Dissipative lattice model with exact traveling discrete kink-soliton solutions: discrete breather generation and reaction diffusion regime.

Abstract

We introduce a nonlinear Klein-Gordon lattice model with specific double-well on-site potential, additional constant external force and dissipation terms, which admits exact discrete kink or traveling wave fronts solutions. In the non-dissipative or conservative regime, our numerical simulations show that narrow kinks can propagate freely, and reveal that static or moving discrete breathers, with a finite but long lifetime, can emerge from kink-antikink collisions. In the general dissipative regime, the lifetime of these breathers depends on the importance of the dissipative effects. In the overdamped or diffusive regime, the general equation of motion reduces to a discrete reaction diffusion equation; our simulations show that, for a given potential shape, discrete wave fronts can travel without experiencing any propagation failure but their collisions are inelastic.

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Dissipative lattice model with exact traveling discrete kink-soliton solutions:
Discrete breather generation and reaction diffusion regime
J. C. Comte,1P. Marquie ´,1and M. Remoissenet2
1Laboratoire d’Electronique, Informatique et Image (LE21) Universite ´ de Bourgogne, Aile des Sciences de l’Inge ´nieur, BP 47870,
21078 Dijon Cedex, France
2Laboratoire de Physique de L’Universite ´ de Bourgogne (LPUB), Faculte ´ des Sciences Mirande, 21011 Dijon, France
?Received 7 June 1999?
We introduce a nonlinear Klein-Gordon lattice model with specific double-well on-site potential, additional
constant external force and dissipation terms, which admits exact discrete kink or traveling wave fronts
solutions. In the nondissipative or conservative regime, our numerical simulations show that narrow kinks can
propagate freely, and reveal that static or moving discrete breathers, with a finite but long lifetime, can emerge
from kink-antikink collisions. In the general dissipative regime, the lifetime of these breathers depends on the
importance of the dissipative effects. In the overdamped or diffusive regime, the general equation of motion
reduces to a discrete reaction diffusion equation; our simulations show that, for a given potential shape,
discrete wave fronts can travel without experiencing any propagation failure but their collisions are inelastic.
?S1063-651X?99?16811-9?
PACS number?s?: 45.05.?x, 05.45.Yv, 63.20.Pw
I. INTRODUCTION
In recent years, the dynamics of kinks in nondissipative
?Klein-Gordon? systems ?for a recent review see Braun and
Kivshar ?1?? or traveling wave fronts in strongly dissipative
or reaction diffusion systems has attracted considerable at-
tention. It has become clear that continuous propagation
equations and reaction diffusion equations provide an inad-
equate description of the behavior of weakly coupled lattices
where the interplay between nonlinearity and spatial dis-
creteness can lead to novel effects not present in the con-
tinuum models. For example, in nondissipative ?or weakly?
lattices such as: ferromagnetic chains ?2?, hydrogen bonded
chains ?3?, or chain of base pairs in DNA ?4?, kink solitons or
domain walls, whose width is of the order of few lattice
spacings, may pin on the lattice owing to discreteness ef-
fects. On the other hand, in strongly dissipative lattices of
coupled excitable cells, which are used as models in neuro-
physiology ?5,6? and cardiophysiology ?7? in order to de-
scribe wave propagation in nerve cells, wave propagation
failure, which also originates from lattice discreteness effects
?8? is an important phenomenon which may often lead to
breakdown of these systems with potentially fatal conse-
quences.
In order to gain understanding of wave motions in dis-
crete systems, for which exact results are scarce even in one
dimension, it is desirable to investigate lattice models with
exact solutions. In this regard, Schmidt ?9? pointed out that if
the double-well on-site potential of the ?4lattice model is
suitably modified, the single kink soliton becomes an exact
solution to the discrete model; later, this model was shown to
be integrable in the static limit and admit exact static solu-
tions into the form of generally unpinned soliton lattices
?Jensen et al. ?10??. On the other hand, Bressloff and Row-
lands ?11?, using an approach which presents some similari-
ties with Jensen et al. ?10?, have recently shown that it is
possible to construct exact traveling wave solutions of a dis-
crete reaction diffusion equation, describing a system of
coupled bistable elements, if the form of the bistable poten-
tial is adequately chosen. In fact, their potential corresponds
precisely to Schmidt’s potential. Very recently, the general
problem of finding kink or pulse shaped traveling waves so-
lutions was considered by Flach and coworkers ?12?. As
have the above-mentioned authors, they have approached the
traveling wave existence problem from the inverse side, that
is, they have shown that for a given wave problem, corre-
sponding equations of motion can be generated, so that these
equations yield the chosen wave profile as a solution. They
have studied conservative lattice models and dissipative ?re-
action diffusion like? models, separately. One might there-
fore wonder if it is possible, in a similar way, to construct a
general discrete model including both inertia and dissipation.
