pulse-train solutions of a spatially-heterogeneous...

Under consideration for publication in Physica D 1

Pulse-train solutions of a spatially-heterogeneousamplitude equation arising in the subcritical

instability of narrow-gap spherical Couette flow

By EDWARD W. BLOCKLEY1†, ANDREW P. BASSOM2 ‡,ANDREW D. GILBERT1

AND ANDREW M. SOWARD1

1 Department of Mathematical Sciences, University of Exeter, Exeter, Devon, EX4 4QE, UK2 School of Mathematics and Statistics, University of Western Australia, Crawley 6009, Western Australia

(Received 15 December 2006)

We investigate some complex solutions a(x, t) of the heterogeneous Complex-Ginzburg-Landau equation

∂a/∂t =[λ(x) + ix − |a|2

]a + ∂2a/∂x2 ,

in which the real driving coefficient λ(x) is either constant or the quadratic λ(0) − Υ2εx

2. This CGL–equation arises in the weakly nonlinear theory of spherical Couette flow between two concentric spherescaused by rotating them both about a common axis with distinct angular velocities in the narrow gap(aspect ratio ε) limit. Roughly square Taylor vortices are confined near the equator, where their amplitudemodulation a(x, t) varies with a suitably ‘stretched’ latitude x. The value of Υε, which depends on sphereangular velocity ratio, generally tends to zero with ε. Though we report new solutions for Υε 6= 0, ourmain focus is the physically more interesting limit Υε = 0.

When λ = constant, uniformly bounded solutions of our CGL equation on −∞ < x < ∞ have someremarkable related features, which occur at all values of λ. Firstly, the linearised equation has no non-trivial neutral modes a(x) exp(iΩt) with any real frequency Ω including zero. Secondly, all evidenceindicates that there are no steady solutions a(x) of the nonlinear equation either. Nevertheless, Bassomand Soward [J. Fluid. Mech. 499 (2004) 277, referred to as BS] identified oscillatory finite amplitudesolutions,

a(x, t) =∑

n∈Z

a(x − xn) exp i [(2n + 1)Ω t + φn] ,

expressed in terms of the single complex amplitude a(x), which is localised as a pulse on the length scaleLPS = 2Ω about x = 0. Each pulse-amplitude a(x − xn) exp(iφn) is identical up to the phase φn =(−1)nπ/4, is centred at xn = (n+ 1

2 )LPS and oscillates at frequency (2n+1)Ω. The survival of the pulse-train depends upon the nonlinear mutual-interaction of close neighbours; self-interaction is inadequate,as the absence of steady solutions shows. For given constant values of λ in excess of some thresholdλMIN(> 0), solutions with pulse-separation LPS were located on a finite range Lmin(λ) ≤ LPS ≤ Lmax(λ).

Here, we seek new pulse-train solutions, for which the product a(x, t) exp(−ixt) is spatially periodic onthe length 2L = NLPS, N ∈ N. The BS–mode at small λ has N = 2, and on increasing λ it bifurcates toanother symmetry-broken N = 2 solution. Other bifurcations to N = 6 were located. Solution brancheswith N odd, namely 3, 5, 7, were only found after solving initial value problems. Many of the largeamplitude solutions are stable. Generally, the BS–mode is preferred at moderate λ, while that preferenceyields to the other symmetry-broken N = 2 solution at larger λ. Quasi-periodic solutions are also common.We conclude that finite amplitude solutions, not necessarily of BS–form, are robust in the sense that theypersist and do not evaporate.

† Email: [email protected], [email protected], [email protected]‡ Email: [email protected]

2 E. W. Blockley, A. P. Bassom, A. D. Gilbert and A. M. Soward

1. Introduction

The problem of the onset of instability of the axisymmetric flow between two differentially rotatingconcentric spheres in the narrow-gap limit has a long history beginning with Walton [1]. The basic flowconsists of a primary zonal flow together with a weaker meridional circulation between the equator andthe poles. When the inner sphere rotates faster than the outer, the centrifugal forces that drive theinstability are greatest in the vicinity of the equator; a stationary outer sphere is generally favoured byexperimentalists. When the ratio ε of the gap between the spheres to the radius of the inner one is small,the local equatorial geometry resembles concentric cylinders. In that small ε limit the expected mode ofinstability takes the form of Taylor vortices of angular length scale θ = O(ε), where −θ is the latitude,as Wimmer’s [2,3] experiments confirm.

The asymptotic methods employed by Walton [1] and subsequent researchers combine a mixture ofWKB and amplitude equation methods. Within the WKB framework solutions are sought with amplitudesproportional to

exp

[i

(ε−1

∫k dθ − ωt†

)](ε ≪ 1) , (1.1a)

where time t† is measured in units of the viscous diffusion time for the gap width, and the dimensionlesswave-number k and frequency ω are of order unity. Locally, in the vicinity of a given latitude, the ordinarydifferential equation and boundary conditions, which determine the radial structure, yield an eigenvalueproblem. The radial structure is the eigenfunction, while the eigenvalue is the local frequency

ω = ω(θ, k, Ta), (1.1b)

where Ta is the Taylor number which measures the angular velocity gradient driving the instability.

Local instability is determined by minimising Ta over real θ and k subject to the constraint that thefrequency is real which fixes critical values

θ = θc = 0 , k = kc , ω = ωc = 0 , T a = Tac . (1.2a)

A Taylor series of the local frequency in the neighbourhood of the critical values determines

ω = θ ω,θ + 12 θ2 ω,θθ + θ(k − kc)ω,θk + 1

2 (k − kc)2 ω,kk + (Ta − Tac)ω,Ta , (1.2b)

where the subscript comma (,) denotes partial differentiation with respect to the variable that follows it;derivatives denoted this way are evaluated at critical. The local minimisation of Ta ensures that

ω,k = 0 although ω,θ 6= 0 ; (1.3)

rather ω,θ is more weakly restricted and is merely real-valued. This behaviour can be traced to twophysical effects – the presence of the meridional circulation and the curvature of the boundary. Thesymmetries are such that

Imω,θ = 0 , Reω,θθ = 0 , Imω,θk = 0 , Reω,kk = 0 , Reω,Ta = 0 , (1.4a)

while the asymptotics of the physical problem with the outer sphere at rest reveals that

Reω,θ < 0 , Imω,θθ < 0 , Reω,θk > 0 , Imω,kk < 0 , Imω,Ta > 0 . (1.4b)

Since Imω,kk < 0, we see from (1.2b) that the maximum growth rate occurs when k = kc, and thenImω > 0 when

− θℓ < θ < θℓ , where θℓ =√

[− 2(Ta− Tac)ω,Ta/ω,θθ] . (1.5)

This defines the domain over which the system is locally unstable.

In order to understand the role played by the various terms in (1.2b) and to set the scene for ourdevelopment in this paper, we rewrite (1.1a) in the alternative form

a†(θ, t†) exp(iε−1kcθ

). (1.6)

Wave-trains in spherical Couette flow 3

We find that the heart of the difficulties which we must face originates from the fact that ω,θ 6= 0. Toisolate its influence, we consider the reduced form ω = θ ω,θ of (1.2b), which leads to the amplitudeequation

∂a†

∂t†≈ − i ω,θ θ a† . (1.7a)

Since the solution

a†(θ, t†) = a†(θ, 0) exp(− i ω,θ θ t†) (1.7b)

possesses a frequency ω,θθ that varies with position, the spatial effect is referred to as phase-mixing [4].From an alternative point of view, the temporal effect is to cause the real wave-number k to vary linearlyin time, k = kc − εω,θt

†, so that the growth rate Imω determined by (1.2b), at fixed θ, is only positivefor a limited period of time and certainly leads to a decaying disturbance as t† → ∞.

Since our use of the dispersion relation (1.2b) relies on the notion that θ, k − kc and Ta− Tac are allsmall, we retain the dominant terms

ω = θ ω,θ + 12 (k − kc)

2ω,kk + (Ta − Tac)ω,Ta (1.8a)

sufficient to describe unsteady dissipative solutions driven by the excess Taylor number Ta − Tac. Theappropriate amplitude equation is

∂a†

∂t†= − i [(Ta − Tac)ω,Ta + ω,θ θ] a† + 1

2ε2 i ω,kk∂2a†

∂θ2. (1.8b)

Despite the presence of the spatially varying coefficient iω,θθ, solutions have a remarkable translationalgroup property linked to (1.7), namely if a†(θ, t†) is a solution of (1.8b) so is

a†(θ − θloc, t†) exp(− i ω,θ θloc t†) , (1.8c)

where θloc is an arbitrary real constant. Since ω,θ is real, the growth rate properties of a solution a†(θ, t†)localised near the origin, are the same as the shifted solution (1.8c) localised near θloc. For Ta > Tac thiscurious spatial homogeneity is consistent with the fact that the system is locally unstable everywhere;i.e. since ω,θθ = 0 in the dispersion relation (1.8a), it follows from (1.5) that θℓ = ∞. The notion oflocal instability, however, is misleading here, as the outcome of any initial disturbance a†(θ, 0) is ultimatedecay to zero! Indeed, irrespective of the value of Ta−Tac, (1.8b) has no separable solutions of the forma†(θ) exp(pt†), which are bounded for all θ, for any value (real or complex) of the eigenvalue p. To seethis, it is sufficient to consider the marginal steady version

0 = − i ω,θ θ a† + 12ε2 i ω,kk

∂2a†

∂θ2(Ta = Tac p = 0) (1.9)

of (1.8b), as non-zero values of p and Ta − Tac can be removed by a spatial translation, similar to(1.8c), albeit into the complex θ–plane. Our (1.9) is Airy’s equation relative to the independent variablei(−2ω,θ/iω,kk)1/3ε−2/3θ, which incidentally sets the θ–length scale O(ε2/3). Unfortunately for the param-eter restriction (1.4) the new variable takes pure imaginary values for real θ. Thus there are no solutionsthat decay to zero both as ε−2/3θ tends to +∞ and −∞. This degenerate feature was noted by Soward[5] in a related Benard convection problem in a rotating self-gravitating sphere.

Curiously the difficulties associated with (1.8b) are not resolved with inclusion, on the right-hand sideof (1.8b), of the nonlinear Stuart–Landau term proportional to −|a†|2a† identified by Davey [6] in thecase of the classical cylinder Taylor–Couette problem. At any rate, all the evidence [Hocking, privatecommunication circa. 1980, 7,8] indicates that even for this nonlinear extension no bounded steadysolutions exist. The absence of nonlinear steady solutions is indeed remarkable, in view of the fact thatthe zero-amplitude state is locally unstable everywhere, once Ta exceeds Tac!

The resolution of the conflicts just described is in part achieved by noting that the reduced systemconsidered is unable to meet the global conditions for instability ∂ω/∂θ = ∂ω/∂k = 0 together withImω = 0 (see [9]). At the very least we have to reinstate the term 1

2θ2ω,θθ in (1.2b) and extend (1.8a)


to

ω = θ ω,θ + 12θ2ω,θθ + 1

2 (k − kc)2ω,kk + (Ta − Tac)ω,Ta (1.10a)

with corresponding amplitude equation

∂a†

∂t†= −i

[(Ta − Tac)ω,Ta + ω,θ θ + 1

2 ω,θθ θ2]a† + 1


∂θ2. (1.10b)

From (1.10a) we see that the global conditions are met at

θ = θg , k = kc , ω = 0 , T a = Tag , (1.11a)

where

θg = −ω,θ/ω,θθ and Tag − Tac = 12 (ω,θθ/ω,Ta) (θg)

2 ; (1.11b)

in view of (1.4), θg is pure imaginary. The folly of attempting an expansion about the local critical valuesis now self-evident as both θg and Tag − Tac (> 0) are of order unity and not small as the validity ofthe expansion procedures require. This difficulty was resolved by Soward and Jones [10], who applied theglobal conditions directly (i.e. no expansions about the local values were invoked). Later Jones, Sowardand Mussa [11] employed these direct techniques to resolve the long standing Benard convection problemin a rotating self-gravitating sphere [5] mentioned above. In both problems, θg took complex values.Despite misgivings about the results depending on the analysis in the neighbourhood of the complexlocation θg, the results were confirmed by the numerical solutions of the complete system of partialdifferential equations that govern the physical system in the small parameter (here ε) limit.

Following the resolution of the linear problem, the challenge became how to obtain the weakly nonlinearextension with the inclusion of the Stuart–Landau term proportional to −|a†|2a†. Though the linearanalysis can be undertaken in the vicinity of the complex location θg, the non-linear extension is fraughtwith difficulties, which have not yet been resolved. To unravel these complications Harris, Bassom andSoward [12] considered the narrow gap spherical Couette flow problem with the outer sphere rotating.Then the angular momentum ratio of the outer to the inner sphere provides another dimensionlessparameter. As that ratio approaches unity, the local frequency gradient ω,θ approaches zero and inconsequence so do θg and Tag − Tac. In that limit, it is legitimate to investigate (1.10) (the parameterrestrictions (1.4b) continue to hold) together with its nonlinear extension.

