line plots show 兩A兩for the last
time output of the surface plot below. The surface plots show the spatiotemporal dynamics of 兩A兩with darker shading indicating smaller 兩A兩
and black corresponding to 兩A兩=0. The dotted lines mark the beginning and end of the plane-wave band as detected using the method
described in the main text. The figure shows that while different parameters can result in contrasting spatiotemporal dynamics behind the
plane-wave band, the constant width of the band is a consistent phenomenon. Simulations are initialized with 兩A兩=0 other than a small
perturbation in x⬍1. The boundary conditions are Ax=0. Our numerical method is semi-implicit finite difference with grid spacing of 0.2
and a time step of 10−3.
MATTHEW J. SMITH AND JONATHAN A. SHERRATT PHYSICAL REVIEW E 80, 046209 共2009兲
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A
˜
共x
˜
,0兲=sech共0.05x
˜
兲. Again, we were able to reproduce their
results by using the “truncated” version of this initial condi-
tion 共we omit details for brevity兲. The rather complicated
dynamics in Fig. 3共a兲of Nozaki and Bekki is partly due to
the truncated initial conditions generating different unstable
plane waves in different parts of the domain. When Nozaki
and Bekki were working, more than 25 years ago, computa-
tional precision was much more limited than today; never-
theless, care is still needed to avoid numerical artifacts when
using initial conditions generating “pushed fronts.” In Fig. 1
and throughout the remainder of this paper, we consider ini-
tial conditions that generate a “pulled front,” which has
asymptotic linear spreading speed v
ⴱ=2冑1+b2关4兴; we write
Q共v
ⴱ兲=Qⴱand A共v
ⴱ兲=Aⴱfor brevity. However, we have also
found the same phenomena in pushed fronts, for which the
propagation speed is faster and the corresponding plane-
wave solution is different.
Since Nozaki and Bekki’s study there have been a number
of real physical experiments that have reported the phenom-
enon of plane waves behind invasion fronts 关4兴. To our
knowledge, however, none have studied the phenomenon in
sufficient detail as to test our prediction that the growth of a
plane-wave band is transient when the wave selected by the
propagating front is unstable.
III. CALCULATING THE WIDTH
OF THE PLANE-WAVE BAND
Our calculation of the width of the plane-wave band is
based on methods we have used in previous studies of the
dynamics behind propagating fronts in the case of b=0 关7,8兴.
We provide only a general overview here, concentrating on
the elements that are different from our previous work, and
refer the reader to our previous publications for detailed de-
scriptions of the methodology.
The key issue underlying our calculation is the absolute
stability of the plane wave Aⴱin a frame of reference moving
with velocity V. If the plane wave is absolutely unstable in a
frame with V⬎v
ⴱ, then perturbations to the plane wave can
outrun the front and the plane wave will not be seen. Con-
versely, if the plane wave is stable then there will be an
uninterrupted expanse of plane waves rather than a band.
However, if the plane wave is convectively unstable in the
frame of reference moving with the front speed v
ⴱthen all
unstable modes will propagate away from the front as they
grow leading to the band in which the plane waves are vis-
ible even though they are unstable. The left-hand edge of the
band occurs when the perturbations present in the plane
FIG. 2. Numerical solutions of the complex Ginzburg-Landau
equation as formulated in Nozaki and Bekki 关6兴with parameters as
in their Fig. 2. The equation is
⳵
t
˜
A
˜
=2A
˜
+共2.2+i兲
⳵
x
˜
2A
˜
−共1+i兲兩A
˜
兩2A
˜
,
which can be converted to Eq. 共1兲by simple rescalings. Shading
corresponds to the value of 兩A
˜
兩, with darker shading indicating
smaller 兩A
˜
兩and black corresponding to 兩A
˜
兩=0. The initial conditions
are as follows: 共a兲A
˜
共x
˜
,0兲=sech共0.05x
˜
兲;共b兲A
˜
=sech共0.05x
˜
兲if
sech共0.05x
˜
兲⬎10−5 and A
˜
=0 otherwise; 共c兲A
˜
共x
˜
,0兲=0 except for a
small perturbation near x
˜
=0. The boundary conditions are
⳵
x
˜
A
˜
=0.
Our numerical method is semi-implicit finite difference with grid
spacing of 0.2 and a time step of 10−3.
