Cent. Eur. J. Phys. • 11(12) • 2013 • 1638-1644
DOI: 10.2478/s11534-013-0281-6
Central European Journal of Physics
Stability bands for moving solitons in uniform and
inhomogeneous damped ac-driven circular Toda
lattices
Research Article
Lautaro Vergara1∗ , Boris A. Malomed2†
1 Departamento de Física, Universidad de Santiago de Chile,
Av. Ecuador 3493, 9170124 Santiago, Chile
2 Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University,
69978 Tel Aviv, Israel
Received 28 March 2013; accepted 16 July 2013
Abstract:
By means of systematic simulations, we study the motion of discrete solitons in weakly dissipative Toda
lattices (TLs) with periodic boundary conditions, resonantly driven by a spatially staggered time-periodic
(ac) force. A complex set of alternating stability bands and instability gaps, including scattered isolated
stability points, is revealed in the parametric plane of the soliton’s velocity and forcing amplitude for a given
size of the circular lattice. The analysis is also reported for the circular TL including a single light- or heavymass defect. The stability chart as a whole shrinks and eventually disappears with the increase of the
lattice’s size and strength of the mass defect. Qualitative explanations to these findings are proposed. We
also report the dependence of the stability area on the initial position of the soliton, finding that the area is
largest for some intersite position. For a pair of solitons traveling in opposite directions, there exist regimes
where both solitons survive periodic collisions in small-size lattices.
PACS (2008): 05.45.-a; 05.45.Yv; 05.10.-a
Keywords:
soliton physics
© Versita sp. z o.o.
1.
Introduction
The relevance of the study of the dynamics of nonlinear
lattices has been shown in modeling many physical systems, such as crystals, polymer molecules, arrays of optical waveguides, Bose-Einstein condensates trapped in
∗
†
E-mail: [email protected] (Corresponding author)
E-mail: [email protected]
deep optical lattices, arrays of coupled Josephson junctions, etc. (see, e.g., recent reviews [1–3] and a topical
collection of articles [4]). In particular, highly localized excitations of nonlinear lattices, i.e., discrete solitons, have
drawn especial attention, being present in all the abovementioned systems.
The homogeneous Toda lattice (TL) [5, 6], being the first
system where exact discrete solitons were found, is one
of universal dynamical-lattice models. It is based on the
chain of equations of motion, mn ün = Fn (rn ) − Fn+1 (rn+1 ),
where the overdot stands for the time derivative, mn and
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Lautaro Vergara, Boris A. Malomed
un are the mass and displacement of the n-th particle, and
the interaction forces are
Fn (rn ) = an [exp (−bn rn ) − 1] , rn ≡ un − un−1 ,
(1)
with constants an and bn pertaining to the n-th nonlinear spring. The inhomogeneous TL, with the constants
depending on the lattice coordinate, n, is not integrable,
but, as well as a number of allied lattice models, it supports solitons (persistently propagating solitary waves) in
numerical simulations [7–17].
Realistic nearly-uniform models of the TL type may include local imperfections, i.e., particles with a different
mass, and defects featuring local variations of the spring
constants. The interaction of solitons with impurities in
the TL were first studied in Ref. [11, 12], and later considered in a number of works [13–17]. Other types of defects
in the TL were studied too, including surface impurities
[18, 19] and interfaces between lattices with different parameters [20–22]. In addition, dissipation (damping) in
real media causes the decay of excitations. It was shown
that stable progressive motion of a soliton in the TL with
friction can be supported by a staggered [which means
it contains factor (−1)n ] time-periodic (ac) driving force,
provided that the amplitude of the drive exceeds a certain threshold value, which, in the first approximation, is
proportional to the dissipative constant [23–25] [similar
results for the damped TL, driven by the staggered multiplicative (parametric) ac drive, was reported in ref. [26]].
This effect was demonstrated experimentally in an electric
LC transmission line [27, 28]. Alternatively, the dissipation without the drive supports the stable propagation of
shock waves in the TL [29].
The aim of this work is to identify an area in the plane
of the soliton’s velocity and amplitude ε of the staggered
ac drive, which admits the stable circulation of solitons
in the TL with periodic boundary conditions. Previous
works [23–26] demonstrated the possibility of the stable
ac-driven motion of TL solitons, but did not report a structure of the stability area. We perform the analysis for the
uniform and inhomogeneous TLs, with a single mass defect
in the latter case, set at n = 0:
mn = 1 + (m∗ − 1) δn,0 .