The preliminary study of Harris, Bassom and Soward [12] extended the earlier investigation by Hockingand Skiepko [13] of the branch of steady finite amplitude Taylor vortices that bifurcate from the basicstate. This branch has low amplitude, the magnitude of which appears to rapidly decrease as the size ofω,θ is increased. The subsequent bifurcation sequence via time-dependent states was later investigatedin considerable detail by Harris, Bassom and Soward [14] (henceforth referred to as HBS). Interestingly,the results obtained were sensitive to the value of ω,θ, which by necessity is small. On increasing ω,θ

within its permitted small value range, HBS discovered relatively large amplitude subcritical solutions,whose existence suggested that, when ω,θ = O(1) appropriate to the outer sphere at rest, finite amplitudesolutions might exist at Taylor numbers Ta close to Tac. The basic idea suggested by HBS and exploredby Bassom and Soward [8] (henceforth referred to as BS) was that, despite the absence of steady solutionsof the nonlinear extension

∂a†

∂t†= − i [(Ta − Tac)ω,Ta + ω,θ θ] a† + 1


∂θ2− |a†|2a† (1.12a)

of (1.8a), the new equation (1.12a) might possess instead time-dependent solutions of the form∑

n∈Z

a†(θ − θn) exp[− i (ω,θ θn t† − φn)] (1.12b)

(c.f. (1.8c)), where Z denotes the set of all integers 0, ±1, ±2, . . ., the φn are constant phase anglesdetermined as part of the solution and the θn are equally spaced, separated by a constant amount

L†PS = θn+1 − θn = O(ε2/3) . (1.12c)


This solution corresponds to a train of pulses a†(θ−θn) each oscillating with frequency ω,θ θn. The O(ε2/3)latitudinal extent of each pulse a†(θ − θn) about θ = θn is determined by the Airy equation balance ofthe terms −iω,θ(θ− θn)a† and − 1

2ε2 i ω,kk∂2a†/∂θ2, as already mentioned. The fact that steady solutionsappear to be impossible suggests that pulses cannot persist in isolation and that their existence dependson nonlinear interaction with their neighbours which, in turn, provides the basis of the estimate (1.12c).In order to support O(ε2/3) length scale pulses, we estimate from (1.12a) that the excess Taylor numberdriving the motion and the pulse-amplitude suppressing it must have magnitudes

Ta − Tac = O(ε2/3) and |a†| = O(ε1/3) . (1.13)

BS found solutions of the type (1.12b), when L†PS lay in some range L†

min ≤ L†PS ≤ L†

max, where L†min

and L†max depend on Ta. Furthermore a minimum value Ta = TaMIN (> Tac) was found at which

L†min = L†

max and below which no finite amplitude solutions exist.

Now analysis of (1.12a) is in some respects an over-simplification, as it avoids the issue of the truefinite extent 2θℓ of the locally unstable region. Indeed we really need to return to the full local dispersionrelation (1.2b) and reinstate all the terms into our amplitude equation:

∂a†

∂t†+ ε ω,kθθ

∂a†

∂θ= −i

[(Ta − Tac)ω,Ta + ω,θ θ + 1

2 ω,θθ θ2]a† − 1


∂θ2− |a†|2a† . (1.14)

With Ta − Tac = O(ε2/3) (see (1.13)1) it follows from (1.5) that θℓ = O(ε1/3), which is large comparedto the O(ε2/3) width of the pulses. This shows that we have a multiple length scale problem for which,in the vicinity of some location θloc, the pulse-train takes on the form

exp(− i ω,θ θloc t†)

∞∑

n=−∞

a†loc(θ− θn) exp[− i (ω,θ θn t†−φn)]

(θ = θ − θloc , θn = θn − θloc

)(1.15)

similar to (1.12b) relative to the local origin θloc (c.f. (1.8c)). This representation is valid on an in-termediate length scale short compared to θℓ = O(ε1/3) but long compared to the pulse-separation

L†PS = O(ε2/3). We may only speculate on how the pulse-train behaves as θloc is varied across the entire

range −θℓ < θloc < θℓ. The amplitude must certainly vary as the local driving parameter is not Ta−Tac

but by (1.5) and (1.14) is (Ta − Tac)[1 − (θloc/θℓ)2] instead; the latter must exceed TaMIN − Tac for

pulse maintenance. In the vicinity of the location at which equality is achieved we expect the pulses toterminate abruptly. Furthermore, the pulse-train is likely to drift slowly at the speed of the local groupvelocity εθlocω,θk, i.e. each θn increases slowly with the consequence that L†

PS does so too (see BS fordetails). These non-uniform features may be disruptive on a long time scale. For example the terminationlocation of the pulse-train may wander, and internally dislocations of the trains may occur as in otherComplex-Ginsberg-Landau investigations [15,16]. Certainly on these longer length scales (and associatedlonger time scales) we anticipate chaotic behaviour.

The upshot of the previous studies and the above estimates is the suggestion that finite amplitudeTaylor vortices of angular length scale O(ε) can occur at values of Ta close to the local critical valueTac. Significantly, they must have finite amplitude to exist (see (1.13)2) consistent with their subcriticalnature (i.e. Tac < Tag). Since from solutions a† of (1.14) we may generate others e−iφa† by any choiceof the constant phase angle φ, our theory does not fix the phase. That means, for example, that weare unable to predict the location of the vortex cell boundary nearest to the equator θ = 0 on theO(ε) length scale. Rather our theory is concerned with conditions for existence of solutions and theirnature on longer length scales. Specifically, the vortex amplitudes are modulated under pulse envelopes|a†

loc(θ − θn)| of lateral extent O(ε2/3), which each oscillates at a different frequency ω,θ θn (see (1.15)).How the pulses themselves are modulated on the longer O(ε1/3) of the locally unstable domain remains amatter of speculation. Nevertheless some limited evidence to support the claims of the previous paragraphis provided by the HBS study [14] of (1.14) within the framework of the approximation ω,kθ = 0 valid intheir |ωθ| ≪ 1 limit appropriate to the case of almost co-rotating spheres. All these issues were discussedin detail by BS [8].


We must stress that, though the local solution (1.15) looks complicated, it is obtained simply bythe transformation (1.8c) of the solution (1.12b,c). So the key to the modulated problem resides in themore primitive system (1.12). Indeed, in view of the disruptive environment outlined above in which thesolutions must exist, the robustness of such solutions is a matter of concern. Now BS only identified asingle family of periodic solutions, which at the time of their discovery appeared to possess a curious andnon-trivial structure. It is therefore of interest to ascertain whether or not the BS–solution is exceptionalor belongs to a wider class. Thus our objective here is to investigate the possibility that other periodicsolutions exist and to categorise them. In addition, we are interested in their stability as well as theirrobustness; i.e. whether or not they are approached and achieved as the consequence of time-steppingnumerical integrations.

The outline of our paper is as follows. In Section 2 we set the scene by summarising the essentialdetails of HBS and BS’s results and report some extensions that help to cement links between them. InSection 3 we introduce the finite Fourier transform in space, which is the tool whereby we are able tofind and construct our pulse-train solutions. In Section 4 we explain how the temporal evolution of theFourier coefficients can be determined relatively straightforwardly by time-stepping nonlinear systemsof first order ordinary differential equations and illustrate the method with the BS–solution. We alsooutline our tests for stability and robustness of our periodic solutions. New periodic solutions, outside theBS–solution class, are described in Sections 5–7, while numerical results indicate that chaotic solutionsalso exist. We close the paper with an overview in Section 8, with the reassuring conclusion that ourpulse-trains are robust albeit with weakly chaotic features likely.

2. A summary of HBS and BS’s results with some extensions

In this section we set up the HBS– and BS–problems in dimensionless units. Though the main thrustof this paper is to obtain new solutions of the BS–equation (2.8) below, they only have meaning asthe asymptotic limit Υε ↓ 0 of the HBS–problem (2.2) below. Therefore for the BS–solutions to haveany significance, we must link them to the HBS–solutions. In Section 2.1, we describe HBS–solutionsfor small Υε, which have a clearly identifiable pulse-structure consistent with the picture developed inthe Introduction. Then in Section 2.2, we find pulse-train solutions of the BS–problem, which agreequalitatively with the HBS–pulses in its locally unstable region. Now the value of the HBS-parameter Υε

is never really small in our numerical integrations. Consequently at the excess Taylor number adoptedfor our Fig. 2(a) below, the locally unstable region −θℓ < θ < θℓ is only about 5 pulse–widths (5L†

PS).Even so the quantitative agreement with the corresponding BS–result is really quite acceptable.

2.1. The HBS–model

In the limit of almost co-rotating spheres, for which |ωθ| ≪ 1, it is first natural to rescale θ as

θ = µε2/3x with µ = (−iω,kk/2ω,θ)1/3 , (2.1a)

and likewise for the other key angles θℓ (real; see (1.5)) and θg (purely imaginary; see (1.11b)1):

θℓ = µε2/3xℓ and θg = µε2/3xg . (2.1b)

Then under an appropriate scaling of time t† and amplitude a†, we may rewrite (1.14) in the form

∂a

∂t= (λ(x) + ix) a +

∂2a

∂x2− |a|2a , (2.2a)

in which the driving coefficient is

λ(x) ≡ λ(0) − Υ2ε x2 , (2.2b)

where

λ(0) = Υ2ε x2

ℓ , Υ2ε = 1

2 i/xg . (2.2c)


In so doing, we have dropped the term involving the group velocity, which is legitimate in the Υε = O(1)parameter range considered by HBS. Note that we assume that µ and Υε are both real and positive, whichis possible for the parameters restricted by (1.4), with the one proviso that Ta > Tac. The objective nowis to find bounded solutions of (2.2a) valid for all x with a → 0 as x → ±∞; we call this the HBS–problem.

In order to classify the solutions of (2.2), it is necessary to understand the nature of the symmetryclasses in which they lie. The most pertinent symmetries at this early stage are the observations thatgiven a solution a(x, t) of (2.2) we can generate others by

Rφa(x, t) ≡ e−iφa(x, t) , Ca(x, t) ≡ a∗(−x, t) , (2.3)

where the phase rotation φ is an arbitrary constant, while the asterisk denotes complex conjugate. Wenoted the former property in the Introduction, while the latter relies on the fact that the real drivingparameter λ(x) and the imaginary phase mixing parameter ix are even and odd functions of x respectively.We will describe the action of C as complex conjugate reflection (c.c.r.). For simplicity, throughout thispaper we will restrict attention to solutions belonging to the c.c.r. symmetry class

Ca(x, t) = a(x, t) (2.4a)

with the convenient properties that a(0, t) is real and ∂a/∂x(0, t) is purely imaginary:

Ima(0, t) = 0 and Re

∂a

∂x(0, t)

= 0 . (2.4b)

Notice that if a(x, t) belongs to the c.c.r. symmetry class (2.4a) then Rφa(x, t) does not unless φ is aninteger multiple of π; however it can be immediately brought back into the fold by the action R−φ.

To appreciate the essential difficulty associated with (2.2a) in the BS–limit Υε ↓ 0, we note (see [12])that the onset of instability is characterised by the steady solution

a(x) = a(0) exp[12

(i Υ−1

ε x − Υε x2)]

, (2.5a)

which belongs to our c.c.r. symmetry class (2.4a) and satisfies the global stability criterion when

λ(0) = λg(0) = (2Υε)−2 + Υε at x = xg = 1

2 i Υ−2ε . (2.5b)

The eigenfunction (2.5a) has a Gaussian envelope of width O(Υ−1/2ε ) under which the solution oscillates.

In the limit Υε ↓ 0, the Gaussian occupies a negligible fraction of the locally unstable region −xℓ < x < xℓ,where xℓ =

√λg(0)/Υε = O(Υ−2

ε ). Even more serious is the fact that λg(0) ↑ ∞ despite the fact that thelocal critical value for instability is λ(0) = 0. These are the bizarre features that make the ensuing finiteamplitude theory so complicated. Indeed, because of the numerical difficulties encountered on decreasingΥε, the smallest value reached in HBS’s study was Υε = 1

4 .

We briefly summarise HBS’s findings for their problem (2.2) when Υε = 14 . Following the initial

supercritical pitchfork bifurcation from the basic state at λg(0) = 4.25 (see (2.5b)1), the ensuing steadyfinite amplitude solution almost immediately undergoes a supercritical Hopf bifurcation to a vacillatingsolution with non-zero-mean. On increasing λ(0), the temporal oscillation period lengthens and tends toinfinity at λ(0) ≈ 4.35. This all happens over a very short range in λ and throughout this interval theamplitude a is extremely small. What happens next provides the underlying theme for our paper.

The infinite period at λ(0) ≈ 4.35 is associated with a subcritical global gluing bifurcation which leadsto an oscillatory zero-mean solution similar to (1.12b). As it belongs to the c.c.r. symmetry class (2.4a)it may be written in the form

a(x, t) =∑

n∈Z

bn (x − xn) exp (ixnt) + c.c.r. with xn =(2n + 1

2

)LPS , (2.6a)

where here and below ‘+ c.c.r.’ means ‘add the complex conjugate reflection C of the preceding terms’,


2 3 4 5 6 7 8 9 100

1

2

3amax

LPS

λ

Figure 1. The maximum amplitude amax (solid) and LPS (broken) vs. λ(0) for the new solutions (2.6) withinthe c.c.r. class (2.4a) of the HBS–problem (2.2a,b) for Υε = 1

4.

i.e. expressed in full (2.6a) is

a(x, t) =∑

n∈Z

bn(x − xn) exp (ixnt) +∑

n∈Z

b∗

n(−x − xn) exp (−ixnt) , (2.6b)

where we have used the C operation in the sense that

Cbn(x − xn) = b∗

n(−x − xn) = Cbn (x − (−xn)) (2.6c)

and so in general Cbn(x − xn) 6= Cbn (x − xn) unless xn = 0. The form (2.6a,b) differs from (1.12b),in as much as each amplitude bn(x−xn) is distinct. To make clear the link with (1.12b), we first note that−x−n = (2n− 1

2 )LPS. Then, since the sums run over all integer n, both positive and negative, the pulsesin the entire two sums form an interlocking sequence of pulses located at x = (2n ± 1

2 )LPS (n ∈ Z). Inother words, the pulses defined by each sum are separated by a distance 2LPS, but together they definea train of pulses separated by the distance LPS (c.f. (1.12c)). It follows that the frequencies in (2.6b)include all of ±Ω ± 3Ω ± 5Ω . . ., where

Ω = 12LPS determines the quarter-period TPS = π/LPS (2.7)

of the solution; over the half-period 2TPS the solution changes sign: a(x, t) = −a(x, t − 2TPS). The valueof Ω (equivalently LPS or TPS) is determined by the solution as a function of λ(0). As the value of λ(0) isreduced, these solutions follow a small amplitude (lower) branch until a minimum value of λ(0) ≈ 3.7 isreached; then as λ(0) increases again, so the solution returns along a larger amplitude (upper) branch (seeHBS, Fig. 12). Most of the upper branch was found to be unstable and so HBS looked at the possibilityof solutions outside the c.c.r. symmetry class (2.4a). Stable periodic solutions of this more complicatedtype were found in various windows of λ–values with much larger characteristic amplitudes.

