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Spiraling solitons: A continuum model for dynamical phyllotaxis of physical systems
Cristiano Nisoli
Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
共Received 30 April 2009; published 11 August 2009兲
A protean topological soliton has recently been shown to emerge in systems of repulsive particles in
cylindrical geometries, whose statics is described by the number-theoretical objects of phyllotaxis. Here, we
present a minimal and local continuum model that can explain many of the features of the phyllotactic soliton,
such as locked speed, screw shift, energy transport, and—for Wigner crystal on a nanotube—charge transport.
The treatment is general and should apply to other spiraling systems. Unlike, e.g., sine-Gordon-like systems,
our soliton can exist between nondegenerate structures and its dynamics extends to the domains it separates;
we also predict pulses, both static and dynamic. Applications include charge transport in Wigner Crystals on
nanotubes or A-toB-DNA transitions.
DOI: 10.1103/PhysRevE.80.026110 PACS number共s兲: 89.90.⫹n, 05.45.Yv, 68.65.⫺k, 87.10.⫺e
I. INTRODUCTION
The topological soliton, the moving domain wall between
degenerate structures, ubiquitously populates systems of dis-
crete symmetry, most notably the Ising model, and appears at
different scales and in many realms of physics 关1兴:inme-
chanical or electrical apparatus 关1–3兴, superconducting Jo-
sephson junctions 关4兴, nonperturbative theory of quantum
tunneling 关5兴, and particle physics 共e.g., Yang-Mills mono-
poles and instantons 关6兴, sigma model lumps, and skyrmions
关7兴兲.
Recently Nisoli et al. 关8兴discovered a novel kind of to-
pological soliton in systems of repulsive particles whose de-
generate statics is dictated by the intricate and fascinating
number-theoretical laws of phyllotaxis. As the phenomena
they describe are purely geometrical in origin, one would
expect this kind of “phyllotactic” soliton to play a role in
many different physical systems at different scales, wherever
repulsive particles in cylindrical geometries are present 关8兴.
Physically, the problem appears to be highly nonlocal, as
energy and momentum are not confined inside the soliton but
flows through it between the rotating domains it separates
共see below兲. Yet, in this paper we show that a minimal, local,
and continuum model for the phyllotactic soliton that sub-
sumes the effect of the kinetic energy of the domains into a
modification of the potential energy in the familiar equiva-
lent Newtonian problem 关5兴can correctly predict its speed,
the transfer of energy between boundaries, the observed shift
in screw angle, and the charge-density variations, and it can
classify its rather complicated zoology.
II. FROM LEVITOV’S TO DYNAMICAL PHYLLOTAXIS
In the early 1990s, while studying vortices in layered su-
perconductors, Levitov 关9兴showed that a system of repulsive
particles on a long cylinder presents a rich degenerate set of
ground states; these can be labeled by number-theoretical
objects, the Farey classes, which arise in phyllotaxis, the
study of dispositions of leaves on a stem, and other self-
organized structures in botany 关10兴. While it is still debated
whether Levitov’s model is the best suited to explain botani-
cal phyllotaxis, it certainly imported the same mathematical
structures into the description of a variety of physical sys-
tems. Already, phyllotactic patterns had been seen or pre-
dicted outside the domain of botany 关11兴, in polypeptide
chains 关12兴, cells of Bénard convection 关13兴, or vortex lat-
tices in superconductors 关9兴.
Dynamical cactus
Recently Nisoli et al. 关8兴showed numerically and experi-
mentally that physical systems whose statics is dictated by
phyllotaxis can access dynamics in ways unknown in botany.
They investigated the statics and the dynamics of a proto-
typical phyllotactic system, the dynamical cactus of Fig. 1,
and found that, while its statics faithfully reproduce phyllo-
taxis, its dynamics reveal new physics beyond pattern forma-
tion. In particular, the set of linear excitations contains clas-
sical rotons and maxons; in the nonlinear regime, both
dynamical simulations and experiments revealed a novel
highly structured family of topological solitons that can
change identity upon collision and posses a very rich dynam-
ics 关8兴. Unlike other topological solitons, the phyllotactic one
separates domains of different dynamics; these domains can
store kinetic energy while rotating around the axis of the
dynamical cactus; energy and momentum flow across the
soliton. Moreover, as those domains are different, the kinks
among them have different shapes, characteristic speeds, and
behaviors under collision. In some simple cases the dynam-
ics of the soliton can be modeled heuristically via conserva-
tion laws, and certain observables—such as its speed—can
be computed in this way. A more general and complete ap-
proach would obviously be desirable.