(2)
Both cases of the heavy (m∗ > 1) and light (m∗ < 1)
defects will be considered, for the inhomogeneous lattice.
In the uniform one, with mn ≡ m, an obvious rescaling
makes it possible to fix m ≡ 1 in Eq. (3), which we assume
below for the uniform part of the TL.
Using the periodic boundary conditions, it is convenient to
rewrite the TL equations of motion in terms of the relative
displacements, see Eq. (1):
r̈n = 2 Fn − Fn+1 − Fn−1 − α ṙn
−1
+ε(−1)n m−1
n + mn−1 cos(ωt),
(3)
where α is the friction coefficient. For simulations of these
equations, a modified version of the fortran code developed by E. Hairer and G. Wanner was employed, with
an explicit Runge-Kutta method of order 8, as defined by
Dormand and Prince [30].
The rest of the paper is organized as follows. The results for the dynamics of a single soliton in uniform and
inhomogeneous lattices are reported in Sections 2 and 3,
respectively. In both cases, the stability area features a
complex structure of bands separated by gaps, and includes sets of scattered isolated points. Overall contraction and eventual disappearance of the structure occurs
with the increase of the size of the lattice and/or strength
of the mass defect. Qualitative explanations to these findings are proposed in Sections 2 and 3 too. In section 4
we consider the dynamics of the soliton in a uniform lattice, for the case when it was initially set between lattice
sites. A particular value of the initial location is found
which gives rise to a largest stability area in the parameter space. Section 5 deals with a pair of counterpropagating solitons. We find an area in the parameter space
where both solitons survive periodic collisions. Section 6
concludes the work.
2. Numerical results for the uniform
lattice
The study was performed for the circular TL with four
different lengths (numbers of the sites): N = 30, N = 100,
N = 200, and N = 400. The initial condition was taken
as per the exact soliton solution for the ideal uniform TL
[5],
#
sinh2 (k0 )
,
= − ln 1 +
cosh2 (k0 (n − n0 ))
"
rn(0)
(4)
where n0 is the coordinate of the soliton’s center. The
amplitude of the soliton, along with its velocity,
v = k0−1 sinh (k0 ) ≥ 1
(5)
(we chose the velocity directed to the right), are determined by free parameter k0 . Note that the velocity is
a monotonously growing function of k0 . The driving frequency used in Eq. (3) corresponds to the first resonance
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Stability bands for moving solitons in uniform and inhomogeneous damped ac-driven circular Toda lattices
with the moving solitons [23, 24], ω = π v (this means that
the soliton passes the cell of the staggered driving-force
field, ∆n = 2, during time ∆t = 2/v, which is equal to
the period of the ac drive, 2π/ω). For the most part of
the present work, the friction coefficient was fixed to be
α = 0.002, which is the same value at which the stabilization of the solitons by the ac drive was demonstrated
in ref. [25].
We intend to compare the results on the stability of solitons for different lattice sizes. Having in mind that the
lattice spacing is set to be 1 in Eqs. (1) and (3), the number of lattice sites N is the same as the lattice length of
the lattice. We have registered the soliton as a stable
one if it would keep moving persistently for a reasonably
long time, completing a large number of round trips in the
circular TL. At extremely long simulation times, the soliton may start decaying due to accummulation if numerical
errors.
The results were generated by changing the forcing amplitude in steps of ∆ε = 0.0001, and parameter k0 of input
(4) in steps of ∆k0 = 0.1 (this is why in the figures that follow below, there appear stripes separated by ∆k0 = 0.1).
Simultaneously with the variation of k0 , the driving frequency was varied according to the above-mentioned relation, ω = πv, to keep ω in resonance with velocity (5).
In the figures following below, points represent parameter values at which the soliton is found to be stable in
the direct simulations. We stress that the figures do not
display a manifold of solutions coexisting in one and the
same system. Instead, the figures show a manifold of the
systems, with different values of k0 corresponding to different values of the driving frequency, as explained above.