Though HBS’s large amplitude results outside the c.c.r. symmetry class (2.4a) pointed the way forwardfor BS, they are unsatisfactory for direct quantitative comparison with BS–solutions. So we have under-taken further numerical investigations of the HBS–problem with Υε = 1

4 seeking solutions within the


c.c.r. symmetry class (2.4a). We considered various initial value problems and were fortunate to discovera new branch of periodic solutions (2.6) apparently disconnected from the HBS–branch portrayed in theirFig. 12. Since a(0, t) is a real oscillatory function (see (2.4b)1), we measured its maximum amplitude amax

over the full period 2π/Ω = 4TPS and plotted amax and LPS versus λ(0) in Fig. 1. In order to compareFig. 1 with HBS Fig. 12, we remark that with Υε = 1

4 (HBS’s κ = 4) our λ(0), amax and LPS correspondto HBS’s 1

4λ, 12amax and 1

2ω respectively (see Appendix C of BS for further details of this relationship).

The essential features that our Fig. 1 illustrates are that our new solutions extend to smaller λ(0) (about3 instead of 3.7) and generally have larger amplitude than HBS’s (our upper branch amplitude is roughly50% bigger than the HBS Fig. 12 solutions). Our lower unstable branch has a complicated structure, whichwe tracked for a while. Eventually we abandoned this tactic because the solutions showed no evidence ofsystematically reducing their amplitude. The results therefore suggest that the entire branch portrayedin Fig. 1 is disconnected from the basic unperturbed state. On increasing λ(0) from its minimum value(roughly 3), solutions on the upper branch were found to be unstable and relaxed to the zero-amplitudestate for λ(0) up to roughly 3.88. As λ(0) increased further so small windows of stability were located;the first two occur for λ(0) between ∼3.88 and ∼4.12 and ∼5.38 and ∼6.25. Elsewhere, yet still withinthe c.c.r. symmetry class (2.4a), quasi-periodic solutions of roughly the same amplitude were preferred;this was similar to HBS’s conclusion for solutions unconstrained by (2.4a).

Despite the likelihood that the majority of the solutions identified in Fig. 1 are unstable, they donevertheless illustrate the simplest type of large amplitude solution of (2.2) for Υε = 1

4 . As an example,we take the solution associated with λ(0) = 10 for which the periodic solution (2.6) has LPS ≈ 2.487. The

amplitude functions |bn(x− xn)| and |b ∗

n(−x− xn)| (see (2.6c)) for n = 0, ∓1 are illustrated in Fig. 2(a).They are centred roughly at ±x0 = ±LPS/2, ∓x−1 = ±3LPS/2, ±x1 = ±5LPS/2 respectively. The outerpair sit on the edges ±xℓ of the unstable region, where xℓ =

√λ(0)/Υε ≈ 2.543 LPS. The amplitudes

of the |bn(x − xn)| and |b ∗

n(−x − xn)| for |n| > 1, are very small over the domain of the figure and areomitted to avoid clutter. The weakening of the solution amplitude with increasing |x| is also evident onthe space-time constant amplitude contours on Fig. 3(a).

Though Υε = 14 is only moderately small, the results just described are of the type that support the

small Υε pulse-train model developed in the Introduction. Specifically Fig. 2(a) illustrates localised pulsesseparated by the distance LPS (c.f. (1.12c)) and a modulation of the pulse-train across the locally unstableregion −xℓ < x < xℓ with relatively abrupt termination at its edges x = ±xℓ. In addition, the findingof λ(0)–windows of stability may be significant because the local unstable region half-width xℓ increasesin concert with λ(0). So we may speculate that the pulses only reside comfortably in the local unstableregion for certain xℓ–windows and any lack of ‘comfort’ may be responsible for the quasi-periodicityelsewhere, outside those windows.

2.2. The BS–model

In order for the narrow gap spherical Couette system to achieve the O(1) values of Υε used by HBS,the inner and outer sphere angular velocities must be very close. The more normal situation typified bythe outer sphere at rest has |ωθ| = O(1), with the consequence that Υε = O(ε1/3). This provides thechallenge, taken up by BS, to bypass the weakly nonlinear theory and subsequent bifurcation sequencestudied by HBS and home straight in on the finite amplitude solutions of (2.2a) with λ taken to be λ(0),a constant. As explained in the Introduction, this only gives the solution on the x = O(1) (θ = O(ε2/3))length scale in the vicinity of θ = 0. To obtain the solution elsewhere in the vicinity of θ = θloc, we mustmove to that new local origin, set θ = θloc + µε2/3x and remove the oscillatory factor exp(− i ω,θ θloc t†)present in (1.15) from the complex amplitude. In this way the amplitude a ∝ a† exp(i ω,θ θloc t†) satisfies

∂a

∂t= (λ + ix) a +

∂2a

∂x2− |a|2a , (2.8)

in which λ is a constant. In other words the local problem anywhere on the domain −θℓ < θ < θℓ is


reduced to finding bounded solutions on −∞ < x < ∞ of (2.2) with Υε = 0 (or simply (2.8)). We callthis the BS–problem and a more thorough justification for its relevance is provided by BS itself.

Encouraged by HBS’s findings, BS successfully sought solutions to (2.8) of the type (2.6). These solu-tions possess a spatial periodicity, reflected by the fact that all the functions bn(x) are identical:

bn(x) = b(x) ≡ eiπ/4 a (x) (n ∈ Z) , (2.9a)

where a (x), like a(x, t) itself, possesses the c.c.r. symmetry

C a (x) = a ∗(−x) = a (x) . (2.9b)

Notice that b(x) lacks this symmetry property but satisfies instead CRπ/4 b (x) = Rπ/4 b (x) . Thesubstitution of (2.9a) into (2.6a) gives

a(x, t) =∑

n∈Z

a (x − xn) exp[i(xnt + 1

4π)]+ c.c.r. (2.10)

with xn = (2n + 12 )LPS. We will postpone explaining how these solutions are obtained until the next

section. Our only point at this stage is that the apparently simple form (2.10) hides a subtle structurehinted at by the distinct phase angles π/4 in the first sum and −π/4 in its c.c.r.. With that caveat theyare, in fact, the simplest form of periodic solution that we identify in this paper.

The λ(0) = 10, LPS ≈ 2.487 HBS–solution (2.6) of (2.2) for Υε = 14 , illustrated in Figs. 2(a) and 3(a),

provides an excellent test of the relevance of the Υε = 0 BS–solutions (2.10) of (2.8). Now unlike theHBS–solutions, which at specified λ(0) only occur at discrete LPS, the BS–solutions at specified λ arefound over a continuous range of LPS. For this reason, we are able to find, also at λ = 10, LPS ≈ 2.487,the BS–solution whose amplitude a (x) is illustrated in Fig. 2(b). An interesting point of comparison isthe similarity in the maximum amplitude a (0) = |b(0)| of the BS–pulse in Fig. 2(b) with that of thefirst HBS–pulse |b0(0)| in Fig. 2(a). The corresponding space-time constant amplitude contours of theBS–solution, which are shown in Fig. 3(b), also compare favourably with the HBS–solution shown inFig. 3(a). These comparisons are reassuring and point to the relevance of the BS–pulse-trains for theHBS–problem in the limit Υε ↓ 0.

Typical of the results uncovered by BS are those in Fig. 4 illustrating the locus of the square root ofthe energy measure

〈E〉 =1

LPS

∫ ∞

−∞

|a(x)|2 dx (2.11a)

of solutions (2.10) for a few values of λ (≤ 5). At these modest values of λ, Fig. 4 shows that there are twosolutions at each LPS over a range Lmin(λ) ≤ LPS ≤ Lmax(λ) loosely consistent with the existence of anupper and lower branch for the HBS solutions exhibited on Fig. 1. The finite LPS–range means that thepulses cannot be over crowded (LPS < Lmin) or separated by too large gaps (LPS > Lmax), i.e. the latternegation implies that neighbour interaction is essential. Furthermore the LPS–range width Lmax − Lmin

grows with increasing λ, while on decreasing λ the width eventually vanishes at a threshold excess Taylornumber associated with λMIN ≈ 2.54074. Below that value, λ < λMIN, wave-trains are unsustainable.The corresponding HBS–solution at the values λ(0) = λ employed on Fig. 4 has a corresponding LPS

identified on Fig. 1. So, for comparison purposes, we mark on Fig. 4, at the appropriate LPS, the valueof the square root of

〈E0〉 =1

LPS

∫ ∞

−∞

|b0(x)|2 dx (2.11b)

for the largest HBS–pulse b0(x) of (2.6a) at some values of λ(0) = λ used to plot the BS–contours.The fact that they systematically lie beneath the corresponding BS–contours can be explained by theproperty λ(x) ≡ λ(0) [1 − (x/xℓ)

2] < λ(0) for x 6= 0 (see (2.2b,c)), i.e. the effective λ determining theHBS–solutions is smaller than that for the BS–solutions.

The BS–solution structure characterised by the√〈E〉 versus LPS curves of Fig. 4 becomes more com-


-6 -4 -2 0 2 4 60

0.5

1

1.5

2

2.5

3

x/LPS

-5 -4 -3 -2 -1 0 1 2 3 4 5-3

-2

-1

0

1

2

3

x/LPS

a(x

)

Figure 2. Pulses for the case λ = 10, LPS ≈ 2.487 (TPS ≈ 1.263). (a) The HBS–style solution (Υε = 1

4; right-hand

edge of Fig. 1 above) identified by the solid dot in Fig. 5 below. The pulse-amplitudes |bn(x−xn)| and |b ∗

n(−x−xn)|

centred at x = xn and x = −xn (n = −1, 0, 1) (2.6) are illustrated. Those centred at ±LPS/2, ±3LPS/2 and±5LPS/2 are drawn as continuous, long-dashed and chain lines respectively. (b) The BS–style solution identifiedby the point on the upper continuous line in Fig. 5 at the same LPS–value as the solid dot. Only the pulse a(x)(2.9a) is shown; the modulus |a|, real part Rea and imaginary part Ima are identified by the continuous,dashed and chain lines respectively.

plicated as we increase λ beyond 5 (the maximum value examined in BS). The first development setsin for λ just above 6, when a new closed

√〈E〉 versus LPS curve emerges inside the original. All theseextra solutions, however, are unstable in the sense that, if the governing equations are marched forwardin time starting with numerical approximations of these new solutions then we invariably lock onto thehigher energy ‘old’ solution. With further increases of λ the lower parts of the two loops coalesce. The


−2 −1 0 1 20

1

2

3

4

5

6

7

8

−2 −1 0 1 20

1

2

3

4

5

6

7

8Rea(x, t) Ima(x, t)

x/LPSx/LPS

t/TPSt/TPS

−2 −1 0 1 20

1

2

3

4

5

6

7

8

−2 −1 0 1 20

1

2

3

4

5

6

7

8Rea(x, t) Ima(x, t)

x/LPSx/LPS

t/TPSt/TPS

Figure 3. Contours of constant Rea(x, t) and Ima(x, t) in the x/LPS–t/TPS plane for the λ = 10, LPS ≈ 2.487case illustrated in Fig. 2. The half-period is 2TPS ≈ 2.526 (see (2.7)), while the time origin is chosen so thatRea(0, 0) = 0. (a) The HBS–style solution (Υε = 1

4). (b) The BS–style solution. (Remember that the time

origin choice means a time shift t → t − qTPS for some non-integer q.)


1 1.5 2 2.5 30

0.5

1

1.5

2

λ = 3

λ = 4

λ = 5

LPS

√〈E

〉

Figure 4. Contours of√〈E〉 vs. LPS for the BS–style solutions at five values λ = 2.55 (innermost curve), 2.6, 3,

4, and 5 (outermost). The values√〈E0〉 and LPS of the related HBS–style solutions with λ(0) = λ are identified

by the labelled solid dots at λ = 3, 4, and 5.

subsequent loop structure is exemplified by Fig. 5 for the case λ = 10 used for the numerical resultsdisplayed in Figs. 2 and 3. Note that the lower part of the main loop appears to achieve zero meanenergy; this is illusory however, since solutions of infinitesimal amplitude are impossible and no solutionbranch can terminate at zero

√〈E〉.

3. Background theory

In Section 3.1, we introduce the Fourier series in space with time-dependent coefficients, which is thetool whereby we are able to find spatially periodic (in a sense that we will define) BS–style solutions. InSection 3.2, the space and time symmetries of the solution class, identified in Section 3.1, are described,while they are cast within a more general group framework in the Appendix. In Section 3.3, we classifythe distinct classes of temporally periodic solutions that we obtain numerically. Then in Section 3.4,we formulate the solutions within the space-time symmetries identified in Section 3.3 as Fourier seriesin time. The coefficient of each temporal harmonic defines a pulse in space. Finally in Section 3.5, wedescribe the Fourier transforms necessary to translate the time-dependent Fourier coefficients introducedin Section 3.3, into the pulses, namely the space dependent Fourier coefficients of Section 3.4.