The dynamical cactus, a simple phyllotactic system de-
picted in Fig. 1, consists of repulsive massive objects holo-
nomically constrained on rings rotating around a fixed axis:
let a=L/Nbe the distance between consecutive rings for a
cylinder of length Land radius Rcontaining Nobjects,
ibe
the angular coordinate of the ith particle, and
i=
i−
i−1 be
the angular shift between consecutive rings. The total poten-
tial energy for unit length is then
PHYSICAL REVIEW E 80, 026110 共2009兲
1539-3755/2009/80共2兲/026110共8兲©2009 The American Physical Society026110-1
V=1
2L兺
i⫽j
U共
i−
j兲,共1兲
where Uis any long-ranged repulsive interaction that makes
the sum extensive and well behaved, such as a dipole-dipole
or a screened Coulomb.
It has been shown that stable structures correspond to spi-
raling lattices 关8兴, where each ring is shifted from the previ-
ous by the same screw angle
i=⍀. The energy of spiraling
structures as a function of ⍀is reported in Fig. 1for different
values of a/R:asa/Rdecreases, more and more commen-
surate spirals become energetically costly, independent of the
repulsive interaction used; for every j, there is a value of a/R
low enough such that any commensurate spiral of screw
angle 2
i/jwith i,jrelatively prime, becomes a local maxi-
mum, as particles facing after jrings become nearest neigh-
bors in the real space. The minima also become more nearly
degenerate as the density increases since, for angles incom-
mensurate to
, each particle is embedded in a nearly uni-
form incommensurately smeared background formed by the
other particles 关8兴.
One-dimensional degenerate systems are entropically un-
stable against domain wall formation 关14兴. Kinks between
stable spirals were found numerically and experimentally
关8兴. Numerically, it was proved that these kinks can propa-
gate along the axis of the cylinder. Experimentally, the dy-
namical cactus was observed expelling a higher-energy do-
main by propagating its kink.
As discussed in Ref. 关8兴such axial motion of a kink be-
tween two domains of different helical angles confronts a
dilemma: helical phase is unwound from one domain at a
different rate than it is wound up by the other. Numerical
simulations show that the moving domain wall solves this
problem by placing adjacent domains into relative rotation
共see supplementary materials in Ref. 关8兴for an animation兲.
Detailed dynamical simulations reveal a complex zoology
of solitons, along with an extraordinarily rich phenomenol-
ogy: kinks of different species merge, decay, change identity
upon collision, and decompose at high temperature into a sea
of constituent lattice particles.
III. CONTINUUM MODEL
A. Lagrangian
To describe domain walls between degenerate structures
of the dynamical cactus, we propose a minimal continuum
model. First, in the continuum limit, we promote the site
index ito a continuum variable: ia→z,
i→
共z兲, the screw
angle is now
i→
共z兲⬅a
z
共later we will just assume a
=1兲. A spiral corresponds to
共z兲=
z
共z兲=const. A kink cor-
responds, in the
,zdiagram, to a transition between two
constant values.
We seek a model that is local and returns the right energy
when the system is in a stable configuration. To express the
cost in bridging two stable domains, we introduce a general-
ized rigidity
. We thus propose the 共linear density of兲La-
grangian as
L=1
2I
˙2−V共
z
兲−
2共
z
2
兲2.共2兲
The first term is the kinetic energy 共I⬅Ia−1, with being the
moment of inertia of a ring兲. The second accounts locally for
the potential energy, so that the Hamiltonian from Eq. 共2兲
returns the right energy for a static 共or uniformly rotating兲
stable spiral like those observed experimentally. The last
term, the lowest-order correction allowed both in the field
and in its derivatives, conveys the rigidity toward spatial
variations of the screw angle and ensures that a kink always
corresponds to a positive-energy excitation.
