Physica D 239 (2010) 1581–1592

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Physica D

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Snaking of radial solutions of the multi-dimensional Swift–Hohenberg equation:A numerical studyScott McCalla ∗, Björn SandstedeDivision of Applied Mathematics, Brown University, Providence, RI 02912, USA

a r t i c l e i n f o

Article history:Received 15 December 2009Received in revised form23 March 2010Accepted 21 April 2010Available online 2 May 2010Communicated by M. Silber

Keywords:Swift–Hohenberg equationRadial localized structuresSnaking

a b s t r a c t

The bifurcation structure of localized stationary radial patterns of the Swift–Hohenberg equation isexplored when a continuous parameter n is varied that corresponds to the underlying space dimensionwhenever n is an integer. In particular, we investigate how1Dpulses and 2-pulses are connected to planarspots and ringswhen n is increased from1 to 2.We also elucidate changes in the snaking diagrams of spotswhen the dimension is switched from 2 to 3.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

We are interested in the formation and parameter depen-dence of localized stationary radial solutions of the variationalSwift–Hohenberg equation

ut = −(1+∆)2u− µu+ νu2 − u3, x ∈ Rn. (1.1)

This equation was first derived by Swift and Hohenberg [1] to de-scribe the effects of random thermal fluctuations on fluid convec-tion just below onset. As shown for instance in [2], the steadySwift–Hohenberg equation is also the normal-form equation forsmall-amplitude radial solutions at Turing bifurcations in re-action–diffusion systems. More generally, the Swift–Hohenbergequation serves as a paradigm for bistable pattern-formingsystems: it exhibits a plethora of interesting localized and non-localized patterns that have also been found in many other bi-ological and physical systems [3–7]. Our interest is in localizedradial steady-state solutions of (1.1). Part of our motivation stemsfrom the observation made in [8,9] that localized stripe, hexagonand rhomboid patches emerge from localized radial solutions viasymmetry-breaking bifurcations. In addition to their relevance tosuch patterned patches, radial solutions are of interest in their ownright in many physical systems, and we refer the reader to [10–12]for references to systems that admit localized patterns of the shapediscussed below.

∗ Corresponding author. Tel.: +1 401 863 3878; fax: +1 401 863 1355.E-mail addresses: sgmccalla@brown.edu, Scott_McCalla@brown.edu

(S. McCalla), b.sandstede@brown.edu (B. Sandstede).

0167-2789/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2010.04.004

We now discuss Eq. (1.1) in more detail. Throughout this paper,we take ν > 0, as the case ν < 0 is obtained upon replacing uby−u. Unless stated otherwise, all computations presented beloware, in fact, done for ν = 1.6. The background state u = 0 isstable for µ > 0 and destabilizes in a Turing bifurcation at µ = 0.The Turing bifurcation gives rise to spatially periodic stationarypatterns with period near 2π , which we refer to as rolls. Rollsbifurcate into the region µ > 0 for ν > ν∗ =

√27/38 ≈ 0.84 and

into the regionµ < 0 otherwise. For fixed ν > ν∗, rolls are initiallyunstable, but when continued towards increasingµ, they undergoa fold bifurcation for sufficiently large µ at which they stabilize.They then return back as stable patterns towards decreasingµ andfinally cross µ = 0 with positive amplitude.As mentioned above, we focus on localized stationary radial

solutions u(x, t) = u(|x|) of the Swift–Hohenberg equation. Suchpatterns satisfy the equation[∂2r +

n− 1r

∂r + 1]2u = −µu+ νu2 − u3, r ∈ R+ (1.2)

together with the boundary conditions ur(0) = urrr(0) = 0 andlimr→∞ u(r) = 0, where r := |x|. To measure and represent thespatial width of localized radial patterns, we use their one- andtwo-dimensional L2-norms given, respectively, by

‖u‖2L2x:=

∫∞

0|u(x)|2 dx, ‖u‖2

L2r:=

∫∞

0|u(r)|2r dr.

In (1.2), we can clearly consider n as a continuous parameter andexamine the dependence of localized patterns on the continuousdimension parameter n. We are particularly interested in solutionprofiles u(r) that exist for µ > 0 and are, in an appropriate sense,composed of the stable roll structures that we discussed above.

1582 S. McCalla, B. Sandstede / Physica D 239 (2010) 1581–1592

Fig. 1. The center panel contains the bifurcation diagramof 1D localized pulses. Thesymmetric profiles that correspond to parameters on the light-colored curve have amaximum at r = 0 as shown in panels (1), (2), and (5), while the symmetric profilescorresponding to the dark-colored branch have a minimum at r = 0 as illustratedin panel (3). As we move up on each branch, a pair of new rolls is added to thesolution profile at every other fold bifurcation. The twodifferent branches discussedabove are connected by ladder branches that correspond to asymmetric profilesas indicated in panels (3)–(5). These asymmetric structures bifurcate at pitchforkbifurcations near each fold from the symmetric pulses.