Figure 1 features well-defined stability regions in the
(k0 ,ε) plane. This structure, that was not reported in earlier works [23, 24, 27, 28], which only demonstrated the
very possibility of the stabilization, is one of the main results of the present work. In particular, the bottom boundary of the stability area clearly implies the presence of the
minimum (threshold) value of the forcing strength, ε, above
which stable solitons exist. An essentially novel feature
revealed by the plots is the alternation of the stability islands with instability gaps, at fixed velocity (fixed k0 ) and
increasing ε. These stability bands are clearly observed
for lower values of the forcing acting in larger lattices,
but they are present too in smaller lattices, although for
larger values of ε. For instance, in the NL with N = 30
the next stability region appears at ε > 0.16 (see also
Figure 7).
An accurate explanation of this feature is a challenging
problem; nevertheless, it is plausible that the increase of
the driving amplitude leads to creation of intrinsic modes
in the traveling soliton (at ε = α = 0, there are no intrin-
Figure 1.
Stability regions for the traveling soliton, starting at initial
position n0 = −9, in parameter space (k0 , ε) for the circular
uniform TLs of length N = 30 (a), N = 100 (b), N = 200
(c), and N = 400 (d), showing the reduction of the stability
zones for larger lattices. The friction coefficient is α =
0.002.
sic modes, as such modes do not exist in solitons produced
by integrable systems). Then, the instability gaps may be
explained by resonances between the driving frequencies
and the plausible intrinsic modes. Further increase of ε
should change the eigenfrequency of the mode and thus
pull it out of the resonance, allowing the stabilization of
the soliton.
Figure 1 also demonstrates gradual shrinkage of the stability area with the increase of the length of the circular lattice (we stress, however, that the stability chart
does not shrink with the increase of the simulation time,
hence this is not a numerical artifact). This finding may
be explained by the effect of “sweeping", similar to that
considered in other models [31, 32]: a soliton circulating
in the system at a sufficiently high frequency suppresses
small perturbations throughout the lattice , while in a very
large lattice the perturbations have enough time to grow
between passages of the solitons, thus destroying the dynamical regime. The conjecture concerning the sweeping
mechanism of the stabilization was verified in the following way: at diametrically opposite points of the circular
lattice with N = 200, we have created two solitons, running in the same direction (so as they will not collide). If
the conjecture is correct, the stability chart for this twosoliton pattern should be the same as for the single soliton
in the lattice of size N = 100. This was observed indeed.
Numerical instabilities become significant for lattices with
N ≥ 600, which does not allow us to find a critical value
of N at which the stability region could disappear. For the
time being, we can only assert that the stability regions
shrink and coalesce for larger lattices such that the lower
bound on ε tends to form a single boundary in the (k0 ,ε)
plane that separates the stability and instability regions,
as seen in Figure 1(d) for N = 400.
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Lautaro Vergara, Boris A. Malomed
Figure 3.
Figure 2.
Stability regions in parameter space (k0 ,ε) for the homogeneous lattice with N = 100 lattice points and friction
coefficient α = 0.004, which is twice as large as in Fig. 1
The resulting shift of the stability region to larger values of
ε is observed, as well as its reduction.
Further, detailed inspection of parts of the stability charts
in the (k0 ,ε) plane reveals, in addition to the stability areas, isolated points (up to the numerical accuracy)
at which the soliton appears to be stable. For example, for the lattice with N = 400, and in the ranges of
2.195 < k0 < 2.203 and 0.067 < ε < 0.0680, we observe a single stability point at (k0 , ε) = (2.2, 0.0674).
The soliton created with these parameters persists for the
simulation time in excess of 8000 periods. It is possible
too that the fine structure of the stability areas is fractal
(we did not aim to study this aspect of the problem). The
presence of the isolated stability points, and the possibility of the fractal structure may be related to the KAM
theory, provided that the ac drive may be treated as a
small perturbation destroying a part of invariant tori of
the integrable TL [33].
2.1.
Larger friction
3. Numerical results for inhomogeneous lattices
As said above, physically relevant lattice models may include local imperfections in the form of particles with a
different mass. In this section we aim to study how such a
single mass defect, inserted into the circular lattice, affects
the soliton dynamics in the damped ac-driven circular TL.
3.1.
Light masses
First, we consider the case of a light-mass impurity, with
m∗ < 1.0 in Eq. (2). In this case, we have scanned the
parameter space, aiming to identify the stability regions,
as was done above. Figure 3 shows that the structure
similar to the one displayed in Fig. 1 is observed here as
well, including the shrinkage of the stability regions with
the increase of the lattice’s size. A new feature revealed
by Fig. 3 is the reduction of the stability region with the
decrease of m∗ , as demonstrated by comparison of Figs.