0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
(b)
c
b
(20) (10)
(5)
(3.3
)
(2.5
)
(50)
(500)
(250)
(100)
(50)
0 5 10 15 20 25
−15
−10
−5
0
5
(a)
c
b
W=0
0<W<∞
W=∞
FIG. 3. 共a兲Wave train bands behind pulled propagating fronts in
Eq. 共1兲are predicted to occur in 共b,c兲parameter space where
0⬍W⬍⬁. The gray broken line is the boundary at which the
selected plane wave is absolutely unstable in the moving frame
of reference with V=v
ⴱ. This line crosses the b=0 axis at
c=18.727 51. The thick solid line is the boundary at which the
selected plane wave is stable. In 共b兲the thin black solid lines show
contours for the bandwidth coefficient W, with the coefficient val-
ues labeled at the edge of the plot, and we overlay the Benjamin-
Feir-Newell curve 共thin black broken line兲, the absolute instability
curve 共thick black broken line兲, and the absolute stability boundary
共when V=0兲for the plane wave selected by a pulled propagating
front 共thick gray line兲. There is a region of multiple Wvalues in the
bottom left of 共b兲located between the two W=50 contours with
b⬍−2.5.
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wave immediately behind the front become sufficiently large
that they dominate the plane wave itself, which we assume to
occur when the perturbations become amplified by a scaling
factor F. For any given frame velocity V, the distance behind
the front at which this amplification first occurs can easily be
calculated: it depends on both the maximum growth rate of
perturbations in that frame of reference and the speed
V−v
ⴱat which the perturbation travels away from the front.
The actual width of the plane-wave band is given by the V
which minimizes this distance. In 关7兴we show that this is
V=Vband, which solves
共v
ⴱ−Vband兲Im关kmax共Vband兲兴 =Re关␭max共Vband兲兴,共2兲
where ␭max is the growth rate of the most unstable
linear mode and kmax is the corresponding spatial
frequency. The bandwidth itself is −log共F兲/Im关kmax共Vband兲兴,
so that all of the parameter dependence resides in
W共b,c兲=1/兩Im关kmax共Vband兲兴兩, which we refer to as the
“bandwidth coefficient.” In general, there can be multiple
solutions of Eq. 共2兲and it is then the smallest Wthat will
determine the width of the plane-wave band.
There are three standard approaches to calculating abso-
lute stability and, hence, ␭max and kmax:共i兲numerical con-
tinuation of saddle points of ␭共k兲−iVk in the complex k
plane 关9兴;共ii兲calculating the sign of the linear spreading
speed 关4兴;共iii兲calculating branch points in the absolute spec-
trum 关10,11兴. We adopt the last approach for which 关11兴
gives a detailed methodological description. For a linear
mode with temporal eigenvalue ␭and spatial frequency k,
the dispersion relation D共␭,k;V兲=0 is a quartic polynomial
in k. We denote the four roots by k1, ... ,k4with
Im k1ⱕIm k2ⱕIm k3ⱕIm k4. “Branch points” are the 共six兲
values of ␭for which ki=ki+1 for some iand they are rel-
evant to absolute stability if their index i=2, which corre-
sponds to the “pinching condition” of 关12兴. Therefore, we
solve Eq. 共2兲for Vband via numerical continuation of known
solutions to
D共␭,k;V兲=
⳵
kD共␭,k;V兲=0 共3兲
monitoring the indices of the repeated roots. Our numerical
codes, which use the software packages MATLAB 关13兴and
AUTO 关14兴, are available from the first author on request.
Note that our approach to calculating absolute stability con-
cerns infinite domains, which is the relevant scenario for the
plane-wave bandwidth. On bounded domains, plane-wave
solutions of the CGLE can exhibit remnant instabilities in
which perturbations grow while being repeatedly reflected
from the boundaries 关11,15兴, but such an instability is not
relevant here.
We used MATLAB 关13兴to solve Eq. 共3兲for given values of
b,c, and Vgiving six values of ␭and their associated values
of k. We used these values as starting points in numerical
continuation of Eq. 共3兲in AUTO 关14兴for varying parameter
values. We first performed continuations in Vlooking for
values that satisfied Eq. 共2兲. This then gave us Vband and,
thus, ␭max,kmax, and Wfor given values of band c. We next
performed continuations in either bor c, while maintaining
equality 共2兲, to allow us to monitor the variation in Wand to
label combinations of band cassociated with specific values
of W. Finally, we used these labeled solutions as starting
points for numerical continuations tracking contours of con-
stant Win b-cparameter space.