In one space dimension, these radial profiles resemble stable rollswith a localized envelope superimposed on them as illustrated inFig. 1, so they can be thought of as localized rolls. In the planarcase, the radial profiles thatwe are interested in appear as localizedtarget patterns; see Fig. 2. We now summarize some of the knownresults about localized radial structures in dimension n = 1, 2, 3for µ > 0.When n = 1, Eq. (1.2) is reversible and Hamiltonian, and

much is knownabout localized radial patterns and their bifurcationdiagrams [13,14,10,15–18]. Localized roll structures, which werefer to as pulses, exist for ν > ν∗. Symmetric pulses that areinvariant under x 7→ −x snake: their bifurcation branch, obtainedby plotting the width of the roll plateau as measured by theirL2x-norm against the parameter µ, resembles a vertical sinusoidalcurve; see Fig. 1. As we move up along the branch, pulses broadenas new rolls are added on either end at every other fold. As shownin Fig. 1, there are two branches of symmetric pulses with eithera positive maximum or a negative minimum at x = 0, andthese branches are connected by horizontal ladder branches thatcorrespond to asymmetric localized roll patterns. Among the otherknown solutions are symmetric 2-pulses,which are bound states oftwo individualwell-separated localized roll structures. Two-pulsesexist along figure-of-eight isolas that lie inside the regions formedby two consecutive ladder branches and the two snaking curvesthat connect them [19,20]. More precisely, symmetric 2-pulsesexist along a two-parameter family of isolas that are parametrizedby (s, `), where ` ∈ N denotes the number of rolls in each of thetwo individual localized roll structures that make up the 2-pulse,and s ∈ N is the number of small-amplitude oscillations near u = 0in between the two individual pulses [21]. Thus, s can be thoughtof as a measure of the separation width, while ` represents theL2x-norm of the 2-pulse. In particular, a countably infinite numberof 2-pulses are expected to exist for each fixed value of their L2x-norm, and these 2-pulses are distinguished from each other by theincreasing separation distance between the two individual pulses.In two dimensions, several different kinds of localized radial

patterns were recently found in [22]. First, for each ν > 0, spotsbifurcate from µ = 0 into µ > 0. As illustrated in Fig. 2, thesespots resemble J0 Bessel functions near r = 0, and they have aninitial amplitude of order

√µ for small µ. From now on, we refer

to these structures as spot A solutions. In addition to these spots,two ring solutions emerge fromµ = 0 for each fixed ν > ν∗. Thesesolutions have an overall sech-like shapewith amaximumof order√µ that occurs at r ' 1/

√µ. For ν > ν∗, spot A and the two rings

appear to snake, as can be seen in Fig. 2. All of these solutions wereproved to exist for 0 < µ� 1 in [22].In three dimensions, numerical evidence for the existence

of spots was presented in [22]; their existence near onset wasrecently proved in [23]. In contrast to the planar case, 3D spotsdo not appear to snake: instead, the L2r -norm along branches oflocalized spots stays bounded.Our goal in this paper is to understand the change in the

behavior of spots and rings when the dimension switches from2 to 3, and to investigate how the 1D, 2D, and 3D structuresdescribed above are related to each other. To elucidate the differentbehaviors of profiles and branches as n varies, we treat n as acontinuous parameter and use numerical continuation techniquesto follow spots and rings from n = 2 upwards to n = 3 anddownwards to n = 1. In particular, the focus of this paper is onnumerical computations, though we will outline some possibleavenues for analysis and rigorous proofs in Section 6. We nowbriefly summarize our results.First, we discovered a second family of planar 2D spots, from

now on referred to as spot B, which seem to exist only for ν > ν∗.In contrast to the spot A structures, spot B solutions have a negativeminimum at r = 0 as shown in Fig. 3. In addition, their amplitudeappears to scale likeµ

38 , so |u(0)| ∼ µ

38 asµ→ 0, and these spots

are therefore not captured by the µ12 -scaling used in the analysis

of spot A solutions in [22].Second, when we follow spots A and B and the two ring

structures down in dimension to n = 1, we find that spotsA and B become, respectively, the symmetric 1D pulses with amaximum and a minimum at r = 0 that we discussed above.The two rings, however, turn into symmetric 1D 2-pulses. Recallthat symmetric 2-pulses exist along a two-parameter family ofisolas, and the mechanism for the production of isolated branchesfrom two connected ring snaking curves turns out to be quitecomplicated. Our numerical continuation results show that eachring curve folds over onto itself several times in a complicatedmanner and then pinches off a number of 2-pulse isolas. On theother hand, we also found 2-pulse isolas that are not connected tothe ring branches upon increasing n but instead shrink to a pointand disappear.Our third result concerns the snaking structure of spots A and B