3(b) and 4, which pertain to m∗ = 0.9 and 0.8, for the same
size of the lattice, N = 100. In fact, no stable solitons are
found for a N = 100 lattice and mass impurities m∗ ≤ 0.6.
It is remarkable that the single mass defect produces such
a strong impact on the behavior of the solitons.
3.2.
In Fig. 2 we demonstrate the effect of doubling the value
of the friction coefficient to α = 0.004. We observe what
is expected to happen: for larger α, larger driving amplitudes are required to compensate the dissipation of energy. We observe too that the stability region is reduced
for larger values of the friction coefficient. This may also
be expected, because a larger value of the ac-drive’s amplitude implies stronger “noise" through which the soliton
must travel.
The same as in Fig. 1, but for the lattice with a single
light-mass defect, m∗ = 0.9.
Heavy masses
The stability chart produced by the simulations of the soliton circulating in the lattice with a single heavy-mass
defect, m∗ > 1.0, is qualitatively similar to the above
pictures, see Fig. 5 for m∗ = 1.1. In particular, the contraction of the stability regions is again observed with the
increase of the lattice’s size. As well as in the case of the
light defect, making the defect stronger, i.e., in the present
case, heavier, leads to the shrinkage of the stability areas
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Stability bands for moving solitons in uniform and inhomogeneous damped ac-driven circular Toda lattices
Figure 4.
Figure 5.
The same as in Fig. 3(b) (for the lattice of size N = 100),
but for mass impurity m∗ = 0.8.
The same as in Fig. 3, but for a single heavy mass impurity, with m∗ = 1.1.
for a fixed size of the lattice. In fact, we have found that
no stable solitons can be driven by the ac force in the
lattice with m∗ ≥ 1.5, for a N = 100 lattice. The case
m∗ = 1.2 (not shown) is qualitatively very similar to the
case m∗ = 0.8, see Fig. 4.
4.
Initially inter-site solitons
The analysis presented above was dealing with the stability of solitons whose initial position coincided with a
lattice site. It is relevant too to consider the soliton whose
center is initially set at a position between two sites, i.e.,
with non-integer n0 in Eq. (4). Systematic simulations
reveal an interval of values of n0 , continually connected
to its integer value, starting from where the soliton moves
as a stable one. In fact, we have found that the stability
set of values of parameters (k0 ,ε) attains its largest span
at particular values of n0 . For the circular lattice of size
N = 30, this occurs at n0 = −9.24, while for N = 100
it is n0 = −9.26. Of course, due to the discrete transla-
Figure 6.
Stability regions for solitons in the uniform lattice of size
N = 100. The initial soliton was placed at n0 = −9.30.
tion invariance the same happens also at n0 → n0 ± 2k,
with integer k (the factor 2 is the effective lattice spacing).
Figure 6 shows an enlarged parameter space for solitons
traveling in the uniform lattice of size N = 100, starting
from the initial position at n0 = −9.30. It is observed
that, for the initial inter-site location, the lower boundary of the stability regions tends to a continuous shape.
At the first glance, it may seem that the stability regions
have merged into a single larger region, but a zoom of
the plot demonstrates that narrow instability stripes still
separate the stability areas. The structure of the distinct
stability islands is also observed at n0 = −9.26, at which
the number of stable points is largest. It is clear too that
the stability region in Figure 6 is larger for faster solitons,
as in that case the soliton gets narrower, which increases
the possibility that the it may be stabilized by the on-site
ac forcing.
We have also compared the shape of the lower stability
boundaries for the lattice of size N = 30, with the initial
solitons located at n0 = −9 or n0 = −9.30, and for the
lattice with N = 100 and n0 = −9.30. This is shown in
Fig. 7, where dots represent data in the (ω,ε) plane, while
curves correspond to a fit provided by expression
εthr = A ω ln(ω/π) + B,
(6)
where ω/π is the soliton’s resonant velocity, while A and B
are fitting parameters. This choice is suggested by analytical approximation proposed in ref. [23, 24] as the threshold value for fast solitons, viz., εthr ≈ 2 α ω ln[ω/(3 π)].
Comparing the upper curve, for N = 30 and n0 = −9,
with the lower dashed one, for N = 30 and n0 = −9.30,
we observe that the position of the lower stability boundary indeed strongly depends on the initial location of the
soliton. We also observe that there is a small effect due to
different lattice sizes, that we have quantified to be < 3%.