This section is the “engine room” of the paper, which may be understood at different levels. Only Section3.1 is needed in order to know how solutions are subsequently obtained numerically. Some understandingof Sections 3.2 and 3.3 is necessary to appreciate which solution class a particular numerical solutionbelongs to. This classification helps to clarify the nature of the bifurcation sequences, which we identify.For the reader to probe the intracacies of the pulse-structure however the details, provided in the lengthycombined Sections 3.4 and 3.5, are pertinent but could be safely omitted on first reading.

3.1. A Fourier series in space

Whereas it is natural to determine temporally periodic HBS–solutions of (2.2a) by means of a Fourier


0 1 2 3 40

0.5

1

1.5

2

2.5

3

LPS

√〈E

〉

Figure 5. As on Fig. 4 the continuous closed curve gives the contour of√〈E〉 vs. LPS for the BS–style solutions

for the case λ = 10. The value√〈E0〉 of the related HBS–solution at LPS ≈ 2.487 illustrated on Figs. 2(a) and 3(a)

is identified by the solid dot. The BS–style solutions illustrated on Figs. 2(b) and 3(b) are located on the uppersection of the continuous closed curve at the same LPS–value as the solid dot. New broken-symmetry solutions,discussed in Section 5 below, are identified by the short section of dashed curve.

time series as in (2.6), this is not the simplest way to determine the corresponding BS–solutions of (2.8),such as (2.10). Instead, we substitute

a(x, t) = exp(itx) b(x, t) (3.1)

into (2.8) and so obtain the equation

∂b

∂t= λ b +

(∂

∂x+ it

)2

b − |b|2b (3.2)

governing b(x, t). The advantage of (3.2) is that it is invariant under translation and so possesses spatiallyperiodic Fourier series solutions

b(x, t) =∑

n∈Z

An(t) exp(− inTx) (3.3a)

of constant wavelength (2L =)2π/T or, equivalently,

a(x, t) =∑

n∈Z

An(t) exp[ i(t − nT )x] . (3.3b)

The substitution of (3.3a) into (3.2) shows that the functions An(t) satisfy

dAn

dt−

[λ − (t − nT )2

]An = −

∑

α,β∈Z

An+α An+β A∗n+α+β (3.4a)

and are subject to the the initial conditions

An(t) → 0 as t − nT ↓ −∞ . (3.4b)


Despite the appearance of the time-dependent coefficients in (3.2), the ensuing initial value problem forthe coupled first order equations (3.4a) is much easier to solve than the corresponding boundary valueproblem for the coupled second order equations governing bn(x) that define the solution (2.6).

We find it useful to measure the solution amplitude by the spatial-mean energy integral (over thewavelength 2L = 2π/T ), which by (3.3b) can be expressed in terms of the Fourier coefficients as

E(t) ≡ 1

2L

∫ L

−L

|a(x, t)|2 dx =∑

n∈Z

|An(t)|2 . (3.5)

BS applied standard energy integral methods to show that finite-amplitude solutions necessarily decaywhen λ < T 2/12. We emphasise that the half–wavelength L, as defined by

LT = π , (3.6)

is not necessarily the same as the pulse-separation LPS = π/TPS (see (2.7)2). In the case of the BS–solution (2.10), they coincide LPS = L (also TPS = T ), which is why the BS–definition (2.11a) agrees,as we will see later, with (3.5). In general, the wavelength 2L is an integer N multiple of LPS. In otherwords, when there are N distinct pulses the total pattern width is NLPS. For (2.10), there are two distinctpulses (N = 2). One family of pulses is identified by the sum involving a(x), while the other is identifiedby the c.c.r. sum involving a ∗(−x). Since a ∗(−x) = a(x), the pulse families only differ subtly throughthe phase angles ± 1

4π.

3.2. Symmetries

We have already noted that when λ is a function of x new solutions of (2.2a) can be generated by thetransformations (2.3). If λ is a constant, other solutions

T qa(x, t) = a(x, t − qT ) , Lpa(x, t) = eipLta(x − pL, t) (3.7a)

of (2.8) can be constructed under the time-shift qT and translation pL, where q and p are arbitraryreal constants. Here the power law notation is suggested by the properties T q1T q2 = T q1+q2 andLp1Lp2 = Lp1+p2 for arbitrary q1, q2 and p1, p2. Note, however, for solutions a(x, t) belonging to thespatial periodicity class (3.3b), T qa and Lpa only remain in the same class when p, q ∈ Z. This empha-sises the importance of the special cases q = 1, p = 1:

T a(x, t) = a(x, t − T ) , La(x, t) = eiLta(x − L, t) . (3.7b)

We introduce the notation

Ga(x, t) =∑

n∈Z

GAn(t) exp[ i(t − nT )x] , (3.8a)

for any of our group actions Rφa, Ca (see (2.3)), T a and La. We may then deduce from (3.3b) that

RφAn(t) = e−iφAn(t) , CAn(t) = A∗n(t) , (3.8b)

T qAn+q(t) = An(t − qT ) (q ∈ Z) , LAn(t) = (−1)nAn(t) . (3.8c)

Note however that two applications of L gives

L2An(t) = An(t) , (3.9a)

which implies the spatial-periodicity on the length 2L:

L2a(x, t) = a(x, t) , (3.9b)

as already mentioned in connection with (3.3). Furthermore the simultaneous application of L and T isoften important for the classification of solutions and the result is sensitive to the order in which theoperations are applied. It is easily deduced from the identities (3.8c) that

T qLAn+q(t) = (−1)nAn(t − qT ) = (−1)q LT qAn+q(t) , q ∈ Z , (3.10a)


in the light of which we note the commutation property

T qLa(x, t) = (−1)q LT qa(x, t) , q ∈ Z . (3.10b)

Finally, our restriction to the c.c.r. symmetry class (2.4a) implies using (3.8b)2 that all A∗n(t) = An(t),

i.e. the Fourier coefficients An(t) are real:

ImAn(t) = 0 , (3.11)

a property which clearly simplifies our numerical solution of the system (3.4a). This also means that

a(0, t) =∑

n∈Z

An(t) and∂a

∂x(0, t) = i

∑

n∈Z

(t − nT )An(t) (3.12)

are real and purely imaginary respectively consistent with (2.4b). These properties make them usefulmeasures of the solution amplitude.

3.3. The temporal periodicity 4TPS

Our numerical integration of the initial value problem (3.4) revealed temporally periodic solutions,which belong to various symmetry groups identified in the Appendix. Here we classify them similarly butfrom a more physical point of view based on whether or not they have zero temporal mean:

T Na(x, t) = a(x, t−2TPS) =

a(x, t), non-zero-mean, period 2TPS, a ∈ P+(T ; N);

− a(x, t), zero-mean, half-period 2TPS, a ∈ P−(T ; N)(3.13a)

(see (A.6)), where

2TPS = NT . (3.13b)

Henceforth we will express the properties (3.13a) by the summary statement

T Na(x, t) = ± a(x, t) , a ∈ P±(T ; N) (3.13c)

and write P(T ; N) to denote the entire set of solutions P−(T ; N) ∪ P+(T ; N). An application of thespace-shift L and use of (3.10b) shows that La(x, t) satisfies

T NLa(x, t) = ± (−1)N La(x, t) . (3.14)

This says that the space-shifted solution La(x, t) has the same periodicity classification as a(x, t) whenN is even but switches to the other when N is odd; more specifically

if a ∈ P±(T ; N) then

La ∈ P∓(T ; N) for N odd;

La ∈ P±(T ; N) for N even.(3.15)

Likewise, trivially we note that T a ∈ P±(T ; N), whenever a ∈ P±(T ; N). In summary therefore, theaction of both T and L on any a in the entire set P(T ; N) of solutions (3.13) keeps the solution withinthe set.

To encompass both periodicity classes (3.13a), we regard them as period 4TPS solutions and write

a(x, t) =

2N−1∑

α=0

aα(x, t − αT ) , (3.16a)

which is determined by the 2N functions

aα(x, t) =∑

n∈Z

Aα(t − 4nTPS) exp[ i(t − 4nTPS)x] , α = 0, 1, . . . , 2N − 1 . (3.16b)

Since by (3.11) the Aα are real, it follows from (3.16b) that aα(x, t), like a(x, t), has the c.c.r. property

Caα(x, t) = aα(x, t) . (3.17)


Comparison of the series (3.16a) with (3.3b) shows that A2nN+α(t) = Aα(t − 4nTPS − αT ) with theimplication for n = 0 that

Aα(t) = Aα(t + αT ) . (3.18)

Furthermore the periodicity property (3.13) and use of (3.8c)1 shows that the Fourier coefficients Aα(t)(α ∈ Z) satisfy

Aα(t − 2TPS) = T NAα+N (t) = ±Aα+N (t) . (3.19)

So for the complete set of 2N functions Aα(t) required in (3.16), those on N ≤ α ≤ 2N − 1 are relatedto those on 0 ≤ α ≤ N − 1 by

± Aα+N (t) = Aα(t)

aα+N (x, t) = ± aα(x, t)

for 0 ≤ α ≤ N − 1 . (3.20)

For a ∈ P+(T ; N) with period 2TPS rather than 4TPS, the notation looks cumbersome. Nevertheless, aswe shall see, this present formulation is compact and versatile, e.g. the application of the shift L betweenP+(T ; N) and P−(T ; N), which occurs when N is odd (see (3.15)), is transparent and clear.

Furthermore, our numerical results reveal that, when N is even and 12N is odd a class of solutions

also exist with quarter-period, space-shifted symmetry, which we will call PQ− (T ; N). We sub-classify it

further as follows:

T N/2La(x, t) = ±a(x, t) quarter-period TPS a ∈ PQ±− (T ; N) (3.21a)

for which we emphasise that

TPS = 12NT with 1

2N odd (3.21b)

(see (A.8)). Hence, since from (3.10b) we have LT N/2a(x, t) = −T N/2La(x, t) for 12N odd, it follows

that T Na(x, t) = (T N/2L)(LT N/2a)(x, t) = −a(x, t). This means that solutions with the properties(3.21a) are subclasses of P−(T ; N). In terms of the Fourier coefficients Aα(t) (α ∈ Z), the periodicityproperty (3.21a) becomes, with the help of (3.10a),

(−1)αAα(t − TPS) = T N/2LAα+N/2(t) = ±Aα+N/2(t) for a ∈ PQ±− (T ; N) . (3.21c)

This can be used to relate the 12N functions Aα(t) on 1

2N ≤ α ≤ N − 1 to those on 0 ≤ α ≤ 12N − 1:

(−1)α Aα+N/2(t) = ± Aα(t) ,

aα+N/2(x, t) = ±(−1)α aα(x, t)

for 0 ≤ α ≤ 12N − 1 . (3.22)

To build the remaining functions Aα(t) on N ≤ α ≤ 2N − 1, it is simplest and safest to use the results(3.20) for the set P−(T ; N). This is because, when the relation (3.22) is used iteratively more than once,it is awkward to keep track of the sign. Note that whenever a ∈ PQ±

− (T ; N) a space- or time-shift will

place La or T a in the opposite class PQ∓− (T ; N). What is left of P−(T ; N) after removing the combined

class PQ− (T ; N) = PQ+

− (T ; N) ∪ PQ−− (T ; N) we will call PH

− (T ; N):

P−(T ; N) = PQ− (T ; N) ∪ PH

− (T ; N) . (3.23)

Let us elaborate on the translational issues just raised in a more general context. New solutions T qaand Lpa simply correspond to a change of time and space origin under the time-shift qT and translationpL, as explained below (3.7a), and so are not new solutions of the physical system. That this is true fortime-shifts is obvious. In the case of space-shifts the issues are more subtle and pivot on the pulse–trainrepresentation (1.15) valid in the vicinity of some location θloc. That constitutes our x–origin, i.e. the

definition (2.1a) is replaced by θ = θ − θloc = µε2/3x. Nevertheless, if we wish to compare our resultswith those for the HBS–problem, it is natural to adopt the equator of the sphere. Then θloc = 0 andthe distinction between a solution a(x, t) and La(x, t) becomes important. On the other hand, since


PQ+− (T ; N) and PQ−

− (T ; N) are related simply by a change of time origin, there is no physical reason to

distinguish them and so their combined classification PQ− (T ; N) is generally adequate.

From another point of view, the characteristics of the sets of 2N Fourier coefficients Aα(t) = Aα(t−αT )(see (3.18)), enable us to classify our temporally periodic solutions obtained numerically into one of the

types described above. We call the qualitative nature of the Aα(t) sets the solution signature (thoughperhaps footprint track is a closer analogy). This categorisation is important because, when we find asolution with a new track, we want to know whether or not it really is a new solution or just one alreadyobtained on application of time T and/or space L shifts. For the record, the pulse–solutions reported byBS [8] belong to PQ

− (T ; 2).

3.4. Pulse–structure, separation LPS; a Fourier series in time

Our objective in this section is to reformulate our functions aα(x, t) (3.16b) as Fourier series in time andthen reassemble them to construct solutions a(x, t) (3.16a) with pulse-structures. From their definition(3.16b), we see that the spatio–temporal periodicities of aα(x, t) are curiously not the same as those ofa(x, t). Instead they satisfy

T 2Naα(x, t) = aα(x, t) , L1/Naα(x, t) = aα(x, t) (3.24)

(see (3.7a)). Thus they are temporally periodic over the time 4TPS (= 2NT ) and spatially periodic onthe length 1

2LPS, where, as usual (see (2.7)),

LPS = π/TPS = 2L/N . (3.25)

which are the symmetries of a function in the class P+(4TPS; 1). All the other relations obtained in Section3.3 relate aα’s with different α’s and shed no further light on their individual properties.