for 2 ≤ n ≤ 3, which turns out to be equally complicated. Recallthat indefinite snaking was predicted in [22] from the numericalcomputations presented there. It turns out that the computationsin [22] were stopped at a value of the L2r -norm that was not largeenough to reveal the more complicated bifurcation structure thatwe report on here. Indeed, as we follow spot A up on its bifurcationcurve, the curve eventually turns around, and the L2r -norm of thespots begins to decrease again. At this point, the profile of theunderlying pattern transforms from a spot to the profile of one ofthe two rings. Similarly, spot B broadens for awhile, but eventuallytransforms into the second ring and follows the ring bifurcationcurve downwards towards decreasing L2r -norm. In particular, spotsand rings are pairwise connected in parameter space. Above thesetwo connected curves lies a family of stacked isolas of localizedstructures,which also terminates for a large enough value of the L2r -norm. Above these stacked isolas, we found a connected U-shapedsolution curve that seems to extend up to infinite L2r -norm. Both ofthe branches of this curve snake and the associated profiles cyclethrough spot A and B solutions. These branches seem to continueindefinitely towards increasing L2r -norm, but the width of thesnaking regions in the µ-direction decreases. We also gain insightinto how the snaking curves above and below the isolas depend onthe parameter µ and will discuss this further in Section 3.The paper proceeds as follows. Section 2 describes the

numerical techniques used. Section 3 details the bifurcationstructures for n = 2. In Section 4, the changes of the bifurcation

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Fig. 2. Shown are profiles, representative color plots, and bifurcation branches of localized planar spot A solutions in the top row and of the two localized planar ringsolutions in the bottom row. Profiles and color plots correspond to solutions at (µ, ν) = (0.005, 1.6).Source: Reproduced from [22].

Fig. 3. The profiles of spots A and B with (µ, ν) = (0.005, 1.6) are compared in the left panel, while an enlarged plot of spot A is shown separately in the right panel. Notethat spot B resembles an inverted spot A but with a much larger amplitude. The zeros of the two profiles appear to align well for r � 1.

structure are explored for when n is increased from 2 to 3,while Section 5 discusses how these structures change when n isdecreased from 2 to 1. Section 6 presents conclusions and openproblems.

2. Numerical algorithms

For the sake of clarity, we briefly outline the numericalprotocols used in the exploration of the snaking diagrams.To continue localized radial profiles, we numerically solvedboundary-value problems that are based on the first-order system

u′ = u1u′1 = u2u′2 = u3

u′3 = −(1+ µ)u+ νu2− u3 − 2

(n− 1ru1 + u2

)+(n− 1)(n− 3)

r2

(u1r− u2

)−2(n− 1)r

u3,

(2.1)

where the prime denotes differentiation with respect to r , on theinterval (0, L), together with the Neumann boundary conditions

u1(0) = 0, u3(0) = 0, u1(L) = 0, u3(L) = 0 (2.2)

at r = 0, L. Unless stated differently, we used ν = 1.6 in allcomputations.We employed auto07p [24] to continue solutions of(2.1), (2.2) in the parameter µ. Computing the connected snakingbranches of symmetric 1D pulses and planar spots and rings is thenstraightforward. To ensure that the results do not depend on thevalue of L and to prevent boundary effects, we checked for eachcomputation that the computedpatterns are sufficiently small nearthe boundary and, in addition, repeated these computations forsignificantly larger values of L (typically at least doubling L). Inthe rest of this section, we outline the changes that are necessaryto continue asymmetric pulses and to find isolas of symmetric2-pulses, planar spots and planar rings.Finding isolas: There are several kinds of isolas that appear in ourcalculations, and it requires different techniques to find them.When an isola lies above a snaking segment, wemustmove aroundin parameter space in order to find the isola. Fortunately, the

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secondarysnaking

structure

Fig. 4. Shown are the connected bifurcation curve of spot A and one of the ring solutions in panel (i) and the bifurcation branch of spot B and the second ring solution inpanel (ii). In the upper right corner of panel (i), the branch oscillates between three folds aligned approximately at µ ≈ 0.18, 0.19, and 0.21, and we refer to the part ofthe branch that oscillates between the two rightmost folds as the secondary snaking structure. Note that the scales of the vertical L2r -axes in panels (i), (ii) are different: inparticular, the spot A branch reaches a larger value of the L2r -norm. The solution profiles at the points labelled (a)–(d) are shown in Fig. 5.

a c

b d

Fig. 5. Panels (a)–(d) contain the solution profiles of spots and rings at the parameter values labelled (a)–(d) on the branches shown in Fig. 4. As the spot and ring branchesare traversed towards increasing L2r -norm, additional rolls are added at the right tail of the localized profiles. The maximal (minimal) amplitude of spot A (spot B) alwaysoccurs at r = 0 along the branch. For rings, u(0;µ) oscillates between positive and negative values as we move from one leftmost fold to the next on the branch; new rollsare created only at the tail but not near r = 0. We refer the reader to the accompanying movies for branch A and branch B for further details on the behavior of spots andrings.