The Euler-Lagrangian equation of motion follows directly
from Eq. 共2兲as
I
t
2
=
zV⬘共
z
兲−
z
4
共3兲
and corresponds to the conservation of the density angular
momentum I
t
, where the density of current of angular mo-
mentum is V⬘共
z
兲−
z
3
共V⬘is the derivative of Vwith re-
spect to its argument and represents the local torque兲.
In the following we will prove that, whatever is the pre-
cise form of the actual potential V, and the geometric param-
eters of the problem, there exist solitonic solutions of Eq. 共3兲
between local minima of V. We will show how to derive
many qualitative and quantitative results by reducing the
a
01
4
1
2
3
4
1
2
5
1
5
1
3
3
5
4
5
2
3
2Π
101
102
103
104
105
V
Ε
FIG. 1. Top: the dynamical cactus, repulsive particles 共here di-
poles兲, on the surface of a cylinder. ais the distance between con-
secutive rings and ⍀is their angular shift. The dashed line is the
so-called “generative spiral” 关10兴. Bottom: for a dipole-dipole inter-
action, energy Vversus screw angle ⍀共semilogarithmic plot兲for
halving values of the parameter a/Rstarting from 0.5 共lowest line兲,
in arbitrary units
⑀
.
CRISTIANO NISOLI PHYSICAL REVIEW E 80, 026110 共2009兲
026110-2
problem to an equivalent one-dimensional Newtonian equa-
tion, as customary for these kinds of problems 关5兴, yet with a
significant twist.
B. Equivalent Newtonian picture
We seek a uniformly translating solution of Eq. 共3兲
=
共z−vt兲+wt,共4兲
where the constant waccounts for angular rotation invari-
ance, reflects the underlying transverse structure in our sys-
tem 共which is lacking in Klein-Gordon-like systems兲and is
the source of much of the unique physics. By defining s=z
−vt,wefind
t
2
=v2d2
/ds2, which leads to the equivalent
Newtonian equation 关5兴
d2
ds2=−⌽⬘共
兲共5兲
whose equivalent potential ⌽is given by
⌽共
兲=1
2v2I共
−
˜
兲2−V共
兲.共6兲
˜
is a constant that is sometimes useful to redefine as
˜
⬅
/共Iv2兲in terms of
, which has the dimension of a torque.
We will find that in studying static solitons it is easier to use
, while for dynamical ones
˜
is preferable.
In the equivalent Newtonian picture a soliton corresponds
to a trajectory
共s兲of a point particle that starts with zero
kinetic energy on the top of a maximum of ⌽共
兲and, in an
infinite amount of “time” s, reaches the neighboring maxi-
mum of equal height as detailed in Fig. 2. A similar treatment
applies to Klein-Gordon-like solitons and tunneling prob-
lems 关5,14兴. Yet the presence of an extra quadratic term
translates in completely different physics and accounts for
the dynamics of the domains.
C. Shift from stability
Because of the quadratic term 1
2v2I共
−
˜
兲2in the expres-
sion of ⌽, and unlike the Klein-Gordon-like soliton, our soli-
ton in general does not connect
1,
2, which are local
minima of V; rather it goes from
¯
1=
1+
␦
1to
¯
2=
2+
␦
2,
bridging spirals whose screw angle is slightly shifted away
from stability, with the shift increasing with the speed of the
kink 共Fig. 2兲.
We find then a most interesting result: a moving soliton
cannot connect two stable structures. At most, it can bridge a
stable spiral
1with one shifted away from equilibrium
¯
2
=
2+
␦
2, if we choose
˜
=
1, but only if V共
1兲⬍V共
2兲.
These shifts away from stability were observed in dy-
namical simulations 关integrating the full equation of motion
for the discrete system of particles interacting via Eq. 共1兲兴 in
the form of precursor waves propagating at the speed of
sound in front of the soliton as in Fig. 3. They correspond to
the physically intuitive fact that, in order to move, the kink
needs a torque to propel it.
Clearly, static solitons 共v=0兲can bridge stable spirals if
these are degenerate, i.e., V共
1兲=V共
2兲. On the other hand,
if two domains are degenerate, and are connected by a mov-
ing soliton, they both need to be shifted.
D. Boundary conditions
The conditions of existence of a soliton between the
asymptotic domains
¯
1,
¯
2are
⌽⬘共
¯
1兲=⌽⬘共
¯
2兲=0,
共7兲
⌽共
¯
1兲=⌽共
¯
2兲.