Another representation of aα(x, t) (0 ≤ α ≤ 2N − 1) satisfying the periodicities (3.24) is

aα(x, t) =1√2N

∑

m∈Z

aα(x − 1

2mLPS

)exp

(i 12mLPSt

), aα ∈ P+(4TPS; 1) . (3.26)

Here the expansion in terms of exp(i12mLPSt) is implied by the periodicity over the time 4TPS, whilethe fact that the amplitudes have identical structures aα(x) relative to appropriately chosen space–originfollows from the spatial periodicity over the length 1

2LPS. Some properties of aα(x), which is the non-zero

time average of√

2Naα(x, t), should be noted. Since aα(x, t) has the c.c.r. property (3.17), so does aα(x):

C aα(x) = aα(x) . (3.27a)

Not all the aα(x) are independent. In particular, since a ∈ P±(T ; N), it follows from (3.20) that

aα+N (x, t) = ± aα(x, t) for 0 ≤ α ≤ N − 1 . (3.27b)

An alternative form of (3.26) obtained on setting m = 2nN + β is the double sum

aα(x, t) =1√2 N

∑

n∈Z

2N−1∑

β=0

aα(x − xβ

n

)exp

(i xβ

n t)

with xβn = 2nL + 1

2βLPS . (3.28)

On substitution into (3.16a) and reversing the order of summation we obtain

a(x, t) =∑

n∈Z

2N−1∑

β=0

bβ(

x − xβn

)exp

(i xβ

n t), (3.29a)

where

bβ(x) =

1√2 N

2N−1∑

α=0

exp

(− i

βαπ

N

)aα(x) =

1 ± (−1)β

√2N

N−1∑

α=0

exp

(− i

βαπ

N

)aα(x) ; (3.29b)


the summation range reduction in the second sum is achieved by use of the property (3.27b). Significantly

bβ

defined by (3.29b) has the symmetry properties

b2N+β

(x) = bβ(x) and

Cb

β

(x) = b(−β)

(x) , (3.29c)

where the latter result depends on (3.27a). We now use these results to determine their implications forthe structure of a(x, t) in those various classes, which turn out to be relevant for the solutions that wefind.

When a ∈ P+(T ; N), (3.29b) implies that bβ(x) = 0 for β odd. Consequently the time average of a(x, t)

is

b0(x) =

√2

N

N−1∑

α=0

a α(x) , (3.30)

which is why we called a ∈ P+(T ; N) the non-zero-mean solution in (3.13). When furthermore N is odd,we change the β–summation range in (3.29a) to −N ≤ β ≤ N−1. Then we remove the special case β = 0,leaving sums over the β–ranges −N + 1, . . . , −4, −2 and 2, 4, . . . , N − 1. Now, since x−2γ

−n = −x2γn , use

of (3.29c) establishes the c.c.r. symmetry property

b(−β)

(x − x−β−n) exp (i x−β

−n t) = Cb

β(x − xβ

n) exp (i xβn t)

, (3.31)

where C. . . is used in the sense defined by (2.6c). On replacing β by 2γ, this result enables us to express(3.29a) in the form

a(x, t) =∑

n∈Z

b

0(x − 2nL) exp (i 2nLt) +

[(N−1)/2∑

γ=1

b2γ(

x − x2γn

)exp

(i x2γ

n t)

+ c.c.r.

], (3.32)

where the first isolated term is the β = 0 case removed and the c.c.r. refers to the terms (3.31) withβ = 2, 4, . . . , N − 1.

At this point we recall that other pulse solutions may be constructed by the space–shift L of (3.29a),namely

La(x, t) =∑

n∈Z

N−1∑

β=−N

bβ+N(

x − xβn

)exp

(i xβ

nt). (3.33)

All that has happened is the pulse bβ

centred at xβn has been replaced by b

β+N, i.e. the superscript

modulo 2N has been increased by N and the pulse pattern has been moved a distance L to the right. So,if a ∈ P+(N) and N is odd as above, then by (3.15) we have that La ∈ P−(N). By this device, we canconstruct La from (3.32) to obtain the corresponding family of P−(T ; N) (N odd) solutions. We omitgiving their explicit form, as it is not particularly illuminating.

When a ∈ P−(T ; N), (3.29b) implies that bβ(x) = 0 for β even. Consequently the time average of

a(x, t) vanishes b0(x) = 0, which is why we called a ∈ P−(T ; N) the zero-mean solution in (3.13). When

furthermore N is even, we replace the β–summation in (3.29a) by distinct β–sums, which separately runover the interlocking ranges β = 1, 5, . . . , 2N − 3 and β = 3, 7, . . . , 2N − 1. Then, as above, we use therelated property xβ

−n−1 = −x2N−βn with (3.29c) to establish

b(2N−β)

(x − x2N−βn ) exp (i x2N−β

n t) = Cb

β(x − xβ

−n−1) exp (i xβ−n−1 t)

. (3.34)

This enables us to write the sum over the latter range β = 2N − 1, . . . , 7, 3 as the c.c.r. of the formerrange β = 1, 5, . . . , 2N − 3 albeit with n → −n− 1; a matter of no consequence as we then sum over all


n. We now replace β by 4γ + 1 to obtain

a(x, t) =∑

n∈Z

(N/2)−1∑

γ=0

b4γ+1 (

x − x4γ+1n

)exp

(ix4γ+1

n t)

+ c.c.r. , (3.35)

where x4γ+1n = 2nL + (2γ + 1

2 )LPS.

When we make the further restriction a ∈ PQ±− (T ; N) with 1

2N odd appropriate to the quarter-period

space-shifted symmetry (3.21), we have from (3.22), in addition to aα+N (x, t) = − aα(x, t) (see (3.27b)),that

aα+N/2(x, t) = ±(−1)α aα(x, t) for 0 ≤ α ≤ 12N − 1 . (3.36)

On substitution into (3.29b), we find that bβ(x) reduces further giving

b4γ+1

(x) =2

N

(N/2)−1∑

α=0

exp

[−i

((4γ + 1)α

N± (−1)α

4

)π

]aα(x) . (3.37)

As an example, the N = 2 BS–solution (2.10) belongs to the class PQ−− (T ; 2); the α = 0 term in the

α–sum (3.37) recovers (2.9a) with a = a 0 and b = b1, while the single γ = 0 term in the γ–sum of (3.35)

recovers (2.10) remembering, of course, that LPS = L.

From a general point of view, the form of (3.29b) exposes a rather surprising property of the solution,

namely that under unit increase of β, each alternate bβ(x − xβ

n) vanishes. This is a feature emphasisedby the specific cases just considered. It means that each pulse is separated by the distance

LPS = xβ+2n − xβ

n (3.38)

rather than the periodicity distance 12LPS of aα(x, t). Since we only find solutions for N ≥ 2, we see from

(3.13b) and (3.25) that TPS ≥ T and LPS ≤ L, with equality for the BS–case N = 2, properties thatwere anticipated earlier below the related statement (3.6). The objective in this paper is to numericallyidentify other solutions in the wider class P(T ; N) for N ≥ 2. That there appear to be no solutions inthe apparently simpler class P(T ; 1) is yet another intriguing aspect of the BS–problem!

3.5. The Fourier space–time link

Though our numerical integrations determine the An(t) in the Fourier space series solution (3.3b), thephysical nature of a(x, t) is more vividly portrayed by the Fourier time series solution (3.29a), which

reveals pulses bβ(x−xβ

n), oscillating at the frequency xβn, with local origins x = xβ

n. To achieve that goal,we must identify the periodicity class that the solution belongs to by inspection. Then we determine theappropriate Aα(t) = Aα(t + αT ) (see (3.18)) and link them by methods explained below to the aα(x);

they in turn determine the bβ(x) pulse-structure from (3.29a).

The crucial link is accomplished via the following two steps. Firstly, use of (3.16b) determines thedefinite integral

2

LPS

∫ LPS/4

−LPS/4

exp(−ixt) aα(x, t) dx = Aα(t) . (3.39)

Secondly, substitution of (3.26) into the left-hand side of (3.39) leads to an infinite sum of finite integrals,which conveniently combine yielding the Fourier transform relation

Aα(t) =1√2L

∫ ∞

−∞

aα(x) exp(−itx) dx . (3.40a)

Its inverse Fourier transform is

aα(x) =1√2T

∫ ∞

−∞

Aα(t) exp(itx) dt . (3.40b)


The mean energy integral (3.5) provides a time-dependent amplitude measure. On substitution of(3.16), we determine its value

E(t) ≡ 1

2N

2N−1∑

α=0

Eα(t − αT ) , where1

2NEα(t) =

2

LPS

∫ LPS/4

−LPS/4

|aα(x, t)|2 dx (3.41)

is the mean energy integral associated with each aα. Here there is no interaction here between differentaα’s because they each possess mutually disjoint spatial harmonics (see (3.16b)). Remember also that,despite having 2N α’s, they satisfy |aN+α| = |aα| implying EN+α = Eα for 0 ≤ α ≤ N − 1 (see (3.27b)).Since from (3.26) the time average of |aα|2 is obviously

1

2N2

∑

m∈Z

|aα(x − 12mLPS)|2 , (3.42)

the time average of the mean energy integral (3.41)1 is

〈E〉 =1

N

N−1∑

α=0

〈Eα〉 , where 〈Eα〉 =1

L

∫ ∞

−∞

|a α(x)|2 dx . (3.43a)

We calculate 〈Eα〉 numerically from the formula

〈Eα〉 =1

T

∫ ∞

−∞

|Aα(t)|2 dt , (3.43b)

which follows either directly from (3.16b) and (3.41)2 or indirectly from (3.40) and (3.43a)2 by Parseval’sTheorem. The value of 〈E〉 is a useful measure of solution amplitude.

Finally for a ∈ PQ− (T ; N), the symmetry property |aα+N/2(x)| = |a α(x)|, determined by (3.22) and

(3.26), enables us to reduce (3.43a) further:

〈E〉 =2

N

(N/2)−1∑

α=0

〈Eα〉 . (3.43c)

For the BS–solutions in PQ− (T ; 2) with LPS = L, this sum has only one term and so recovers the result

(2.11a) quoted in Section 2.2.

4. Methodology

When all An(t) have small amplitude we can linearise the system (3.4a). Then the modes decouple andwe notice that there is only a finite range nT−√

λ < t < nT +√

λ of time t over which the individual modeAn(t) can grow, dAn/dt > 0; outside this interval the decay rate is ever increasing as |t− nT | → ∞. Theonly way to achieve a sustainable solution a(x, t) is through the finite amplitude nonlinear interactions.Roughly speaking, for modest λ, a mode An(t) is forced off its zero-amplitude state by nonlinear couplingof its largest finite amplitude predecessors (usually the coupling An−1(t)An−1(t)A

∗n−2(t) in (3.4a)) at the

early end of An(t)’s unstable time range. Provided An(t) reaches a sufficient magnitude it can be used ina subsequent similar nonlinear neighbour interaction to excite the next mode An+1(t).

The above picture suggests that when nT < t ≤ (n + 1)T the only Am(t) that matter are those withm inside the range n−M ≤ m ≤ n + M for some sufficiently large M ∈ N; outside the range we assumethat Am(t) is negligible and can be set to zero. In this way our infinite set of equations is reduced to afinite set of 2M + 1 equations. The problem, of course, is that the set of Am(t) alters as t crosses nT ! So,to take care of this difficulty, on arrival at the end t = nT of the preceding interval (n−1)T < t ≤ nT , forwhich the m–range is n−M − 1 ≤ m ≤ n + M − 1, we drop the term An−M−1(t) (i.e. set it to zero) andinclude An+M (t) instead, where it now takes the initial value An+M (nT ) = 0 for the following interval.This means that any particular An(t) only remains non-zero for (n − M)T < t ≤ (n + M + 1)T and


outside that interval it is assumed to vanish. This technique was introduced to solve a related problemin [17].

In Section 4.1 we expand briefly on the numerical method by which we determine periodic solutionstaking the BS–family of solutions a ∈ PQ

− (T, 2) discovered in [8] as an illustrative example. We discusslimited aspects of the stability of the periodic solutions that we find in Section 4.2.

4.1. Numerical solution

In the light of the above remarks we reformulate the mathematical problem (3.4) into one over thefinite interval 0 ≤ t ≤ T , by first noting that Am+n(t+nT ) = T −nAm(t) (see (3.8c)1). So to determinethe numerical solution on the interval nT < t ≤ (n + 1)T , we solve

d

dtT −nAm −

[λ − (t − mT )2

]T −nAm = −

∑

α,β∈Z

T −nAm+α T −nAm+β T −nAm+α+β (4.1a)

on 0 < t ≤ T instead with T −nAm(t) = 0 for |m| ≥ M + 1, subject to the initial condition

T −nAm(0) =

0 for m = M ;

T −(n−1)Am+1(T ) for −M ≤ m ≤ M − 1(4.1b)

obtained by solving the earlier problem for T −(n−1)Am(t). Note, of course, that in view of our c.c.r. prop-erty all our T −nAm(t) are real (see (3.11)), which is why we have dropped the complex conjugationoriginally in (3.4a).

The numerical strategy adopted by BS, and again by us here, was to fix λ and T together with plausibleinitial data for Am(0) with |A0(0)| largest and |Am(0)| decreasing rapidly to zero with increasing |m|.A standard routine was then used to march the solution forward in time from t = 0 until t = T .Then the next problem for T −1Am(t) was solved and so on. Often quasi-periodic behaviour was found.Nevertheless, sometimes the transients died away leaving a periodicity identified by the mapping propertyT −n+NAm(T ) = ±T −nAm(T ) holding true as n → ∞. This provides the evidence that we havedetected a solution belonging to P±(T ; N). When N is even and 1

2N is odd, the sub-classification of

a P−(T ; N) solution into either PQ− (T ; N) or PH

− (T ; N) is best achieved by inspection. Once a periodicsolution was isolated, we used the AUTO program (see [18]) to track it in parameter space under variationsof the parameters λ and T .