Actually, generalizing Schmidt’s approach, we show in this
paper, that a lattice model with on-site double-well potential,
with additional external force and dissipation terms, can also
admit exact kink or traveling wave front solutions in any
regime: nondissipative or dissipative.
The paper is organized as follows. First, we present our
specific lattice model and show analytically that it can admit
exact discrete kink solutions if the double-well on-site poten-
tial is adequately chosen. Then, in Sec. III, we study numeri-
cally the propagation and collisions of discrete kinks and
antikinks, in the nondissipative and dissipative regimes. In
Sec. IV, we consider the reaction diffusion regime and in-
vestigate numerically the properties of traveling discrete
wavefronts solutions. Section V is devoted to concluding re-
marks.
II. MODEL AND EQUATION OF MOTION
We consider a chain of harmonically coupled particles of
mass m, lying in a double well on site potential U. This
system is modeled by the general discrete equation of motion
PHYSICAL REVIEW EDECEMBER 1999VOLUME 60, NUMBER 6
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md2un
dt2??dun
dt?k?un?1?2un?un?1??F?dU?un?
dun
,
?2.1?
where unis the nth particle displacement, ? is a dissipative
coefficient, and k is a coupling term. F is a constant term
which may be an external force and the energy gain due to F
is compensated by the loss mechanism. We assume that a
traveling wave front solution of ?2.1? has the following kink
shape:
un?t??u0tanh??t??na?.
?2.2?
Here, u0is the amplitude, a the lattice spacing, and ? and ?
are two constants such that the ratio c??/? represents the
velocity of the front. Following Schmidt ?9?, we construct a
potential, including the external potential, which has the
form:
V?un??Fun?U?un?
?Fun?A
un
2?B0
2
un
3?B
3
un
4?C0
4
un
5?C
5
un
6?D0
6
un
7
7
?D
un
8?¯ ,
8
?2.3?
such that the expression ?2.2? becomes an exact discrete so-
lution of ?2.1?. Here, A,B0,B,C0,C,D0,D, represent con-
stant coefficients. The detailed calculations are presented in
the Appendix. We obtain the new equation of motion
m
k
d2vn
dt2??
k
dvn
dt??dV?vn?
dvn
??vn?1?2vn?vn?1?,
?2.4?
where we have introduced the dimensionless variable vn
??(un/u0), with ??tanh(?a). Here V(vn) is given by
V?vn???vn??vn
2??vn
3??vn
4?? ln?1?vn
2?,
?2.5?
with ??????/k, ??(m?2/k?1), ????/(k?), ??
?m?2/(2k?2), and ???2(?2?1). The propagation veloc-
ity of the front is given by
c??
??
a?
arctanh???.
?2.6?
The general discrete equation of motion ?2.4? with potential
?2.5? and solution
vn?? tanh??t??na?,
?2.7?
is valid for any value of F and ?. At this stage, three differ-
ent cases can be considered.
?i? When m?0 and ??0, the coefficients of vnand vn
?2.5? become zero, the symmetry breaking due to F disap-
pears and the potential becomes symmetric with two degen-
erate minima. One recovers the conservative system studied
by Schmidt ?9?; the total energy is constant and to each ??,
?? combination corresponds a different well shape and ve-
locity.
3in
?ii? When m?0, ??0, this is the general case where both
inertial and dissipative effects play a role. The value of ? is
imposed ?see Eq. ?A7?? by the relation F???u0?. It turns
out that the potential well shape and velocity both depend
uniquely on the choice of ?, that is ?.
?iii? When ??0 and m?0, the lattice dynamics becomes
overdamped or diffusive. In ?2.5?, ?, the coefficient of vn
reduces to ?1 and ?, the coefficient of vn
potential is asymmetric with two nondegenerate minima.