Now, from Eq. 共4兲, we obtain the expression of the angular
velocity for a soliton of speed v,
˙共s兲=−v
共s兲+w,共8兲
which implies that, in general, as the soliton travels along the
axis, one or both of its two domains must be set into rotation
and at different angular velocities. As mentioned, this rota-
tion of the domains is indeed observed in dynamical simula-
tions 关8兴as the mechanism with which the soliton unwinds
and rewinds spirals of different gaining angles.
Ω
1Ω1∆1Ω
2Ω2∆2
Ω
1
Ω
2
VΩ
Ω
1
2Iv2ΩΩ
2
Kink: ssΩ
FIG. 2. 共Color online兲The equivalent Newtonian picture, stan-
dard for Klein-Gordon-like solitons or for tunneling problems 关5兴:a
soliton is a solution of Eq. 共5兲that corresponds to a trajectory
共s兲
关red solid line 共light gray兲兴 of a point particle that starts and ends in
two equally valued maxima 关horizontal dotted red lines 共light gray兲兴
of the equivalent potential energy ⌽共
兲=1
2v2I共
−
˜
兲2−V共
兲共black
solid line兲. As the kinetic energy must be zero on those maxima, it
takes the Newtonian trajectory an infinite amount of time s共i.e.,
“time” in the Newtonian picture, space in the soliton picture兲to
reach the “top of the hill” with asymptotic speed equal to zero 关5兴.
Unlike the Klein-Gordon-like case, the presence of the quadratic
term 1
2v2I共
−
˜
兲2共dashed line兲, which accounts for the kinetic en-
ergy of the domains, slightly shifts the asymptotic value of the
kinks 共red dotted lines兲away from stable static structures
1,
2
共black dotted lines兲, the minima of the real potential energy V共
兲
共dashed line兲. Instead, the soliton connects spirals of angle
¯
1
=
1+
␦
1,
¯
2=
2+
␦
2, which in general are not stable structures of
V. The extra quadratic term also allows solitons among nondegen-
erate configurations and brings the speed of the soliton vinto the
picture. 关Everything is in arbitrary units. The soliton is obtained by
numerical integration of the Eq. 共5兲using the energy profile por-
trayed in the figure.兴
SPIRALING SOLITONS: A CONTINUUM MODEL FOR…PHYSICAL REVIEW E 80, 026110 共2009兲
026110-3
Unlike Klein-Gordon-like one-dimensional topological
solitons, which separate essentially equivalent static domains
and can travel at any subsonic speed 关14兴, we see that phyl-
lotactic domain walls instead separate regions of different
dynamics: energy and angular momentum flow through the
topological soliton as it moves, rather than being concen-
trated in it; as mentioned above, this was used to show heu-
ristically that its speed vis tightly controlled by energy-
momentum conservation, phase matching at the interface,
and boundary conditions 关8兴. We will show how a more pre-
cise version of those heuristic formulas can be deduced
within the framework of our continuum model and in a more
general fashion.
The angular speed of rotation of the domains depends on
the parameter w, which along with
˜
is a constant to be
determined. On the other hand, we can show that energy
conservation constrains wto
˜
. In fact, when the two do-
mains are shifted from equilibrium, and thus subjected to a
torque, energy flows through the boundaries of the system as
the domains rotate. By imposing no energy accumulation in
the kink, we fix win Eq. 共4兲in the following way: the power
entering the system at the asymptotic boundaries can easily
be deduced via the Noether theorem to be
j⫾⬁ = − lim
s→⫾⬁
V⬘„
共s兲…
˙共s兲,共9兲
which is, not surprisingly, the torque times the angular speed.
By requiring j+⬁=j−⬁an equation for w,
˜
is found as
w=v共
¯
1+
¯
2−
˜
兲,共10兲
which will come in hand in many practical cases.
IV. ZOOLOGY
A. Static kinks
Let us explore static kinks first. Equations 共7兲reduce to
=V⬘共
¯
1兲=V⬘共
¯
2兲,
共11兲
共
¯
1−
¯
2兲=V共
¯
1兲−V共
¯
2兲.