To obtain some understanding of the ideas, we summarise quickly BS’s principal findings. For theirPQ− (T ; 2) solution (3.16a), the Aα(t) (0 ≤ α ≤ 3) in (3.16b) satisfy

A 0(t) = −A 1(t) = − A 2(t) = A 3(t) ≡ A(t) (say) . (4.2)

Evidently the solution has the properties (3.20), (3.22) and so relative to the time origin chosen belongs

to the symmetry class PQ−− (T ; 2). Since each amplitude Aα(t) = Aα(t − αT ) (see (3.18)) is localised

in the vicinity of t = αT , we label them [A,−A,−A,A] sequentially (α = 0, 1, 2, 3). This provides asignature of this PQ−

− (T ; 2) class of solutions as illustrated in Fig. 6(a). The aα(x) (3.26) correspondingto (4.2) determined by (3.40b) are a 0(x) = −a 1(x) = −a 2(x) = a 3(x) ≡ a(x) illustrated in Fig. 7(a)(see also Fig. 2(b) for the same λ but a slightly different L = LPS value to that used in Figs. 6 and 7).As explained below (3.37), this result determines the BS–solution (2.10).

The role of the shift operations is worth noting. According to (3.8c)1 the new solution successivelygenerated by the time-shift T permutes the symbols cyclically to [A,A,−A,−A] ∈ PQ+

− (T ; 2); thiscorresponds to pulse-amplitudes Aα(t) (α = −1, 0, 1, 2) and is effected in Fig. 6a by moving the viewingwindow one unit t/T to the left. On the other hand, according to (3.8c)2 the space-shift L alternatesthe signs of the odd n entries so that [A,−A,−A,A] becomes [A,A,−A,−A] ∈ PQ+

− (T ; 2). These groupactions, when applied in succession, are consistent with the property

T La(x, t) = −LT a(x, t) = ± a(x, t) for a ∈ PQ±− (T ; 2) (4.3)


-2 0 2 4 6-3

-2

-1

0

1

2

3

t/T

An(t

)

-2 0 2 4 6-3

-2

-1

0

1

2

3

t/T

An(t

)

Figure 6. The functions Aα(t) localised in the vicinity of t = αT for successive integers α = −1, 0, 1, . . . , 6,

separated by time T , for the N = 2 case λ = 10, L ≈ 2.094 (T = 1.5): (a) the higher energy, PQ−− (T ; 2) solution

with single amplitude function - denoted [A,−A,−A,A] and found near the highest point of the continuouscurve on Fig. 5 at LPS ≈ 2.094; (b) the lower energy, symmetry-broken PH

− (T ; 2) solution (see Section 5) withtwo distinct amplitude functions - denoted [A,−D,−A,D] and found on the dashed curve on Fig. 5 also atLPS ≈ 2.094.

(see (3.21)).

4.2. Stability tests

Testing the stability of temporally periodic solutions (3.3) in the most general sense is not a straight-forward task. It involves the introduction of a Floquet exponent kT and consideration of a perturbed


0 1 2 3 4-1

0

1

2

3

4

0 1 2 3 4-1

0

1

2

3

4

0 1 2 3 4-1

0

1

2

3

4

(a) (b) (c)

x/LPSx/LPSx/LPS

a(x

)

Figure 7. The pulses aα(x) for the N = 2 case λ = 10, L ≈ 2.094 of Fig. 6. The modulus |a α|, real partReaα and imaginary part Ima α are identified by the continuous, dashed and chain lines respectively: (a) A:

a 0(x) = a(x) for the PQ−− (T ; 2) solution in Fig. 6(a); (b) A: a 0(x)[= −a 2(x)] and (c) D: −a 1(x)[= a 3(x)] for the

symmetry-broken PH− (T ; 2) solution in Fig. 6(b).

solution in the form

a(x, t) = exp(itx)∑

n∈Z

[An(t) exp(− inTx) + A+

n (t) exp(− i(n + k)Tx) + A−n (t) exp(− i(n − k)Tx)

], (4.4)

where 0 ≤ k < 1. The ensuing linear equations for the small perturbations A±n (t) must then be solved

numerically to determine the appropriate return map over the temporal period of the original solution. Adisconcerting aspect of this approach is that a neighbouring periodic solution with T replaced by (1+k)T ,where 0 < k ≪ 1, is not captured by the representation (4.4), at any rate not as a small perturbation.

We did not attempt the general strategy suggested by (4.4), but largely focused on the special casek = 0 – in other words we limited our tests to structures possessing the form (3.3) at fixed T . AUTOwas sometimes able to test stability within this framework but often the results were inconclusive. Ourusual practice was to implement a time-stepping program to find whether under a small disturbance thesolution relaxed to its unperturbed periodic form.

A modification of this k = 0 strategy with a rational k = p/q (p, q ∈ N) led to some useful results inthe cases q = 2 and 3. The idea is to first rewrite (3.3) as a series with T and n replaced by T/q and qnrespectively. Then the new An are the old An/q, when n is an integer multiple of q, but zero otherwise.In this way a periodic solution in P(T ; N) is embedded in P(T/q; qN). Regarded as a member of thelarger class P(T/q; qN) based on T/q rather than T , stability may be explored by the k = 0 methods.The obvious limitation of this technique is that to obtain accurate solutions the number of An equations


that need to be investigated increases by a factor q. Given our machine restrictions on the size of problemthat could be considered, it followed that the (admittedly small) q = 3 was the largest value that gavereliable answers. Despite these limitations, the method enabled us to identify many bifurcations such asthe even N solution branches illustrated on Fig. 15 below. Other bifurcations were found but here andbelow only a few of the more significant and interesting cases are discussed. Finally we remark that allour P±(T ; N) odd N solution branches were identified in the first instance by the initial value problemapproach outlined in Section 4.1.

5. Symmetry broken PH− (T ; 2) solutions

Before λ even reaches 5, the N = 2 quarter-period TPS (= T ) BS–solutions, a ∈ PQ− (T ; 2), display

bifurcations which break the quarter-period space-shift symmetry (3.21) spawning new symmetry-brokensolutions, a ∈ PH

− (T ; 2). Like the BS–solutions, the new PH− (T ; 2) solutions still have zero-mean and half-

period 2TPS but their amplitude functions Aα(t) only satisfy the P−(T ; 2) requirement (3.20), namely

A 0(t) = − A 2(t) , − A 1(t) = A 3(t) with A 0(t) 6= ±A 1(t) (5.1)

and no longer fulfill the PQ±− (T ; 2) requirement (3.22). The original BS–style solutions and the new broken

symmetry solutions are compared in Figs. 6(a) and (b) respectively, for the case λ = 10, L(= LPS) ≈ 2.094(T = 1.5). (This L-value differs slightly from 2.487 fixed by the HBS–solution and used in Figs. 2 and 3.Though the symmetry broken solution exists at L ≈ 2.487, it is indistinguishable from the BS–solutionat those parameter values, which is why we have adopted a slightly different L-value for our presentcomparisons.) The labelling [A,−D,−A,D] of the amplitude sequence Aα(t) = Aα(t−αT ) (α = 0, 1, 2, 3)in Fig. 6(b) provides the signature of the new symmetry broken PH

− (T ; 2) solutions.

Unlike the BS–solutions, which only determine one independent aα(x), namely a 0(x) = a(x) illustratedin Fig. 7(a), the symmetry broken forms (5.1) aided by (3.40b) determine two. They are a 0(x) (= −a 2(x))and −a 1(x) (= a 3(x)) illustrated in Figs. 7(b) and (c) respectively. The amplitude function a 0(x) cor-

responds to the smaller Fourier amplitude A0(t) labelled A, while a 1(x) corresponds to the larger A1(t)identified by −D. According to (3.35), in which the γ–sum only has one term γ = 0, these amplitudefunctions determine the pulse-train

a(x, t) =∑

n∈Z

b (x − xn) exp (ixnt) + c.c.r.(xn ≡ x1

n =(2n + 1

2

)L

), (5.2a)

where from (3.29b)

b(x) ≡ b1(x) =

(a 0(x) − i a 1(x)

)/√

2 . (5.2b)

The modulus of the complex amplitudes b(x − 12L) and b

∗(−x − 1

2L) closest to the origin x = 0 areillustrated in Fig. 8. Other pulse pairs are generated by shifts of distance 2L to both the left and the right.Note that unlike the BS–style solution amplitude a(x), shown in Fig. 2(a), whose modulus is symmetricabout x = 0 (|a(x)| = |a(−x)|), this symmetry is lost for our new solutions so |b(x)| 6= |b(−x)|.

As for the BS–solutions, the new solution generated by the space-shift L alternates the signs of the oddn entries of [A,−D,−A,D, ] to [A,D,−A,−D]. The subsequent T -time-shift (therefore giving a combinedoperation T L) permutes the symbols cyclically to [−D,A,D,−A]. This interchange of the order of thefirst two entries A and −D is manifest by the change in the symmetry of the pulse-structure illustratedin Fig. 8 about the origin (x = 0) under the shift of the origin to x = L. Specifically (5.2) becomes

T La(x, t) = −∑

n∈Z

i b∗(−x + xn) exp (ixnt) + c.c.r. (5.3)

and so only when A = D do we have i b∗(−x) = b (x); then the PQ−

− (T ; 2) symmetry property (4.3) ismet. Otherwise, when A 6= D, it is broken.

As remarked on at the start of this section, a PH− (T ; 2) mode branch is spawned at smallish λ just


-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

x/LPS

Figure 8. The modulus functions |b(x − 1

2LPS)| (which is centred at x = 1

2LPS) and |b ∗

(−x − 1

2LPS)| (centred

at x = − 1

2LPS) for the pulses corresponding to the symmetry-broken PH

− (T ; 2) wave-train solution (5.2) withsignature [A,−D,−A,D] for the case λ = 10, LPS = L ≈ 2.094 of Figs. 6(b) and 7(b,c).

less than 5. By λ = 5 it is manifest as a new solution branch, which bifurcates at both its ends from thePQ− (T ; 2) mode by breaking the latter’s A = D symmetry, a matter which we discuss further in Section

7 (see particularly the dashed curve on Fig. 15 below). As λ is increased a number of further PH− (T ; 2)

branches appear. Since at low values of λ they characterise solutions that are all unstable, we madeno attempt to catalogue them thoroughly. Nevertheless, by about λ = 8 a new branch emerges whichat λ = 10 is just visable as the small section of dashed curve in Fig. 5 close to the energy maximumof the main solution loop. It provides the first evidence of stable symmetry-broken PH

− (T ; 2) solutionsand so we followed their development with increasing λ more closely. Indeed the LPS-interval over whichthese modes exist widens with λ. Simultaneously, the structure of the branch becomes more complicatedand develops ‘steps’; the cases λ = 16 and 22 are illustrated in Fig. 9. Even for λ as small as 10 thesymmetry-broken solutions identified in Fig. 5 are generally the preferred modes. That is meant in thelimited sense of given an L (and corresponding T ) at which they exist, any perturbation of the PQ

− (T ; 2)mode grows and the solution relaxes with time to the other PH

− (T ; 2) mode, despite the fact that thelatter symmetry-broken solutions have a slightly lower energy and more complicated structure. For thesymmetry-broken solutions identified in Fig. 9 at λ = 16 and 22, the situation is less clear cut. The largeramplitude states at smaller L are generally stable, whereas those of lower amplitude at larger L tend tobe unstable. Whether perturbed versions of these larger L symmetry-broken solutions return to the mainsolution loop, find a new period, become quasi-periodic or even evaporate is a sensitive matter.

6. Non-zero- and zero-mean P±(T ; N) solutions with N odd

Our search for alternative solutions was most successful at values of T smaller than those at whichN = 2 solutions exist. The probable reason may be traced to the fact that the pulse-separation isLPS = 2L/N = 2π/NT (see (3.25) and (3.38)) and, since the pulse width is controlled by the balancebetween (λ + ix)a and ∂2a/∂x2 in (2.2a), it is essentially independent of N . However, as neighbouring


1.5 2 2.5 3 3.52.8

3

3.2

3.4

3.6

3.8

4

4.2

LPS

√〈E

〉

1.85 1.9 1.95 2 2.05

3.92

3.94

3.96

3.98

4

LPS

√〈E

〉

Figure 9. (a) Contours of√〈E〉 vs. spatial pulse-separation LPS for the cases λ = 16 (lower) and 22 (upper). The

solid curves determine the largest energy 〈E〉 sections of the PQ− (T ; 2) solutions. The symmetry broken PH

− (T ; 2)solutions that bifurcate off them are drawn broken. The P(T ; 3) solutions are the chain line closed loops (seeSection 6.1). In the case λ = 22, PH

− (T ; 6) solutions bifurcate off the upper P−(T ; 3) solution ‘kiwi-shaped’ chain

loop onto the short convoluted continuous curve, which at its other end bifurcates off the PQ− (T ; 2) solutions (see

Section 7). (b) To clarify the connections, the short convoluted continuous PH− (T ; 6) solution curve is expanded

and shown dashed .

pulses must interact, in some loose sense, the order of magnitude of the pulse-separation LPS shouldremain roughly constant for different N solutions. So, on increasing N , we might expect T to decreaseto keep TPS = 1

2NT (and whence LPS) roughly constant. That is why we use LPS rather than L on ourbifurcation diagrams to accommodate possible changes in the value of N , as we discuss in Section 7. Atthese small T values solutions generally revealed quasi-periodic behaviour. Nevertheless, isolated pockets


of stable periodic P(T ; N) solutions with N odd (but not even) were located. We begin by describing indetail the simplest N = 3 case and then comment briefly on other odd N values.