This regime corresponds to the model studied by Bressloff
and Rowlands ?11?.
2
4becomes zero: the
In the following we consider these three cases successively.
III. NONDISSIPATIVE AND DISSIPATIVE REGIMES
A. Nondissipative regime
In this case (??0), the potential, represented in Fig. 1,
becomes
V?vn???m?2
k?1?vn
2?m
?2
2k?2vn
4?2??2?1?ln?1?vn
2?.
?3.1?
The barrier height is given by ?3.2?,
E0???2?1?
?2
2?0
2???1??2?ln?1??2??,
?3.2?
with ?0
2?k/m. The equation of motion ?2.4? is written as
d2vn
dt2??0
2?vn?1?2vn?vn?1?
2?
?0
??0
2?2
2?2vn?vn
2??2??2vn?2?1??2?
vn
1?vn
2?.
?3.3?
FIG. 1. Symmetric potential V(vn) in the nondissipative regime
with its two degenerate minima. The shape is obtained with the
parameters ??0.0493 and ??0.761. The dashed line represents the
asymmetric oscillations of the central particles of the breather ?with
an amplitude ?vn,max?1.36) created by the kink-anti-kink colli-
sion. Emaxrepresents the maximum of energy of the particles during
the oscillations while E0is the energy barrier height.
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We have checked by numerical simulations that an exact
discrete kink solution given by ?2.7? can propagate freely,
that is, without experiencing any discreteness effect. In Fig.
2, we have represented, at different times, a kink K with a
width of five lattice spacings traveling in a lattice of 2048
cells. The parameters of the lattice and the potential are,
respectively, k?0.1 and m?1, ??0.0493 and ??0.761.
Under these conditions, the velocity of the kink is c
?0.023cells?1. Note that no radiation effects are observed
which confirms that ?2.7? is an exact solution.
Then we have studied the possible generation of nonlinear
localized modes or discrete breathers via kink ?K? and anti-
kink (K¯) collisions. This investigation was motivated by the
interesting properties of discrete breathers: these nontopo-
logical excitations can exist in a large variety of nonlinear
lattices and their existence is associated with a localization of
energy ?for recent reviews see ?13,14??. Specifically, we have
studied numerically a K?K¯collision where the two entities
travel at velocity c?0.023cells?1and ?c, respectively. As
shown in Fig. 3?a?, a discrete stationary breather and a small
amplitude radiation background emerge from the weakly in-
elastic K?K¯collision. In Fig. 3?b?, we have represented this
breather at three different times, t?500 s, t?510 s, and t
?520 s, respectively, such that we can observe its behavior
over one period TBof oscillation: TB?20 s. The oscillations
of the central particules of the breather are asymmetric with
an amplitude ?Vn,max?0.96???0.4??1.36, as represented
by the dashed line in Fig. 1. Indeed, during the oscillations
from Emax?0.364 to E??0.086, the central particles over-
come the potential barrier E0???0.214?, given by ?3.2?. In
the breather spectrum, the fundamental angular frequency
?B?2?/TB?0.314 rads?1is represented by a in Fig. 4,
while the second and third harmonics are, respectively, rep-
resented by b and c. Although the breather looks like very
stable, it radiates phonons very slowly. The radiation fre-
quency corresponds to the third harmonics in the breather
spectrum ?c in Fig. 4? which actually represents only 2% of
the energy of the first harmonic. Specifically, the frequency
of this third harmonic lies in the phonon band given by ?
??0?2?(1??)?cos(?a)?, as shown in Fig. 4, where ?
?1?2(?2/?0
tion can propagate. Nevertheless, in spite of these weak ra-
diation losses, this discrete breather has an important lifetime
and presents a physical interest. We have also investigated
the collision of a static kink (c?0) with a kink moving at
velocity c?0.023cells?1. Such a collision results in a dis-
crete breather with the same properties as the static one, but
moving at mean velocity cb?0.013cells?1.