The first of the two Eqs. 共11兲tell us that
is a torque applied
at the asymptotic boundaries of the system, while the second
shows that no torque is necessary if the topological soliton
connects degenerate structures, as we anticipated. Unlike to-
pological solitons of the Klein-Gordon class, here static
kinks are allowed between nondegenerate domains through
an applied torque that shifts the two structures out of the
minima of V.
B. Solitons moving between nondegenerate domains
Unlike the Klein-Gordon-like Lagrangian, our Lagrangian
is not invariant under Poincaré group. Our traveling soliton
is not just a boost of the static one, as it is clear from Fig. 2.
In particular the shifts of the domains from stability increase
with the speed v. By expanding V共
兲around its local
minima and using Eq. 共6兲, one finds easily that small shifts
are proportional to the square of the speed of the soliton, or
␦
1,
␦
2⬀v2.
Let us consider a particular case of practical importance.
Kinks between nondegenerate structures can sometimes be
found in the experimental settings, because of static friction,
after having annealed mechanically the cactus with free
boundaries. If a small perturbation is given, enough to over-
come that friction, the dynamical cactus will expel the
1 50 100 150 200
S
p
ace
1.43
1.39
Ω2
Ω
2
0 50 100 150 200
Space
5.1
0
2.9 Θ
2
Θ
2
Ω1Ω2Ω3
Ω
4
.75
4
.45
VmJ
0
50
100
150
200
Space
12
6
3
9
0
Ts
0
50
100
150
Space
FIG. 3. 共Color online兲Dynamical simulation for the collision or
conversion of two solitons, emitted from free boundaries of a dy-
namical cactus. Top panel: screw angle vs space 共in number of
rings兲and time 共in seconds兲; note the elastic wave preceding the
soliton. Second panel: energy per particle 共mJ兲vs screw angle 共rad兲
for our system 共
1=1.79 rad,
2=1.43 rad,
3=2.40 rad兲; the two
kinks—before collision—connect a high 共inner:
1兲to a low 共outer:
2兲energy domain; after collision a low 共inner:
3兲to a high 共outer:
2兲. Third panel: angular speed 共s−1兲vs space at a given time: the
speed of the soliton can be extracted as v=⌬
˙/⌬
=22.1 s−1 共pre-
dicted: 23.4 s−1兲. Fourth panel: the precursor; plot of the screw
angle vs space at different times while the soliton and its preceding
wave advance. The amplitude
␦
1of the precursor is predicted via
Eq. 共13兲as
␦
1=
1−
¯
1=−0.043 rad, in excellent agreement with
numerical observations. The simulation uses the experimental den-
sity and the magnetic interaction as in Ref. 关8兴.共See supplementary
materials in Ref. 关8兴for an animation.兲
CRISTIANO NISOLI PHYSICAL REVIEW E 80, 026110 共2009兲
026110-4
higher-energy domains by propagating the kink along the
axis 关8兴. Figure 3reports a dynamical simulation of this case
obtained by integrating the full equation of motion for the
discrete system of particles whose interaction is given in Eq.
共1兲.
Let us show how our continuum model can reproduce a
soliton between nondegenerate minima and provide quanti-
tative predictions. Free boundaries imply j⫾⬁ = 0: since v
⫽0, V⬘cannot be zero on both boundaries, as we saw before,
nor can the angular speed of rotation of the domains be. The
only possible free boundaries solution is that one has V⬘=0
on one side and
t
=0 on the other. Hence the soliton must
connect, say,
¯
2=
2to
¯
1=
1+
␦
1, with V共
2兲⬍V共
1兲,
which fixes
˜
=
2=
¯
2. Equations 共7兲then fix the speed of
the soliton as
vk
2=2⌬V
¯
I⌬
¯
2共12兲
with ⌬V
¯
=V共
¯
1兲−V共
¯
2兲and ⌬
¯
=
¯
2−
¯
1共in this particular
case,
2=
¯
2兲. Equation 共12兲corrects an earlier formula
found heuristically via energy conservation, which neglected
the shifts 关8兴. From the angular velocities of rotation of the
domains in the dynamical simulation 共Fig. 3, third panel兲,a
speed of 22.1 s−1 can be extracted and compared with the
value 23.4 s−1 predicted by Eq. 共12兲: the small discrepancy
is likely due to energy dissipation into phonons—a known
effect for solitons in discrete systems.
From Eq. 共10兲,wefindw=v
¯
2which together with Eq.