We must stress that when we discuss stability of a solution in Sections 6.2 and 6.3, we are onlyconsidering behaviour within the class of solutions of the form (3.3) at given T . We will explore in thelater Section 7 the possibility of instabilities of the form (4.4) with T replaced by T/q (q ∈ N), as explainedin Section 4.2. Then the importance of LPS will become clear.

6.1. The case N = 3

The smallest value of λ at which we identified N = 3 solutions is λ ≈ 14.45 with a correspondingT ≈ 0.87. At this λ no N = 2 solution exists with this relatively small value of T . On increasing λ to16, that isolated point expands into the small closed chain loop illustrated on Fig. 9(a), in the vicinity ofLPS = 2L/3 = 2π/3T ≈ 2.3 (T ≈ 0.9) and

√〈E〉 ≈ 3.3. With yet further increases of λ beyond roughly20, a second set of solutions emerges detached from the main loop with roughly the same energy butwith slightly larger values of T . By λ = 22 these loops coalesce to form the ‘kiwi-shaped’ chain loop alsoshown on Fig. 9(a).

The non-zero-mean half-period 2TPS = 3T solution a ∈ P+(T, 3) of the form (3.16) is characterised by

the six amplitudes Aα(t) = Aα(t − αT ) (α = 0, 1, . . . , 5), although only three of these are independent

since Aα−3(t) = Aα(t) for α = 0, 1, 2 (see (3.20)). A typical robust solution at λ = 22, LPS ≈ 1.821(T = 1.15) is illustrated in Fig. 10(a). As in the previous section we label the amplitudes Aα(t) (α =0, 1 . . . , 5) sequentially as [A, C,−E ,A, C,−E ], where the sign reflects the dominant sign of the amplitudedrawn. The corresponding plots for the space-shifted zero–mean solution La ∈ P−(T, 3) of the amplitudes

LAα(t)[= (−1)αAα(t − αT )] are illustrated in Fig. 10(b) with signature [A,−C,−E ,−A, C, E ].

As mentioned at the end of Section 3.2, the x = 0 values of a and its derivative ∂a/∂x, determinedfrom (3.12), provide a useful solution diagnostic. So we plot the time series of the P+(T ; 3) solutions±a(0, t) = ±∑

α∈ZAα(t) and the P−(T ; 3) solutions La(0, t) =

∑α∈Z

(−1)αAα(t) in Fig. 11(a). Thecorresponding phase portraits ± Im∂a/∂x(0, t) versus ±Rea(0, t) and Im∂La/∂x(0, t) versusReLa(0, t) are plotted in Fig. 11(b). It is interesting to compare these figures with their counterpartsobtained by HBS [14] and BS [8].

The spatial-amplitude functions a 0(x), a 1(x) and −a 2(x) illustrated in Fig. 12 are determined via

(3.40b) from A 0(t) (i.e. A), A 1(t) (C) and −A 2(t) (E) respectively illustrated in Fig. 10. Of the three,

A 0(t) is the largest and leads to the large amplitude sharply focused pulse a 0(x). The other smaller A 1(t)

and −A 2(t) have off-centred peaks that suggest the possible manifestation of a non-zero wave-number inthe pulse-amplitudes a 1(x) and −a 2(x). This is particularly pronounced for the two-hump structure of

A 1(t) for which there are sizeable amplitudes of the oscillations in the tail of a 1(x).

The three amplitudes aα(x) (α = 0, 1, 2) are used to construct our P+(T, 3) pulse–train solution.According to (3.32) with 2L = 3LPS it is

a(x, t) =∑

n∈Z

b

0(x − 3nLPS) exp (i 3nLPS t)

+[b

2(x − (3n + 1)LPS) exp (i (3n + 1)LPS t) + c.c.r.

], a ∈ P+(T ; 3) , (6.1a)

in which

b0(x) =

√2

3

(a 0(x) + a 1(x) + a 2(x)

)(6.1b)

is the non-zero time average (3.30) of a(x, t), while from (3.29b)

b2(x) =

√2

3

(a 0(x) + e−i2π/3 a 1(x) + e−i4π/3 a 2(x)

). (6.1c)

The amplitudes of the pulses b−2

(x + LPS) [= Cb 2(x + LPS) by (3.29c)], b0(x) and b

2(x − LPS) are


-2 0 2 4 6 8 10

-4

-2

0

2

4

t/T

An(t

)

-2 0 2 4 6 8 10

-4

-2

0

2

4

t/T

An(t

)

Figure 10. The functions Aα(t) and LAα(t) = (−1)αAα(t) for successive integers α = −1, 0, 1 . . . , 10 for thecase λ = 22, T = 1.15. The functions with α = 3k, α = 1 + 3k and α = 2 + 3(k − 1) (k = 0, 1, 2, 3) are drawncontinuous (A), dotted (C) and dashed (E) respectively: (a) The P+(T ; 3) solution amplitudes Aα(t) – denoted[A, C,−E ,A, C,−E ]; (b) The P−(T ; 3) solution amplitudes LAα(t) – denoted [A,−C,−E ,−A, C, E ].

illustrated in Fig.13. Finally, the zero-mean solution obtained by the space-shift L is

La(x, t) =∑

n∈Z

b

0(x − (3n + 3

2 )LPS) exp(i (3n + 3

2 )LPS t)

+[b

2(x − (3n + 5

2 )LPS

)exp

(i (3n + 5

2 )LPS t)

+ c.c.r.]

, a ∈ P−(T ; 3) . (6.2)

The space-time constant amplitude solution contours are illustrated on Fig. 14. We plot x/LPS–t/TPS con-tours of Rea(x, t) and Ima(x, t) in Fig. 14(a), and ReLa(x, t) and ImLa(x, t) in Fig. 14(b).


-2 0 2 4 6 8 10

-4

-2

0

2

4

t/T

a(0

,t)

-4 -2 0 2 4-20

-10

0

10

20

Im

∂a/∂

x(

0,t)

Rea(0, t)

Figure 11. The x = 0 behaviours of the P(T ; 3) solutions for the case λ = 22, T = 1.15 (L ≈ 2.732) displayed inFig. 10. The P+(T ; 3) solutions ±a(x, t) (dashed lines) and the P−(T ; 3) solution La(x, t) (continuous line). (a)The time series of ±a(0, t) = ±P

α∈ZAα(t) and La(0, t) =

Pα∈Z

(−1)αAα(t) vs. t/T . (b) The phase portraitsof ± Im∂a/∂x(0, t) and ±Rea(0, t) and Im∂La/∂x(0, t) vs. ReLa(0, t). As time proceeds eachloop is followed an anti-clockwise sense.

6.2. The search for odd N solutions

Our basic strategy used to search for odd N periodic solutions was to time–step initial value problemsat fixed T and λ. Sometimes the solution evapourated but, for certain values of T and λ, the solutionevolved and settled onto a periodic solution characteried by some value of N . Now, as remarked atthe beginning of this section, we expect that the pulse-separation LPS will be roughly the same (sayLpref = π/Tpref) irrespective of the value of N possessed by the realised solution. Since T ∼ 2Tpref/N


0 1 2 3 4

-2

0

2

4

6

0 1 2 3 4

-2

0

2

4

6

0 1 2 3 4

-2

0

2

4

6(a) (b) (c)

x/LPSx/LPSx/LPS

a(x

)

Figure 12. The amplitudes aα(x) (α = 0, 1, 2) corresponding to the functions bAα(t) portrayed in Fig. 10(a)for the P−(T ; 3) case λ = 22, T = 1.15. The modulus |a α|, real part Reaα and imaginary part Ima α areidentified by the continuous, dashed and chain lines respectively. The functions plotted are (a) A: a 0(x); (b) C:a 1(x); (c) E : −a 2(x). Note that the pulse-separation length LPS = 2L/3 ≈ 1.821.

(see (3.13b)), we were able to locate N ≥ 3 solutions simply by searching at small T values (less thanTpref), where the N = 2 solutions are either absent or cease to dominate. Once an odd N solution islocated it can be tracked in T –λ space by use of the AUTO program mentioned in Section 4.1 and someresults are portrayed on Figs. 9 and 15.

Some care is needed with the interpretation of the results portrayed on Figs. 9 and 15 because√〈E〉

is plotted versus the pulse separation LPS (= 2L/N) rather than L. The reason that we choose LPS isbecause it turns out to be the appropriate parameter for the identification of bifurcations into cases withdistinct N–values. Nevertheless, it is more illuminating in the present context, where in any particularnumerical integration L (= π/T ) is fixed, to bear in mind what happens on the figures if L (= NLPS/2)rather than LPS is the abscissa. For then both the N = 3 results on Fig. 9(a) and the N = 5, 7 resultson Fig. 15 are moved significantly to the right. Thus they are located at relatively large values of L,equivalently small T , where the N = 2 mode is either unstable or does not exist at all. As remarkedabove, this appears to explain our success at finding those stable members of the odd N–families.

The smallest value of λ at which odd N solutions were located lies just below λ = 3.5 with correspondingN = 7. On increasing λ rather complicated N = 7 solution loci emerge as illustrated by the dotted curveon the left of Fig. 15 for the case λ = 5. Even on this curve the only stable solutions were found on avery small curve section in the vicinity of the

√〈E〉 maximum at LPS ∼ 0.9 (T ∼ 1.0). This correspondsto a largish value of L (> 3), at which there are no N = 2 solutions. Curiously, the N = 7 solution withmaximum energy appears to have T very close to unity over a considerable range of values of λ.


-6 -4 -2 0 2 4 60

1

2

3

4

5

x/LPS

Figure 13. The pulse-amplitudes b0(x) (broken line) and b

∓2(x ± LPS) (continuous lines) for the P−(T ; 3) case

λ = 22, T = 1.15, LPS ≈ 1.821 (see Figs. 10 and 12).

On increasing λ the next odd N solution discovered was for N = 5 at λ ∼ 5.019, LPS ∼ 1.8 (T ∼ 0.7).This corresponds to L ∼ 4.5, well into the region where N = 2 solutions are absent. Its associatedenergy is identified by the point labelled II on Fig. 15. We made no effort to follow its development withincreasing λ as it was generally found to be unstable. If fact, neither the N = 5 nor the N = 7 solutionsseem to be particularly interesting but we have mentioned them to illustrate possibilities that appear toexist.

The N = 3 solutions described in Section 6.1 do not emerge until λ ≈ 14.45. Once they appear, however,they exhibit definite finite intervals of T where they are stable, which makes them more interesting thanthe N = 5 and 7 cases. By λ = 16 a small set of N = 3 solutions exists, identified by the lower andsmaller chain loop on Fig. 9(a) in the vicinity of LPS ∼ 2.4, for which L ∼ 3.6. At such large values ofL the N = 2 solutions exist but are unstable. Again this explains why we were able to discover N = 3stable solutions in the first instance.

Finally, we emphasise that the N = 5 and 7 solutions always appeared to have extremely small stableT –intervals, which made them very hard to locate in contrast to the case for N = 3. During our searcheswe never found periodic solutions for any other odd value of N .

7. Zero-mean P−(T ; N) solutions with N even

In Section 5 we considered zero-mean P−(T ; 2) solutions. Here we explore the possibility of otherP−(T ; N) solutions (3.35) for even N (> 2). Their pulse-structures determined by the pulse-amplitudes

b4γ+1

(0 ≤ γ ≤ 12N − 1) defined by (3.29b) simplify further to (3.37) for the quarter period space–shift

symmetry class PQ− (T ; N).

In Section 6, P(T ; N) solutions with N odd were discovered at fixed T and λ by solving initial valueproblems numerically. Since, at the small values of T chosen, no periodic solutions were known in advance,the initial data was generally chosen arbitrarily. Nevertheless, when T lay in a range where PQ

− (T ; 2)solutions exist, we were able to use them as initial data instead. If stable, the solution persisted but, ifunstable, it evolved into a new state, which might be periodic, quasi-periodic or simply the zero-amplitude


Figure 14. Contours of constant amplitude for the P(T ; 3) solution in the x/LPS–t/TPS plane for the case λ = 22,T = 1.15 (L ≈ 2.732, pulse-separation LPS ≈ 1.821). Note that the time origin is not that used on the other figuresbut has been shifted t → t− qT with q a non-integer chosen such that for the zero-mean solution Rea(0, 0) = 0:(a) The P+(T ; 3) solution, Rea(x, t) (left) and Ima(x, t) (right); (b) The P−(T ; 3) solution, ReLa(x, t)(left) and ImLa(x, t) (right).


I

II

0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

LPS

√〈E

〉

Figure 15. Contours of√〈E〉 vs. LPS for a variety of solutions at λ = 5. The outer solid loop (denoting the BS

PQ− (T ; 2) solutions) and the solid dot labelled I (denoting the HBS–solution) are redrawn from Fig. 4. The dashed

curve identifies the symmetry-broken PH− (T ; 2) solutions. The chain curves identify the PQ

− (T ; 6) (see Section 7),

while the interior solid curve defines the symmetry-broken PH− (T ; 6) solutions. The dotted curve identifies the

P(T ; 7) solutions. Lastly, the solid dot labelled II marks the point in parameter space from which the P(T ; 5)solutions emerge – this occurs at λ ≈ 5.019 only very slightly larger than that used here.

trivial state. Indeed, at largish λ, that is how we identified some of the stable PH− (T ; 2) solutions illustrated

by the broken curves on Fig. 9(a). That said, the initial value approach using arbitrary or known datanever revealed any new even N > 2 solutions for any choice of T and λ. Nevertheless using AUTO we wereable to reach such elusive solutions via bifurcations to them. The bifurcation method relies on embeddingsolutions from one class into a wider class. So for example, some of the PH

− (T ; 2) solutions described inSection 5, particularly at smallish λ, were located after known quarter period space–shift symmetry classPQ− (T ; 2) solutions were embedded in the wider P−(T ; 2) class. Then on following a PQ

− (T ; 2) solutioncurve, AUTO either flagged up a bifurcation or simply followed the bifurcated PH

− (T ; 2) branch throughchoice. For example, the method was used successfully at smallish λ = 5 to locate the unstable PH

− (T ; 2)solutions illustrated by the dashed curve on Fig. 15.