bifurcation structure provides an easy solution. As n is decreased,the height of the connected snaking curve is found to increase.Thus, we initially continue a solution in n for fixed µ towardsan appropriate smaller value of n, and then fix this value of nand follow the snaking curve in µ towards increasing L2r -norm.Afterwards, we fix the parameter µ and continue in n towards

increasing n until we reach its original value. If we continue highenough in the L2r -norm in the second step, the final solution willlie on an isola, which we can now trace out by continuing in µfor fixed n. When we continue ring structures from dimension 2to dimension 1, isolas of 2-pulses are pinched off the bifurcationcurves. To find these isolas, we continue a large number of

S. McCalla, B. Sandstede / Physica D 239 (2010) 1581–1592 1585

Fig. 6. Panel (i) shows in blue the connected snaking branch of the spot B and ring B solutions from Fig. 4(ii) together with a stack of isolas, plotted in red and alternatelydashed and solid, along which profiles resemble those of spot B and ring B. Panel (ii) contains the spot A curve (in dark cyan) and the spot B branch (in blue) from Fig. 4togetherwith the stacked isolas (in red) frompanel (i). Note that the isolas alignwell with the secondary snaking structure visible near the top of the spot A branch, indicatingthat they pinch off from the spot A branch as n is changed. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

solutions with starting data in a single period of the snakingstructure towards decreasing n.Computing asymmetric 1D pulses: The computation of asymmetricpulses for n = 1, when the system (2.1) is autonomous, requiresan additional phase condition to fix the location of the localizedpattern somewhere inside the interval (0, L). We use the usualintegral phase constraint∫ L

0uold1 (x)(u(x)− u

old(x)) dx = 0, (2.3)

where uold refers to the solution evaluated at a previouscontinuation step. In order to solve the phase constraint, we addthe term cu1 to the last component in (2.1), so that c can be thoughtof as awave speed: theoretically, c should vanish identically duringcontinuation; in practice we found that c is typically of order10−12 and certainly never exceeds 10−6. To find starting data,we break the pitchfork bifurcation through which the asymmetricstates appear, which will allow us to obtain asymmetric pulses bycontinuing the known symmetric pulses. To break the reflectionsymmetry r 7→ −r present for n = 1, we add the term δ sin rto the fourth component of (2.1). Thus, to find asymmetric pulses,we solve (2.1) with the expression (0, 0, 0, cu1 + δ sin r)t addedto its right-hand side, together with (2.2), (2.3). We start witha symmetric 1D pulse away from the pitchfork bifurcation andcontinue initially in δ up to a fixed small value, typically near δ =0.05. Afterwards, we continue in µ for fixed δ until we encountera fold bifurcation. Once we have passed the fold bifurcation, wecontinue in δ for fixed µ until δ becomes zero. The resultingstructure is then the desired asymmetric profile on a ladder branch,which can be validated by continuing again inµ. During the above

Fig. 7. Shown is the first isola (in green) of a second family of stacked isolas thatappears above the spot A branch (plotted in dark cyan). (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

computations, we allow c to vary, although, as explained above, itsvalue will stay close to zero.

3. Localized 2D states

In this section, we focus on the bifurcation diagram of spots andrings for Eq. (1.2) with n = 2. We emphasize that other localizedstructures may exist but these are not considered here. We fix ν =1.6 and note that ν exceeds the critical value ν∗ =

√27/38 ≈ 0.84

belowwhich rings do not exist.We consider exclusively the regimeµ > 0, where u = 0 is stable for (1.1).As already mentioned, the existence of three solution branches

associated with small-amplitude spot A structures and two ring

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patterns was proved in [22] for 0 < µ � 1 in the regions ν > 0for spot A and ν > ν∗ for rings. The numerical evidence presentedin [22] indicated that these branches begin to snake indefinitely asin the one-dimensional case. Indeed, the three solution brancheswere continued in µ a significant distance away from the origin,and convincing snaking was seen with the associated foldsapproaching two vertical asymptotes. As in the one-dimensionalcase, additional localized rolls are added at every other fold alongthe branch near the tail of these localized structures.It turns out, however, that this picture changes drasticallywhen

a large enough number of localized rolls has been added to theunderlying pattern or, in other words, when the L2r -norm hasbecome sufficiently large. In Fig. 4(i), we present computationsthat indicate that the spot A branch and one of the ring branchesare connected in parameter space. In other words, if we continuespot A solutions towards increasing L2r -norm, then the branchwill reach a maximal L2r -norm near which the spot A profilestransform into rings, and the branchwill then continuedownwardstowards decreasing L2r -norm along the ring branch. At themaximalL2-norm, the underlying profile consists of around 20 rolls. Fig. 4(ii)shows the results of a similar computation, where we continuedthe second ring along its bifurcation branch. At the top of thesolution branch shown in Fig. 4(ii), the ring profiles transforminto spot-like profiles, with the maximal amplitudes occurringnear the core at r = 0, and the associated branch descendstowards lower values of the L2r -norm. This second spot (spotB) has not been observed before, and we will comment on itsproperties in more detail below. From now on, we refer to thering structures connected to spots A and B as the ring A andring B patterns, respectively. Representative profiles of spots andrings are shown in Fig. 5 and in the movies provided in thesupplementary materials.As indicated in Fig. 4(i), a secondary snaking structure is visible