2)?(1??2)/(1??2). As a result, this small radia-
B. General dissipative regime
In the general regime, where both inertial and dissipative
terms are present, we have analyzed numerically a K?K¯
collision. As pointed out in Sec. II, the potential well shape
and velocity both depend on the choice of ?. In this case, as
it should be expected, we have observed that the number N
of breathing oscillations depends on ?/m. For example, we
have the following results: N?27 for ?/m?0.001, N?8 for
?/m?0.01, and N?2 for ?/m?0.1. For ?/m?0.1, the
number of oscillations decreases and tends rapidly to zero.
Our results suggest that in real lattices, where dissipation
FIG. 2. Traveling kink K with a width of five lattice spacings
and a constant velocity c?0.023cells?1in the nondissipative re-
gime.
FIG. 3. ?a? Creation of a discrete stationary breather with a
small amplitude radiation background by a K?K¯ collision. ?b?
Representation of the breather at three different times (t?500s, t
?510s, and t?520s) allowing to observe its behavior over one
period of oscillation TB?20s: the oscillations of the central particle
are asymmetric with an amplitude ?vn,max?1.36 ?see Fig. 1?.
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cannot be ignored, the existence of breathers is relevant only
in the case of weak dissipation. Actually, this general re-
gime, which should present unexpected features for some
parameters range, remains to be explored carefully and will
be discussed elsewhere.
IV. STRONGLY DISSIPATIVE OR DIFFUSIVE REGIME
Let us now focus on the overdamped or diffusive case
(m?0). Equation ?2.4? then reduces to
?dvn
dt?k?vn?1?2vn?vn?1?
??????2kvn???
?
vn
2?2k??2?1?
vn
1?vn
2?.
?4.1?
In order to compare our results ?see hereafter? to those of
Bressloff and Rowlands ?11?, we perform the following
transformations. We divide the two members of ?4.1? by ?
and set D?k/?, ?D??/(?D) and ?0?2(1??2), to obtain
dvn
dt?D?vn?1?2vn?vn?1?
?D??D??1??0/2??vn
2??
?0vn
1?vn
2?2vn?.
?4.2?
Equation ?4.2? is a discrete reaction diffusion ?7? equation of
the form:
dvn
dt?D?vn?1?2vn?vn?1??f?vn?,
?4.3?
with kink-shaped solution ?2.7? provided that, the potential
V(vn) is well chosen. Indeed, the potential corresponding to
f(vn) is
V?vn??D??D?1??0/2?vn?vn
2
??D
vn
3???0/2?ln?1?vn
3
2??.
?4.4?
Note that, for D?1, relation ?4.3? is identical to the equation
?1.3? of Bressloff and Rowlands in ?11?. The potential ?4.4?,
represented in Fig. 5 as a function of vn, is asymmetric with
two nondegenerate minima ?a and b in Fig. 5?. The value of
? is imposed by F and the velocity of the discrete front
solution of ?4.2? depends on the choice of ??tanh(?a). The
velocity of the front expressed in terms of the new param-
eters ?0and ?Dis
c??/???DD
?
?
?DD
?1??0/2.
?4.5?
This velocity c is proportional to the diffusion coefficient D,
as represented in Fig. 6 by the continuous linear curves ?ob-
tained for different values of coefficient ??. In this figure are
also superimposed the results obtained by numerical simula-
tions. They have been performed on a lattice of 2048 par-
ticles, for different values of the diffusion coefficient D and
the ?0coefficient. The initial condition consists in a front
given by relation ?2.7?. One can observe that the velocities
measured during the simulations ?plus signs in Fig. 6? are
very well fitted by the theoretical curves. The fundamental
result, in the diffusive case, is the nonexistence of propaga-
tion failure, contrary to the systems described by a discrete
reaction diffusion equation of the Fitzhugh-Nagumo type
?8,15,16?. Indeed, in these systems, there exists a critical
value of the coupling constant ?or the diffusion coefficient?
under which propagation of diffusion of fronts becomes im-
possible ?15?. Considering the discrete lattice with the poten-
tial given by relation ?4.4?, we observe numerically that, for
any value of the diffusion coefficient D ?except for D?0
when all the particles are independent?, there exists a travel-
ing wave front. In fact, for a lattice with a given coupling
constant, we have constructed a potential for which a propa-
FIG. 4. Breather spectrum: the fundamental angular frequency
?B?0.314 rads?1is represented by letter a while the second and
third harmonics are respectively represented by b and c. The third
harmonics ?c? lies in the phonon band ?represented by the continu-
ous line? leading to small radiations of the breather.