To find new even N > 2 solutions, we use the Floquet theory approach based on (4.4) with theFloquet exponent k restricted to rational ratios p/q (p, q ∈ N). Actually only the values q = 2, 3 wereconsidered because of the practical numerical limitations mentioned in Section 4.2. As explained there, atemporally periodic solution based on the spatial Fourier series representation (3.3) for given T is recastas another based on T/q rather than T . Now, since the pulse spacing LPS remains the same after theembedding, it is appropriate to adopt the invariant TPS = π/LPS as our fixed time rather than T . In thisway a periodic solution belonging to P−(T ; N) = P−(2TPS/N ; N) is embedded inside the wider classP−(2TPS/(qN); qN), into which bifurcations may then be located.

As an example of the procedure, we note that a PQ−− (TPS; 2) solution with signature [A,−A,−A,A] can

be expressed as a PQ−− (1

3TPS; 6) solution with signature [A, 0, 0,−A, 0, 0,−A, 0, 0,A, 0, 0]. When tracked

as such at λ = 5, a number of bifurcations from PQ−− (TPS; 2) to genuine PQ−

− (13TPS; 6) solutions with


signature

[A, −B, −C, −A, −B, C, −A, B, C, A, B, −C] (7.1a)

were located. (Since no Aα(t) are plotted, we are abandoning the convention that the signs in (7.1a) reflect

the dominant sign of each Aα(t). This is so that we can encompass the many solutions found, e.g. anotherwith B replaced by −B, without classifying them individually.) Of the many found we have only indicateda representative sample of three that follow along the chain curves on Fig. 15. For such quarter-periodspace-shift symmetry solutions (1

2N = 3), the 3 independent amplitudes Aα(t) (α = 0, 1, 2) linked to A,

−B, −C determine the aα(x) in (3.37), which in turn determine the three b4γ+1

(x) (γ = 0, 1, 2) neededfor the pulse-train solution (3.35). Like the P±(T ; 3) solutions (6.1) and (6.2) of Section 6.1, the solution

only possesses 3 independent pulse profiles, which here are b1(x), b

5(x) and b

9(x).

On two of the PQ−− (1

3TPS; 6) curves, a symmetry breaking bifurcation occurs leading to PH− (1

3TPS; 6)solutions with signature

[A, −E , −F , −D, −B, C, −A, E , F , D, B, −C] . (7.1b)

Furthermore, the two bifurcation points are connected as indicated by the interior solid curve joiningthem on Fig. 15. Despite the increase to six distinct amplitudes Aα(t) (α = 0, . . . 5), the number of

b4γ+1

(x) (γ = 0, 1, 2) remains the same as that prior to the symmetry breaking bifurcations.

Already at λ = 5 it is clear that letting the value of N increase appears to open up the possibilityof a multitude of different modes. So on tracking a PH

− (13TPS; 6) mode to larger λ, it came as no great

surprise to find that at λ = 22 its solution curve illustrated by the short convoluted continuous curve onFig. 9(a) made different connections to those discovered at λ = 5 (the interior solid curve on Fig. 15).The new situation is clearly visible on the blown up curve now dashed on Fig. 9(b). At its right-handend, the PH

− (13TPS; 6) solutions bifurcate directly from the PQ

− (TPS; 2) solutions without following first

the intermediate PQ− (1

3TPS; 6) solutions occurring at λ = 5. The bifurcation at the left-hand end from aP−(2

3TPS; 3) solution is interesting, as we have not encountered it before. Cast in PH− (1

3TPS; 6) form theyhave the signature [A, 0,−E , 0,−C, 0,−A, 0, E , 0, C, 0]. Though the PH

− (13TPS; 6) solution branch found at

λ = 22 is rather short it is significant because of the co-existence of three other classes, namely PQ− (TPS; 2),

PH− (TPS; 2) and P−(2

3TPS; 3). Despite the competition, time stepping revealed that the PH− (1

3TPS; 6)solution was preferred and in that sense robust.

8. Discussion

Our work shows that the structure of the various possible wave-train solutions of (2.8) is remarkablycomplicated. One reason for this is that there is no bifurcation sequence from the the trivial zero-amplitudestate. Indeed, even our solutions of the HBS–problem (2.2) with Υε = 1

4 , which closely resemble BS–solutions, appear to lie on a solution branch disconnected from the zero-amplitude state.

Despite the apparent complexity of the solution structures identified in this paper, it must be stressedthat all solutions found are spatially periodic in the sense of (3.3) at fixed wavenumber T with corre-sponding wavelength 2L = 2π/T . Temporally periodic solutions have been found belonging to classesP(T ; N) (N ≥ 2) consisting of pulse-trains; each pulse is separated by a distance LPS = 2L/N . At givenλ any realisable solution at given N tends to have roughly the same pulse-separation LPS. The pulse-trainpattern repeats itself over the longer wavelength 2L = NLPS. Our solution sets are best summarised bythe

√〈E〉 versus LPS plots shown in Figs. 5, 9 and 15. Taken together, these diagrams illustrate vari-ous possibilities for N = 2, 3, 5, 6, 7 modes, where the range of N taken was limited by computationalconstraints.

Even very modest values of λ reveal a morass of solutions as Fig. 15 attests. Indeed even within therestricted classes N ≤ 7 considered only a representative set of solutions and bifurcations is illustrated.The abundance of solutions probably reflects the fact that the pulse-structures in different classes onlydiffer slightly in subtle ways, which in some sense our classification over emphasises. That said, almost


all solutions portrayed on Fig. 15 are unstable except for the simplest large amplitude solutions PQ− (T ; 2)

solutions found by BS [8].

As λ increases, the PQ− (T ; 2) solutions lose their dominance. In this respect Fig. 9(a) is important as it

illustrates that the symmetry-broken PH− (T ; 2) solutions at λ = 16 and 22 are preferred over the original

PQ− (T ; 2) solutions for a broad range of LPS, generally the highest ‘step’ of the broken curve. Not only

that, but Fig. 9(b) identifies P−(T ; 6) solutions which over a very narow ranges of LPS are preferredover other possibilities. Fig. 9(a) also reveals another curious feature. For whereas the stable solutions ina particular symmetry class are generally those with the largest pulse energy E , that is not necessarilyso when solutions outside that class are allowed for. As an example, the PH

− (T ; 2) solutions, which are

sometimes preferred, have lower energy than their PQ− (T ; 2) counterparts.

Prior to our study, the only known persistent finite amplitude solutions were the PQ− (T ; 2) solutions.

Since they had proved so elusive prior to BS’s investigation [8], we were concerned that they might beexceptional and not realisable in practice. Our present study has dispelled those doubts and revealed amultitude of periodic solutions. That said, there are also quasi-periodic solutions that appear to persistrobustly in certain parameter ranges.

From the point of view of the motivating narrow-gap spherical Couette flow problem, these conclusionsare reassuring. It means that our solutions should provide the building blocks of the multiple length scalesolution of the amplitude equation (1.14). As explained in Section 1, on long time and length scales weneed to take account of the fact that λ is not constant. The spatially varying group velocity will alsobe disruptive causing the local pulse-separation to systematically increase. So it is important that as alocal train of pulses encounters such unfavourable conditions, either in space or time, it can lose stabilityand recover as a new train with a more favourable pulse-separation distance. These disruptions suggestthat on the long length and time scales the system will be weakly chaotic. Regions of coherent pulse-trains are likely to persist for limited times over limited space ranges, perhaps connected by regions ofquasi-periodic behaviour. Pattern dislocations mentioned in Section 1 are also a possibility. Since our rigidspatial Fourier structure (3.3b) essentially prohibits the detection of pattern dislocations, it is not an issuethat our investigation can shed any light on. On the other hand, the time stepping of numerical solutionsof (2.2) by [14] were not constrained in this way and have the possibility of capturing dislocations. Itmust be stressed, however, that with [14]’s modest value of Υε = 1

4 only about 5 pulses were identifiable,not really enough to reveal dislocations and none were detected.

The central point, which we need to keep sight of, is that the existence of persistent solutions of (1.14)means that finite amplitude states are possible for Taylor numbers close to the local critical value atthe equator θ = 0. Those Taylor numbers are O(1) smaller than the true global critical value for theonset of instability. From an asymptotic point of view, this is a remarkable property and a much strongerstatement than simply saying that solutions are subcritical on some asymptotically small range.

Other than the early results of Wimmer [2, 3] very little has been done on the narrow gap limit. Thenarrow gap work of Dumas and Leonard [19] is, however, of interest. They identify non-axisymmetricspiral vortices, which have also been found in the medium gap limit [20, 21, 22, 23]. In their results wavyvortices are seen near the equator while, at higher latitudes, the vortices spiral away from the equator inthe negative longitudinal (azimuthal) direction. If we tentatively identify the azimuthal coordinate withtime, the spiral vortex states found in the papers cited immediately above resemble a wave propagatingtowards the equator, as our waves under their pulse envelopes (see Fig. 14) suggest.

Acknowledgements

E.W.B. acknowledges the support of EPSRC via a research studentship.


Appendix Group properties

In this appendix we discuss briefly some of the discrete symmetries of the amplitude equation (2.8) withλ = constant and consider how they can be embedded within a more general group-theoretic framework.Any symmetry of this PDE takes the form GpL, qT, φ or GpL, qT, φ C, where

GpL, qT, φ a (x, t) ≡LpT qRφa

(x, t) = a(x − pL, t − qT ) exp[i(pLt − φ)] (A.1)

(see (2.3) and (3.7)) and C is the complex conjugate reflection symmetry (c.c.r.) defined in (2.3). Theelement GpL, qT, φ is a space-time translation through (pL, qT ) and a phase shift through φ, which isdefined modulo 2π. We have

GpL, qT, φ Gp′L, q′T, φ′ = G(p+p′)L, (q+q′)T, φ+φ′+p′qπ , (A.2)

so that these elements commute up to a phase shift and where without loss of generality we have chosenLT ≡ π, as throughout this paper. Also for the c.c.r., we have

C2 = I, C GpL, qT, φ = G−pL, qT,− φ C . (A.3)

Note that

GpL, 0,0 ≡ Lp , G0, qT, 0 ≡ T q , G0,0, φ ≡ Rφ . (A.4)

A solution a is invariant under a subgroup of these symmetries, the isotropy subgroup of a. Clearly fora non-zero solution a, the isotropy subgroup cannot contain any non-trivial phase shifts Rφ for φ 6= 0:we say that the subgroup must be ‘phase compatible’. We consider here only discrete symmetry groupswhich have this property and also possess the two independent space-time translations, (pL, qT ) and(p′L, q′T ). We need only classify these up to conjugation by a group element (see Golubitsky, Stewartand Schaeffer [24] for background).

The main part of the paper considers only solutions invariant under the c.c.r. symmetry C, and thereare two classes of subgroups in which this property is enforced:-

• The first is generated by a space-shift, a time-shift and the reflection:

PL, T R, C (pL, qT, φ) = 〈Lp, T qRφ, C〉 , 12pq = N ∈ N , φ = 0, π . (A.5)

Consider a solution a of spatial periodicity 2L fixed by setting p = 2, q = N . Then according to (3.13),we have

P+(T ; N) ≡ PL, T R, C (2L, NT, 0) and P−(T ; N) ≡ PL, T R, C (2L, NT, π) . (A.6)

The translational properties of each set under L is detailed in (3.15), and under T in remarks that follow.

• The second class of subgroups is generated by a space-time translation and the reflection:

PQG,C(pL, qT, φ) = 〈GpL, qT, φ, C〉 , pq = 1

2N ∈ N , φ = 0, π , (A.7)

where by necessity N is even. We set p = 1, and consider odd integers q = 12N . Then according to (3.21)

(note that T qL = GL, qT, π for odd q), we have

PQ+− (T ; N) ≡ PQ

G,C (L, 12NT, π) and PQ−

− (T ; N) ≡ PQG,C (L, 1

2NT, 0) . (A.8)

We showed below (3.21a) that solutions invariant under these subgroups have spatial periodicity 2L andso belong to P−(T ; N). As argued below (3.21b), the members of each set shift to the other under theaction of either L or T .

If we drop the constraint that solutions are to be invariant under the reflection C, then there are twofurther classes of subgroups:-

• The first subgroup involves a time-translation combined with reflection

PL,T C(pL, qT ) = 〈Lp, T qC〉 , 12pq = N ∈ N . (A.9)


If we fix the spatial periodicity as p = 2 and let q = N then invariance of a solution under this symmetryclass requires An(t − NT ) = A∗

n+N (t) in the Fourier representation (3.3). Since we assumed that thecoefficients An(t) are real, we are unable to access such solutions except for the restricted c.c.r. class(A.6) or its subclass (A.8).

• The second subgroup does not involve C at all, and is

PG,G(pL, qT, p′L, q′T ) = 〈GpL, qT, 0, Gp′L, q′T, 0〉 , 12 (pq′ − p′q) = N ∈ N. (A.10)

The conditions ensure phase compatibility, and mean that the two space-time translations must span aparallelogram whose area is a multiple of 2π. A general solution with this symmetry has a space-timelattice of pulses that need not be aligned with the coordinate axes. However a spatial periodicity of 2Lcan be enforced by setting p = 2, q = 0, q′ = N and p′ = k unrestricted. In this case solutions of theform (3.3) have Fourier coefficients with the property An(t − NT ) = An+N (t) exp[−i(N + n)kπ].


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