near the upper right part of the spot A branch, where the branchoscillates between three distinct limits rather than two: we referto the part of the branch that oscillates between the two rightmostfolds as the secondary snaking structure. Its presence appears tobe related to a stacked family of isolated branches of spot B andring B structures that fill the region in between the secondarysnaking structure thatwe just discussed and the spot B branch fromFig. 4(ii). These isolas along with the spot B branch are shown inFig. 6(i) and together with both spot branches in Fig. 6(ii). Abovethe spot A branch, we found a second family of stacked isolas. Thefirst of these isolas is shown in Fig. 7, while some of the remainingisolas are presented in Fig. 8(i). Along each isola, the solutionprofile changes in an intricate way between the four spot and ringprofiles.The second family of isolas ends at a value of the L2r -norm that

corresponds to profiles that contain around 38 rolls. Above thisvalue, we found another connected solution branch that consistsof two intertwined arms, each of which snakes as indicated inFig. 8. Note that both of these arms oscillate back and forth betweenfolds that align themselves along four distinct curves. As we followeither vertical branch of the snaking curve up, the amplitude of thepattern near the core at r = 0 oscillates up and down between themaximum of spot A and the minimum of spot B. These oscillationscreate new rolls near the core as we move up on the branch,while no new rolls are formed near the right tails of the localizedstructures: this is in sharp contrast to the situation along the lowerspot A and B branches or the situation for n = 1. We referthe reader to the accompanying movie for further details of thebehavior of spots along the upper branch. Note that Fig. 8 alsoindicates that the width of the top snaking branches decreases aswe move up on the branch. Fig. 9 contains log–log plots of theL2r -norm against the difference of theµ values at which folds occurfrom the Maxwell point µ = 0.2 at which the fully nonlinear

1D roll patterns with vanishing Hamiltonian have zero energy.1These results suggest that the width shrinks to zero as the L2r -normgoes to infinity. Note though that we do not knowwhether the topbranch continues upwards indefinitely or whether it, too, ends ata finite value of the L2r -norm.Finally, we comment in more detail on the planar spot B

solutions that we encountered. Recall that their profiles are shownin Fig. 3. These spots differ in variousways from the spot A patternsfound in [22]. First, spot B resembles the Bessel function −J0 nearits core, and its amplitude is therefore negative near r = 0.More importantly, Fig. 10 shows that the supremum norm of spotB appears to scale like µ

38 as µ approaches zero, so spot B is

significantly larger than spot A, whose amplitude scales with µ12 .

Another significant difference is that spot Awas proved to bifurcateat µ = 0 from the trivial background state u = 0 for each fixedν > 0 [22]. In contrast, as shown in Fig. 11, we were not ableto continue spot B below ν = ν∗ =

√27/38 ≈ 0.84, which

is the value at which rings cease to exist. Thus, we believe thatthe bifurcation mechanism that leads to the existence of spotB solutions depends crucially on the far field even though theirprofile envelopes appear to decay to zero monotonically in r .

4. The connection between 2D and 3D branches

The bifurcation diagram for 2 ≤ n ≤ 3 is similar to that forthe 2D case, except that the height and width of the isolas andthe snaking branches decrease significantly as n is increased. Toillustrate these behaviors, we show in Fig. 12 the lower snakingbranches of the two spot–ring pairs and, in Fig. 13, the uppersnaking branch of the two spots for different values of n. We do notshow our computations for the family of stacked isolas betweenthe spot A and spot B branches or for the second family of isolasthat exist between the lower and upper snaking branches shown inFigs. 12 and 13, respectively. These isolas look qualitatively similarto those for n = 2, but they are narrower and there are fewer ofthem as the height of the overall bifurcation diagram decreases.In Section 3, we found that the width of the upper snaking

branch for n = 2 shrinks as we move up along the branch. Fig. 13indicates furthermore that the overall width of these branchesdecreases as n increases. In addition, the two arms of the uppersnaking branch that overlap significantly for n = 2 becomeseparate for n = 3. Even though both of these arms lie to the leftof the Maxwell pointµ = 0.2, fitting the folds using a log–log plotindicates that the spine of these branches aligns itself with a curveof the form µ = 0.2− C‖u‖−1.38

L2rfor some constant C > 0.

Note that we used the L2r -norm in Figs. 12 and 13. It might bemore natural to use the n-dependent norm

‖u‖2L2n:=

∫∞

0|u(r)|2 rn−1 dr, (4.1)

or appropriate scalar multiples thereof, but since using this normdid not reveal any features not already visible in the L2r -norm, wedecided to use the latter.