FIG. 5. Asymmetric potential V(vn) in the strongly dissipative
or diffusive regime. Letters a and b represent the two nondegener-
ate minima.
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7487DISSIPATIVE LATTICE MODEL WITH EXACT . . .

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gative discrete solution of the form ?2.7? exists. But if we
modify the coupling constant of this lattice without changing
the potential shape, then expression ?2.7? is no more a solu-
tion of the reaction diffusion equation ?4.2?. As a result, in
this case, we find that propagation failure occurs, as shown
also by Bressloff and Rowlands ?11?. We have then observed
that the collision between two wave fronts traveling in op-
posite directions results into their annihilation.
V. CONCLUDING REMARKS
The generalized dissipative nonlinear Klein-Gordon lat-
tice model that we have introduced, interpolates between the
Schmidt’s model ?9? in the conservative regime and the re-
action diffusion model of Bressloff and Rowlands ?11? in the
overdamped regime. Like Flach and coworkers ?12? we have
used an inverse method. However, contrary to these authors
who have considered the conservative and dissipative lattice
models separately, we have investigated a lattice model
which includes both inertia and dissipation, like many sys-
tems in the real world. We have shown analytically that our
lattice equations admit exact discrete kink or traveling wave
fronts solutions if the double-well on-site potential is ad-
equately chosen. In the nondissipative regime, we have
checked numerically that discrete kink ?antikink? solutions
can propagate freely without experiencing any discreteness
effects. Kink-antikink collisions reveal that static or moving
discrete breathers with finite but with physically interesting
lifetimes can be generated. In the general dissipative regime,
discrete kinks can propagate freely and the lifetimes of the
discrete breathers that can be created by collisions between
these kinks depend on the importance of the dissipation.
These results are interesting: they suggest that, for the real
lattices where dissipation cannot be ignored, discrete kinks
can travel and combine to generate discrete breathers with
reasonable lifetimes, if dissipative effects are weak enough
compared to inertial effects. In the overdamped or diffusive
regime, discrete kinks or wave fronts can travel without ex-
periencing any lattice effects or propagation failure. How-
ever, in this case, their collisions are totally inelastic. We
would like to point out again that our model and results are
relevant for physical systems in which the discreteness of the
lattice is important. Obviously, further studies are necessary
in the general dissipative regime to determine all the proper-
ties of these kinks with exceptional mobilities. In conclusion,
we believe that the understanding of discrete nonlinear mod-
els is an active and attractive topic of the current research.
Since realistic physical models are rather complicated, it is
extremely important to develop the basic concepts with the
help of simple lattice models with exact solutions.
APPENDIX: CALCULATIONS DETAILS
Starting from ?2.1?, we seek a total potential V(un)?Fun?U(un) under the form ?2.3? such that ?2.2? is an exact solution
of ?2.1?. Setting Tn?tanh(?t??na) and ??tanh(?a) we then get
dun
dt?u0?1?Tn
2??,
?A1?
d2un
dt2??2u0?2?1?Tn
2?Tn,
?A2?
un?1?un?1?u0?tanh??t???n?1?a??tanh??t???n?1?a???2u0?1??2?Tn?1??2Tn
2??4Tn
4?¯?,
?A3?
and
dV?un?
dun
?F?Aun?B0un
2?Bun
3?C0un
4?Cun
5?D0un
6?Dun
7?... .
?A4?
Substituting relations ?A1?, ?A2?, and ?A3? in ?2.1? yields
?2mu0?2Tn?1?Tn
2????u0?1?Tn
2??2u0k?1??2?Tn?1??2Tn2??4Tn
7?¯?0.
4?¯??2ku0Tn?F?Au0Tn?B0u0
2Tn
2?Bu0
3Tn
3
?C0u0
4Tn
4?Cu0
5Tn
5?Du0
6Tn
6?Du0
7Tn
?A5?
FIG. 6. Theoretical velocity of the front ?Eq. ?4.5?? in the diffu-
sive case for different values of the parameter ? ?continuous lines?:
?a? ??1; ?b? ??0.75; ?c? ??0.5; ?d? ??0.25; ?e? ??0.01. The
measured velocity during simulations ?? signs? are very well fitted
by theoretical curves given by Eq. ?4.5?.
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