共8兲implies that the region of lower energy rotates uniformly,
while that of higher energy, which is shifted, remains still:
since there is no energy flow through the boundaries, the
kink transforms the potential-energy difference between the
two domains into the kinetic energy of rotation—and vice
versa, depending on its direction of propagation. 共Only
propagation toward lower potential energies was observed
experimentally 关8兴.兲
By expanding Varound
1,V⯝1
2Ic1
2共
−
1兲2, where c1is
the speed of sound in the spiral of angle
1, we obtain an
approximate expression for the shift of the screw angle in the
region of higher energy
␦
1
⌬
=
v2/c1
2
1−v2/c1
2共13兲
that clearly holds for vⰆc1. As the system is prepared in a
stable nonshifted configuration of angle
1, a precursor in
front of the soliton propagates to accommodate the shift
␦
1,
as it was seen yet not understood in the dynamical simulation
reported in Fig. 3关8兴. Equation 共13兲applied to the experi-
mentally measured V共
兲employed in the simulation predicts
a shift
␦
1=−0.043 rad, which fits the results of the simula-
tion well 共Fig. 3, last panel兲.
We will not deal here with collisions between solitons.
Yet, even without the knowledge of the precise form of V,a
few considerations can be made on asymptotic states of the
collision depicted in Fig. 3: the domains beyond the ap-
proaching pair of kink-antikink rotate with opposite angular
velocity
˙=v⌬
, with vgiven by Eq. 共12兲. After the colli-
sion vchanges sign, and so must ⌬
. The infinitely long
domains cannot invert angular velocity instantaneously: the
emerging asymptotic configuration must be that of a pair of
different kink-antikinks, connecting the old asymptotic do-
main with a different nearest neighbor, hence the collision
metamorphosis already discussed in Ref. 关8兴.
C. Solitons moving between degenerate domains
To investigate the degenerate case, typical of dynamical
phyllotaxis of high R/aratios 共Fig. 1兲, we consider for sim-
plicity the potential Vof Fig. 4; degeneracy requires now that
˜
=共
1+
2兲/2, and if the potential is symmetric as in the
figure, both boundaries are equally shifted in opposite direc-
tions and subjected to a torque of opposite sign and equal
intensity 关from Eqs. 共7兲兴
兩
兩=I
2
v2共
¯
2−
¯
1兲.共14兲
Exactly, as in the case of the boosted Klein-Gordon-like
one, our symmetric soliton becomes shorter as vincreases
共Fig. 2兲, although via a completely different mechanism. In
the Klein-Gordon-like soliton, shortening is a consequence
of relativistic contraction of the Poincaré group. Instead, in
our case, vraises the difference between the maxima of ⌽
and the minima among them. That means higher ‘‘kinetic
energy’’ in the “valley” of ⌽for the equivalent trajectory,
thus higher values of d
/ds, and hence a shorter soliton.
Equations 共8兲–共10兲tell us that the two domains rotate in
opposite direction with angular velocity of the same intensity
兩
˙兩=v共
¯
2−
¯
1兲.共15兲
In practice, the system acts like a mechanical inverter of
rotation that transmits a power
˙=Iv3⌬
¯
2/2 along the tube.
For any given velocity, many solitons between degenerate
Ω
1Ω
2
Ω
3
Ω
4
Ω
1
Ω
2
Ω
3
Ω
4
VΩ
Ω
1
2Iv2ΩΩ
2Large Kink
Small Kink
FIG. 4. 共Color online兲Equivalent Newtonian diagram 共see cap-
tion of Fig. 2for explanation兲for symmetric solitons between de-
generate spirals. As the quadratic term raises all the maxima of
−V共
兲, solitons with the same speed but connecting different do-
mains are possible under different applied torques. The smallest
kink 关solid red line 共light gray兲兴 and a larger kink 关solid blue line
共dark gray兲兴 are shown.
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structures are allowed, corresponding to different applied
torques, as in Fig. 4.
D. Pulses: Propagating and frozen in
The equation of motion also predicts pulses, both dy-
namic and static 共frozen in兲. The top panel of Fig. 5shows
the pulse soliton in the equivalent Newtonian picture as a
trajectory falling from a local maximum of ⌽, say
1, and
coming back to it without reaching the neighboring
¯
2be-
cause of a higher potential barrier.