5. The connection between 1D and 2D branches

In this section, we investigate to which of the localized 1Dpulses the planar states connect when we decrease n. Thus, westart with the planar spots and rings that we found in Section 3and continue them in n towards n = 1. Note that the spot and

1 Eq. (1.1) is variational for n ≥ 1, and (1.2) is Hamiltonian for n = 1; see [13,14]and references therein for details.

S. McCalla, B. Sandstede / Physica D 239 (2010) 1581–1592 1587

Fig. 8. The lower parts of both panels contain the connected snaking branch of spot A and ring A (in dark cyan) from Fig. 4. Above this branch, we found a family of stackedisolas (plotted in green) that include the isola shown in Fig. 7. The stack of isolas extends only up to a value of the L2r -norm at which the profiles consist of approximately 38rolls. Above this value, we found a single connected solution curve (drawn in brown) that consists of two intertwined branches that both snake, seemingly indefinitely. Forclarity, we show only one of the two intertwined branches in the upper part of panel (ii). The solution profiles along the upper snaking curve are explained in the main textand visualized in the accompanying movie. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. The two panels show log–log plots of the two leftmost and two rightmost folds of the high snaking branch shown in Fig. 8, indicating that the snaking branchconverges algebraically to the Maxwell point µ = 0.2 of 1D rolls.

ring branches are connected at the top for some large value of theL2r -norm and that this value increases as n decreases. Furthermore,the associated profiles change from spots to rings only near thetop of their respective branches, and we can therefore distinguishthese two branches easily and continue them separately towardsn = 1.Fig. 14 contains the solution branches at n = 1 that we obtain

when we continue the planar spots and rings in n from twodimensions to one dimension. As expected, the planar spot Aand spot B patterns connect to the symmetric 1D localized roll

structures shown in Fig. 1 that have, respectively, a maximum anda minimum at r = 0. The situation for rings is more complicated,and the limiting set that we obtain at n = 1 when continuing eachof the two ring branches towards decreasing n is actually a set ofisolated branches that correspond to symmetric 1D 2-pulses. Themechanism that leads froman initially connected bifurcation curveat n = 2 to a family of isolas at n = 1 is elucidated in Fig. 15.As we decrease n, each ring curve becomes entangled with itselfand begins to pinch off isolas. In the left panel of Fig. 15, we showthree isolas for n = 1 that are formed from the ring A branch

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Fig. 10. Panel (i), reproduced from [22], indicates that the amplitude of spot A scales asµ12 asµ approaches zero. As shown in panel (ii), the amplitude of spot B appears to

scale approximately like µ0.374 .

Fig. 11. To delineate the existence region of spot B, we continued spot B in the parameter ν for several fixed values ofµ and visualize the resulting solution branches in twodifferent ways: in the left panel, we plot ν versus the squared L2r -norm (the values ofµ decrease from right to left), while the right panel shows logµ versus ν. Note that thesolution branches stay above the critical value ν =

√27/38 and that the L2r -norm of the associated profiles goes to infinity as ν approaches the lower end of each branch.

Fig. 12. The bifurcation curves of spot A and spot B solutions are presented in panels (i) and (ii), respectively, for different values of the dimension parameter n. The insetsshow the branches for n = 3 in more detail.

and correspond to 2-pulses with similar L2r -norm. Observe that the2-pulse branches look quite different in the right panel of Fig. 15,where we plot them in the one-dimensional L2x-norm.We remark that plotting solution branches in the n-dependent

norm from (4.1) actually obfuscates the relation between thebranches for n > 1 and their limits at n = 1. The reason is thatthe solutions change near r = 0 during continuation, and varying

the power of r in the norm during continuation can hide or amplifythe effect of these changes. Thus, solution branches appear betterrepresented by using a fixed norm.Before we address the relation between planar rings and

symmetric 1D 2-pulses in more detail, we briefly summarize someof the results for 2-pulse isolas from [21] as these will be usefulin the forthcoming discussion. As shown in [21], for each pair

S. McCalla, B. Sandstede / Physica D 239 (2010) 1581–1592 1589

Fig. 13. Panel (i) contains the upper snaking branches of spots for n = 2 (in brown), n = 2.3 (in cyan), and n = 3 (in black). Panel (ii) contains the two arms of the snakingbranch for n = 3 to illustrate that they do not overlap. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

Fig. 14. The two curves plotted in cyan diamonds correspond to the limits at n = 1of the lower planar spot A and spot B branches when continued in n. The profilesalong these branches for n = 1 coincidewith the 1Dpulses shown in Fig. 1. The solidfigure-of-eight isolas plotted in red arise when we continue the two ring branchesfrom n = 2 down to n = 1 using the methods outlined in Section 2. The profilesalong each isola are symmetric 1D 2-pulses. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