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Figure 7.
5.
Threshold values of ε for different lattice sizes and different initial locations of the soliton. Dots represents the numerical results, and curvesthe respective fit (6), see the
text. The upper long-dashed curve corresponds to N = 30
and n0 = −9, while the lower curves represent εthr (ω) for
N = 30 and n0 = −9.30 (short-dashed) and N = 100 and
n0 = −9.30 (full).
Counterpropagating solitons
We have also considered the configuration with two solitons traveling in opposite directions and repeatedly colliding in the uniform lattice (as said above, the staggered
ac drive may support the motion of the discrete solitons
with velocities +ω/π and −ω/π). We have initially set
both solitons at the same lattice site and searched for
stable regimes in lattices of sizes N = 30 and N = 100,
in parameter intervals 1.0 < k 0 < 3.0 and 0.02 < ε < 0.2.
The data have been obtained by allowing the solitons to
travel around the ring-shaped lattices in the course of 300
circulation periods, that is, colliding 600 times. The numerical solution of the equations of motion was performed
this time with a modified version of the RADAU5 integrator developed by Hairer and Wanner [34]. It uses a fully
implicit Runge-Kutta method (Radau IIa) of order 5, with
the stepsize control and continuous output. This integrator was found to be much more stable for the collision
problem than the integrator previously used.
For the lattice of size N = 30, we have found a region in
the parameter space where both solitons survive the periodic collisions, as shown in Figure 8. On the contrary,
for the lattice with N = 100 lattice no stable soliton
pairs where found; this is an indication to the fact that
lattice radiation (phonons) generated by each collision is
an important source of the destabilization. In addition,
by randomly choosing values of k0 and ε from the data,
we have tested the persistence of the solitons up to at
least 490 circulations periods (that is, almost 1000 collisions) for large values of ε (> 0.13), including in this
test the isolated point at k0 = 2.8. For smaller values
Figure 8.
Stability regions for the pairs of counterpropagating solitons in the circular uniform TL of length N = 30.
of the forcing amplitude, ε < 0.12, we have found stable
solitons for at least 1000 circulations periods. A simple
explanation for this behavior may be that the effect of the
collision-generated radiation background is enhanced by
the forcing.
6.
Conclusions
This work aims to revisit the stability problem for discrete solitons traveling in the damped circular TL (Toda
lattice) under the action of the staggered ac drive. The
lattice may also include a mass defect. In previous works,
the possibility of the stable motion of solitons in this system was demonstrated, but systematic results were not
reported. For four different sizes of the lattice, N = 30,
N = 100, N = 200 and N = 400, stability regions for
driven solitons have been identified in the space of the
soliton’s velocity, v [parameterized by constant k0 , see Eq.
(5)], and the forcing amplitude, ε. It is found that, in addition to the previously known minimum (threshold) value of
ε, which corresponds to the bottom border of the stability
area, it features a complex structure of alternating stability regions and instability gaps. In addition to the (quasi-)
continuous stability regions, isolated (up to the numerical
accuracy) points of the stable motion of the ac-driven solitons were found too, suggesting that the genuine structure of the stability chart may be fractal. The stability
chart as a whole shrinks with the increase of the lattice’s
size, and eventually disappears for very large values of
N. Qualitative explanations to these remarkable features
were proposed. We have also found that the of stability
region is largest for solitons starting at particular intersite initial positions. Finally, a region in parameter space
was identified where two solitons moving in the opposite
directions survive periodic collisions between them.
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Stability bands for moving solitons in uniform and inhomogeneous damped ac-driven circular Toda lattices
It may be interesting to extend the analysis to the acdriven regimes corresponding to higher-order resonances,
i.e., v = nω/π with integer n. Another natural generalization would be to consider dense chains of traveling
solitons, i.e., periodic (cnoidal) waves, which will be reported elsewhere. It may also be interesting to analyze
similar effect in the damped TL driven by a multiplicative,
rather than direct, force, cf. Ref. [26].
Acknowledgements
L.V. appreciates discussions with Prof. R. Labbé and hospitality of the Department of Physical Electronics of the
Tel Aviv University, where part of this work was carried
out. This work was partially supported by FONDECYTCHILE under grant No. 1085043. B.A.M. appreciates
hospitality of the Faculty of Engineering at Universidad
de los Andes (Santiago de Chile).
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