We tested our predictions of Wwith numerical simula-
tions of Eq. 共1兲. We used a standard semi-implicit finite dif-
ference method to solve the equations 共in MATLAB 关13兴兲. Nu-
merical tests with this method showed that our simulations
were accurate to about 0.1%. We used an automated method
共implemented in MATLAB 关13兴兲 for detecting the width of the
plane-wave band in simulations. We defined the observed
bandwidth as the region immediately behind the invasion
front at which 兩
⳵
A/
⳵
x兩⬍1⫻10−3. This condition gives a ro-
bust measure of bandwidth although the size of the threshold
means that the resulting values are slightly smaller than 共but
directly correlated with兲estimates suggested by visual in-
spection of space-time plots such as those shown in Fig. 1.
We estimated the derivative numerically after applying a
smoothing algorithm followed by a polynomial fit over a
moving window of 9 grid points. By implementing this
method we could then compare actual measures of band-
width in numerical simulations with our predicted values of
Wfrom numerical continuation. We used standard linear re-
gression to compare the predictions with the simulations. We
allowed for nonzero intercepts in the regression lines because
our method for measuring the width of the plane-wave band
in simulations excludes some regions on either side.
Note that our defined threshold of 兩
⳵
A/
⳵
x兩⬍1⫻10−3 is
larger than in our previous study of the b=0 case 关7兴, where
we used 兩
⳵
A/
⳵
x兩⬍5⫻10−7. We chose the new threshold be-
cause it resulted in a closer correspondence between the au-
tomated measurement and the size of the plane-wave band as
visible by eye 共as illustrated in Fig. 1兲. One consequence of
the new threshold is that, as expected from our theory, it
causes a change in the slope and intercept of the fitted linear
relationship between observed bandwidth and W; hence,
these differ from those given in 关7兴. Repeating the analyses
in 关7兴with the larger threshold results in estimates of the
slope and intercept of the linear regression that are similar to
those found here. This is consistent with our theory, which
predicts that 共for a given threshold兲the observed bandwidth
and Wshould be linearly related, with slope independent of
the parameter values band c.
IV. RESULTS
Our results indicate that a plane-wave band behind propa-
gating fronts will occur for a wide range of parameter values
in the CGLE 关Fig. 3共a兲兴. As expected, one boundary to the
parameter region in which a plane-wave band occurs is the
contour at which the wave train band behind the propagating
front becomes stable. Another boundary is the point at which
the plane-wave band is absolutely unstable in all moving
frame reference velocities, V. Along this curve, Vband=v
ⴱand
W=0.
Our analysis of Win 共b,c兲parameter space revealed the
expected pattern of high bandwidth coefficients close to the
plane-wave stability boundary with lower values further
away 关Fig. 3共b兲兴. It also revealed that the bandwidth varies
nonmonotonically with parameters 共see also Fig. 4兲. We also
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plot in Fig. 3共b兲the Benjamin-Feir-Newell curve 共beyond
which all plane-wave solutions are unstable兲, the absolute
instability curve 共beyond which all plane-wave solutions are
absolutely unstable when V=0兲, and the absolute stability
boundary for the plane wave selected by a pulled propagat-
ing front 共when V=0兲. This highlights that these curves pro-
vide no real information about the bandwidth.
In Fig. 4we compare the width of the plane-wave band
observed in numerical simulations with Wfor two slices in
the b-cparameter plane. The fit is extremely good. The fig-
ure also shows the nonmonotonic dependence of Won pa-
rameters.
We also discovered a region of multiple Wvalues in the
b-cparameter plane 关Fig. 5共a兲兴which allowed us to test and
confirm our prediction that it should be the smallest value of
W关Fig. 5共b兲兴that determines the size of the plane-wave
band. Although we did not find any other regions of multiple
Wvalues during our investigation, we do not know whether
other such regions occur elsewhere in parameter space.
V. DISCUSSION
We have identified a phenomenon in the one-dimensional
CGLE: fixed-width bands of plane waves behind propagating
fronts. We have calculated the bandwidth as a function of
parameters obtaining very good agreement with simulations.
The constancy of the bandwidth over time and the ability to
predict its value precisely make it a natural target for experi-
ments. The widespread applicability of the CGLE means that
there are a number of candidate systems that could be used.