(s, `) of sufficiently large integers, there are four isolas alongwhich symmetric 1D 2-pulses exist: the parameter ` representsthe number of large-amplitude rolls in each of the two pulses thatmake up the 2-pulse, while s is the number of small-amplitudeoscillations in between the two 1-pulses. Thus, s measures theseparation distance of the two pulses, while ` represents thewidthof each pulse. For each of the four 2-pulses that exist for a givenpair (s, `), let u(r) be its profile and denote by j the quadrant inwhich the pair (u(0), urr(0)) lies; then the integer j ∈ {1, . . . , 4}characterizes the 2-pulse uniquely among the four 2-pulses. Inother words, the jth 2-pulse u(r) has (u(0), urr(0)) in the jthquadrant, where j ∈ {1, . . . , 4}.In Fig. 16, we plot the four isolas that belong to the same

pair (s, `) in the left panel and the associated symmetric 2-pulseprofiles in panels (a)–(d), which correspond respectively to j =1, . . . , 4. The 2-pulses in panels (a), (b) arise when we continuering A towards n = 1, while the 2-pulses in panels (c), (d)come from the planar ring B. This is consistent with the precedingdiscussion as the rolls contained in the two planar rings differ bya phase shift of half their period (in the limit µ → 0, the ringsare given by the Bessel function ±J0). Fig. 17 shows the profilesof 2-pulses for different values of s that we obtained by followingrings towards dimension 1.Note that we cannot be sure whether all isolas of symmetric

2-pulses are obtained from continuing the two planar ring patternsto dimension 1. In fact, it is hard to envisage that each ring branch

folds up on itself infinitely often to generate a countably infinitenumber of isolas for each given ring width `, as this would requirethe ring structures to move away from r = 0 to generate 2-pulsesfor all possible separation distances s. To test whether there areisolas that do not connect to the ring branch, we computed oneof the asymmetric localized 1D pulses that exist for n = 1 alongthe ladder branches shown in Fig. 1. We then placed this pulseat position r = r0 inside the interval [0, r1], where 1 � r0 �r1 are chosen so that the profile is close to zero for r = 0and r = r1. Afterwards, we continued this profile in n usingNeumann boundary conditions. The choice of Neumann conditionsguarantees that we can view the resulting profile as a 2-pulse forn = 1 and a ring solution for n > 1. Furthermore, the choiceof r0 guarantees that the 2-pulse corresponds to a large value ofthe separation parameter s. The resulting bifurcation diagrams forthree different values of n are shown in Fig. 18. Thus, it appearsas if these 2-pulse isolas shrink to a point and disappear withoutconnecting back to one of the ring branches.We did not investigatesystematically which of the 2-pulse isolas shrink to zero in thesame fashion but believe that this happens for all 2-pulses withlarger values of s.

6. Discussion

The numerical explorations that we reported on in this paperindicate how the localized radial patterns that exist in dimensions1 to 3 are related to each other when the dimension parametern is treated as a continuous variable. In particular, we found thatplanar spots connect to symmetric 1D pulses, while planar ringsbecome the symmetric 1D 2-pulses.We also resolved the apparentdiscrepancy between snaking in 2D and non-snaking in 3D thatwas reported in [22]. Our results show that neither planar nor 3Dspots snake; instead, the bifurcation diagram is similar in bothcases and consists of branches that snake over a long but finiteinterval which are followed by stacked isolas for sufficiently largevalues of the L2-norm of the underlying patterns.We also found a new family of localized radial structures of

the planar Swift–Hohenberg equation that we have referred to asspot B solutions. These spots do not seem to obey the expectedµ12 -scaling whenµ goes to zero and instead seem to scale likeµ

38 .

They also appear to exist only for values of ν above the criticalvalue ν∗ =

√27/38. We believe that the analytical techniques

used in [22] to prove the existence of planar rings and spot A statescan also be utilized to investigate the existence of spot B solutions:a preliminary formal analysis that we carried out corroborates thatspot B solutions exist only for ν > ν∗ and predicts an amplitude

1590 S. McCalla, B. Sandstede / Physica D 239 (2010) 1581–1592

Fig. 15. The left panel contains the ring A branch for different values of n plotted in the planar L2r -norm. The curve for n = 1.2 is connected but clearly shows structuresthat will pinch off to become individual isolas for smaller values of n. These isolas continue to form and pinch off as the dimension is decreased further, thus leading to isolasof 2-pulses with a given L2r -norm and an arbitrary separation between the pulses. The right panel shows the ring A branch for n = 1.2 and n = 1.3 but now plotted in theone-dimensional L2x -norm. Note that the curve for n = 1.2 appears to cover an entire family of what will later become separate 2-pulse isolas at n = 1.

Fig. 16. The left panel contains four isolas at n = 1 that are found from the two planar ring branches through continuation in n. The right panel contains the solution profilesat the topmost intersection of these isolas with the line µ = 0.195: the profiles in panels (a), (b) and B come from ring A, while the profiles in panels (c), (d) arise from ringB. Since these profiles were computed with Neumann boundary conditions at r = 0, they can be reflected across r = 0 and therefore correspond to 2-pulses.