Clearly, pulses possess less inertia than kinks since only
the region occupied by the pulse rotates during propagation.
Theory suggests that these pulses should be able to propagate
with free boundaries and that no applied torque is
necessary—although solutions corresponding to different
boundary conditions, and hence applied torque, can be found
as well by choosing
˜
⫽
1.
Let us consider pulses in stable structures and thus
˜
=
1. When v⬎c,共the speed of sound in the spiral of angle
1兲then
1becomes a minimum for ⌽and we obtain a
solution that oscillates around
1:the speed of sound is the
upper limit for the velocity of the pulse soliton.Asvde-
creases three things can happen.
If V共
1兲⬍V共
2兲, then pulses exist for vk⬍v⬍c, with vk
given in Eq. 共12兲.Asvapproaches from above the critical
value vk, the pulse asymptotically stretches to a pair of kink
and antikink between domains
1and
¯
2, placed at infinite
distance from each other, traveling at the same speed in the
same direction. If V共
1兲=V共
2兲, the situation is the same as
above, yet the critical value for asymptotic stretch into kink
and antikink is v=0. If V共
1兲⬎V共
2兲, we have a pulse soli-
ton for every value of v⬍c.Asvgoes to zero, the pulse
freezes into a static one. Hence, in the absence of torque at
the boundaries, our theory predicts that static pulses can ex-
ist in every structure, except in the lowest-energy one.
These frozen-in kinks that were indeed observed experi-
mentally 共but not understood兲in higher-energy spirals per-
haps arise from a kink-antikink symmetric collision 关15兴at
the interplay with friction. Figure 5, bottom, shows in solid
blue 共dark gray兲line the data from the experimental appara-
tus, along with our frozen-in soliton calculated within our
continuum model, and the dotted red 共light gray兲line shows
the data for the energy ⌽=V共solid black line兲empirically
measured in the experimental apparatus.
V. AXIALLY UNCONSTRAINED CASE
We have since now considered only an axially constrained
case where the repulsive particles were allowed to rotate
around an angular coordinate, but not to translate along the
axis. Nonlocal optimization via a structural genetic algorithm
has shown that the more general case of axially uncon-
strained particles on a cylindrical surface reproduces the
same fundamental statics of the axially constrained one, i.e.,
the same spiraling lattices 关15兴. Numerical simulations have
also shown kinks propagating among the spirals 关15兴. The
most significant difference with the axially constrained case
is a drop in density in the region of the kink to locally relieve
the mismatch. If these particles were charged, the drop in
density would endow the soliton with a net charge: the phyl-
lotactic soliton could function as a charge carrier. In fact,
Wigner crystals of electrons on large semiconducting tubes
are candidate environments for the phyllotactic soliton at
nanoscale as discussed in Ref. 关8兴. A crystal pinned by the
corrugation potential and/or impurities will not slide along
the tube under a weak enough external field.
A simple extension of the continuum model can easily
incorporate variations in the linear density and allow inves-
tigation of the possibility of charge transport by the phyllo-
tactic soliton. We rescale the interaction as
V→
冉
o
冊
2
V,共16兲
where is the linear density in the presence of axial dis-
placement, whereas o=N/L. Let
=dz/abe the relative
axial displacement; then, if
z
Ⰶ1 we have /o=1/
共1+
z
兲. The new Lagrangian reads
Ω2Ω32ΠΩ
2
ΩTheor. Exp.
Ω2
Ω
1Ω1Ωmax
VΩ
Ω
1
2Iv2ΩΩ
2
Pulse ssΩ
FIG. 5. 共Color online兲Equivalent Newtonian diagram 共see cap-
tion of Fig. 2for explanation兲for pulses. Top: a traveling pulse
among degenerate structures is possible because the nonzero speed
vraises the maximum of ⌽. Bottom: theoretical 关red dashed 共light
gray兲兴 and experimental 关blue solid 共dark gray兲兴 frozen-in pulse on
the measured energy curve of the dynamical cactus 关8兴.
CRISTIANO NISOLI PHYSICAL REVIEW E 80, 026110 共2009兲
026110-6
L=1
2I
˙2+1
2IR−2
˙2−V共
z
兲
共1+
z
兲2−
2共
z
2
兲2.共17兲
To gain insight regarding the shape of the soliton, we restrict
ourselves to the simpler static case with no applied torque.