One such is convection in binary miscible fluids. At rela-
tively high Rayleigh numbers, localized perturbations 共in
temperature兲to the quiescent homogeneous conductive state
can generate propagating fronts behind which are plane
waves 关16兴. To our knowledge, only stable plane waves have
been reported, but the relevant amplitude equation is the 共cu-
bic兲CGLE 关17兴and, thus, our results suggest that plane-
wave bands, followed by spatiotemporal chaos, would be
found as control parameters are varied. Another possibility is
the Taylor-Couette system with through flow 关3兴for which
the 共cubic兲CGLE is again the relevant amplitude equation.
3456789
100
140
180
220
W
Observed band width
(a) c=3
−4 −2 0 2 4
3
4
5
6
7
8
9
b
W
(c) c=3
0 20 40 60
0
400
800
1200
1600
2000
W
(b) b=−4
0 1 2 3 4 5
0
10
20
30
40
50
60
70
c
(d) b=−4
FIG. 4. Comparison of the bandwidth coefficient Wwith nu-
merical simulations of Eq. 共1兲. The initial and boundary conditions
for the simulations are as in Fig. 1except that the domain lengths
and run times were set so that the region of plane-wave solutions
behind the propagating front had reached its limiting width.
The crosses and error bars show the mean and standard deviation
of the observed plane-wave bandwidth for 100 solution times
spaced one time unit apart. We also performed simulations with
共b,c兲=共−4 , 4.5兲and 共−4,5兲, which generated no plane-wave band,
as predicted. 共a兲and 共b兲show the relationship between Wand
estimated plane-wave bandwidth from simulations. We superimpose
the best-fit linear regression lines, which have slopes of 31.15 and
31.88, intercepts of −21.04 and −22.86, and correlation coefficients
of 0.9921 and 0.9983, for 共a兲and 共b兲, respectively. In 共c兲and 共d兲we
have rescaled the measured plane-wave bandwidth using the linear
regressions in 共a兲and 共b兲. We chose equally spaced values of bor c
for our simulations with the exception of one additional value at
c=0.2 in 共b兲and 共d兲, which we add to include a point close to the
maximum in W. A close-up of the region to the top left of 共d兲is
given in Fig. 5共b兲. The lines in 共c兲and 共d兲are Wvalues that were
derived using the numerical continuation methods described in the
main text. We measured the observed plane-wave bandwidth using
the methods described in the main text.
0 0.1 0.2 0.3 0.4 0.5 0.6
−5
−4
−3
−2
−1
(
a
)
W=0
0<W<∞
single W
0<W<∞
multiple W
0<W<∞
single W
c
b
0.1 0.15 0.2 0.25 0.3
50
60
70
80
90
c
W
(b) b=−4
FIG. 5. 共a兲Wave train bands behind pulled propagating fronts in
Eq. 共1兲are predicted to occur in 共b,c兲parameter space where
0⬍W⬍⬁. However, there are multiple Wvalues in the region
marked “multiple W.” In such situations we predict that it is the
smallest Wvalue that determines the width of the plane-wave band.
We include the contour at which Wbecomes zero to facilitate com-
parison with Fig. 3.共b兲Numerical simulations confirm our predic-
tion that, in the case of multiple Wvalues, it is the smallest W
value that determines the width of the plane-wave band. This is a
close-up view of the region of multiple Wvalues shown in Fig. 4共d兲
but with additional simulation results. See the legend of Fig. 4共d兲
for definitions. We used the same regression line as in Fig. 4共d兲to
rescale the measured bandwidth.
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Localized perturbations can be generated by a sudden change
in the inlet boundary location and lead to plane waves behind
a propagating front. Previously, this has been used to locate
the convective instability boundary, but its wider application
would provide a natural test of our results. Potential non-
physical test systems include oscillatory chemical reactions
关5兴and oscillatory microbial interactions 关18兴, both of which
have been studied using the 共cubic兲CGLE. In all of these
various cases, the relevant CGLE coefficients have already
been derived, so that our results can be applied directly to
predict the dependence of the width of the plane-wave band
on the system parameters.
ACKNOWLEDGMENTS
We thank Jens Rademacher 共CWI, Amsterdam兲for many
valuable discussions, Eric Hellmich, Robin Freeman, and Ri-
chard Mann 共all Microsoft Research兲for technical assistance,
and Des Johnston 共Heriot-Watt兲for comments on the paper.
J.A.S. was supported in part by the Leverhulme Trust.
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