S=1 S=2 S=3

Fig. 17. The profiles shown here at n = 1 were found through continuation from rings. Due to the Neumann conditions imposed at r = 0, these solutions correspond tosymmetric 2-pulses with different separation distances represented by the number s of small oscillations near r = 0.

scaling µ38 ; making this formal study rigorous is work in progress

[23].While our numerical computations give a quite detailed picture

of the planar bifurcation diagram, we do not understand thecomplicated structure of different connected solution branchesthat alternate with the families of stacked isolas that theyrevealed. Perhaps the best approach for gaining a theoreticalunderstanding of these diagrams is to carry out a perturbationanalysis of symmetric 1D pulses in the continuous bifurcation

parameter n near dimension 1. Indeed, the mechanism that leadsto the one-dimensional bifurcation structure shown in Fig. 1 iswell understood, and the recent dynamical-systems investigationin [13] may allow us to carry out a perturbation analysis of[∂2r +

ε

r∂r + 1

]2u = −µu+ νu2 − u3, r ∈ R+ (6.1)

in ε := n − 1 near ε = 0. We remark though that sucha perturbation analysis may turn out to be difficult, given thecomplexity of the bifurcation diagrams for n > 1.

S. McCalla, B. Sandstede / Physica D 239 (2010) 1581–1592 1591

lowerfoldlower

fold

upperfold

upperfold

upper folds

lower folds

Fig. 18. When we continue an asymmetric 1D pulse that is centered some distance away from x = 0 in n, we obtain the isolas in panel (i) which shrink and eventuallydisappear. Panel (ii) contains continuation results in (µ, n) of the two upper and lower folds along the isolas. As n increases, the lower folds disappear in a cusp, thus makingthe isola more circular, while the collision of the remaining upper folds corresponds to the point at which the isola disappears.

In the remainder of this section, we briefly outline a formalargument that one could utilize when attempting to understandhow radial spots of (6.1) for 0 < ε � 1 emerge from symmetric1D pulses. First, we remark that the perturbation from ε = 0 is,despite its appearance, a regular perturbation, as the singularity atr = 0 for ε > 0 can be resolved by choosing logarithmic variablesnear r = 0. Thus, we expect each given symmetric 1D pulse topersist for sufficiently small positive ε > 0 provided we stay awayfrom pulses that undergo fold or pitchfork bifurcations. Thus, thekey issue is to understand pulses whose L2-norm is large as wecannot guarantee uniformity of the persistence interval in ε forsuch pulses (nor do we anticipate uniformity given the complexbifurcation structure that we expect to find for n > 1). To discussthe persistence of such solutions, observe that (6.1) is Hamiltonianat ε = 0 with the Hamiltonian given by

H(u) = ururrr −u2rr2+ u2r +

(1+ µ)u2

2−νu3

3+u4

4.

To gain an initial understanding of the behavior of localizedsolutions of (6.1) for ε > 0, it is natural to compute the changeof the Hamiltonian along such a solution. Assuming that u(r; ε)is a family of solutions of (6.1) that is bounded in ε ≥ 0 andsatisfies u(r; ε) → 0 as r → ∞ uniformly in ε, we find via astraightforward calculation that

H(u(∞; ε))− H(u(0; ε)) =∫∞

0

ddrH(u(r; ε))dr

= −2ε∫∞

0[urrr + ur ]

urrdr + O(ε2).

Next, assume that u resembles a spatially periodic roll patternwithwavenumber κ for r ∈ (1, R) with R � 1. Arguing now formallyby assuming that these rolls are of the form cos κr and proceedingas above, we find that

H(u(R; ε))− H(u(0; ε)) ≈ −2ε∫ R

1[urrr + ur ]

urrdr

≈ εκ2(κ2 − 1) log R.Thus, as R increases, the wavenumber κ can stay within a boundedinterval only if ε = 0 or else κ approaches unity like 1/ log R. Onthe other hand, [14, Figure 12] shows that 1D front solutions thatconnect a roll patternwithwavenumber κ to the trivial state u = 0exist only when κ = κc for a certain κc < 1. Hence, if we fix0 < ε � 1, then the wavenumber of rolls inside an extendedlocalized pattern needs to change along the spatial profile fromκ = 1 to κ = κc, where the solution can return to u = 0. Thissuggests that an analysis of spots for dimension near 1 needs toaccount for roll patterns for wavenumbers κ in an entire interval[κc, 1].

Acknowledgements

This research was supported partially by the NSF throughgrant DMS-0907904. We are grateful to the referees for manyconstructive comments that helped to improve the presentationof the paper.

Appendix. Supplementary data

Supplementary data associated with this article can be found,in the online version, at doi:10.1016/j.physd.2010.04.004.

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