By variation in the Lagrangian in Eq. 共17兲, one obtains
d2
dz2=− V⬘共
兲
1+
z
,
共18兲
z
冋
2V共
兲
共1+
z
兲3
册
=0,
which, along with the normalization condition for , returns
the density of a static soliton 共kink or pulse兲as a function of
共z兲,
共z兲
o
=
冉
Vo
V关
¯
共z兲兴
冊
2/3
.共19兲
Here Vois the asymptotic value of Vat the boundaries. The
static kink
¯
共z兲can be found as the solution of the equivalent
Newtonian equation 共5兲with potential
⌽共
兲=−Vo
2/3V共
兲1/3.共20兲
Now, V1/3and Vhave the same set of local minima, and the
same ordering among their values: the extension to axial
displacements does not alter any of the conditions for the
existence of kink and pulse solitons, at least for the static
case.
Equation 共19兲implies, as expected, a drop in density simi-
lar to a dark soliton in the region of the kink, and thus, for a
crystal of charges, a net charge. In particular, the higher the
potential barrier between the two domains, the lower the
density at the center of the kink. Also, from Eq. 共20兲we can
see that allowing an extra degree of freedom makes the kink
longer. In practice the kink can take advantage of density
reduction to relax the potential barrier between the two do-
mains and can thus be longer. All of this is detailed in Fig. 6,
where the solitons for both the axially constrained and un-
constrained cases 共along its variation in density兲are reported
for the same interaction among particles. Analogous consid-
erations apply to static pulses. The case of the axially uncon-
strained traveling soliton is more complicated and will not be
treated here.
VI. CONCLUSION
We have introduced a minimal, local, and continuum
model for the phyllotactic soliton and showed that its predic-
tions are in excellent agreement with numerical data, that it
provides a tool to calculate otherwise elusive quantities, such
as charge transport, energy-momentum flow, speed of the
soliton, and angular velocities, and that it could be used in
the future to develop its thermodynamics.
Our continuum model should be applicable to many dif-
ferent physical systems. As discussed in Ref. 关8兴, dynamical
phyllotaxis is purely geometrical in origin, and thus the rich
phenomenology of the phyllotactic soliton could appear
across nearly every field of physics. Indeed phyllotactic do-
main walls have already been seen, but not recognized, in
simulations 关16兴of cooled ion beams 关17兴where the system
self-organizes into concentric cylindrical shells. Colloidal
particles on a cylindrical substrate provide a highly damped
version 关18兴, and polystyrene particles in air 共as used to in-
vestigate 关19兴the Kosterlitz, Thouless, Halperin, Nelson, and
Young theory of two-dimensional melting 关20兴兲 have reason-
ably low damping and long-range interaction.
Yet it should be noticed that our model has a range of
application much wider than pure dynamical phyllotaxis.
Domain walls in any spiraling system whose energy depends
on the screw angle of the spirals, which manifest different
stable spiraling structures and where a generalized rigidity
can be reasonably introduced to describe transitions between
different spirals, should be described by such formalism.
Kinks in spiraling proteins and in particular transitions be-
tween A- and B-DNA might be approachable this way.
ACKNOWLEDGMENTS
The author would like to thank Vincent Crespi and Paul
Lammert 共Penn State University, University Park兲for useful
discussions and Ryan Kalas and Nicole Jeffery 共Los Alamos
National Laboratory兲for helping with the manuscript. This
work was carried out under the auspices of the National
Nuclear Security Administration of the U.S. Department of
Energy at Los Alamos National Laboratory under Contract
No. DE-AC52-06NA25396.
Ω1Ω2
VΩ
Ω
FIG. 6. 共Color online兲Equivalent Newtonian diagram for both
axially constrained and unconstrained soliton. In blue dashed 共dark
gray兲curve, the axially constrained soliton among the stable struc-
tures
1,
2is shown. The two red solid curves 共light gray兲depict
the soliton in the axially unconstrained case, which is longer, and its
corresponding drop in density .
SPIRALING SOLITONS: A CONTINUUM MODEL FOR…PHYSICAL REVIEW E 80, 026110 共2009兲
026110-7
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