Front propagation into unstable states in discrete media

Abstract

Non-equilibrium dissipative systems usually exhibit multistability, leading to the presence of propagative domain between steady states. We investigate the front propagation into an unstable state in discrete media. Based on a paradigmatic model of coupled chain of oscillators and populations dynamics, we calculate analytically the average speed of these fronts and characterize numerically the oscillatory front propagation. We reveal that different parts of the front oscillate with the same frequency but with different amplitude. To describe this latter phenomenon we generalize the notion of the Peierls-Nabarro potential, achieving an effective continuous description of the discreteness effect.

Full text

Front propagation into unstable states in discrete media
K. Alfaro-Bittnera,∗, M. G. Clercb, M. Garc´ıa-˜
Nustesa, R. G. Rojasa
aInstituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Casilla 4059,
Valpara´ıso, Chile
bDepartamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de
Chile, Casilla 487-3, Santiago, Chile
Abstract
Non-equilibrium dissipative systems usually exhibit multistability, lead-
ing to the presence of propagative domain between steady states. We inves-
tigate the front propagation into an unstable state in discrete media. Based
on a paradigmatic model of coupled chain of oscillators and populations
dynamics, we calculate analytically the average speed of these fronts and
characterize numerically the oscillatory front propagation. We reveal that
different parts of the front oscillate with the same frequency but with differ-
ent amplitude. To describe this latter phenomenon we generalize the notion
of the Peierls-Nabarro potential, achieving an effective continuous description
of the discreteness effect.
Keywords: Fronts propagation, discretization, Peierls-Nabarro potential,
FKPP fronts
1. Introduction
Macroscopic systems under the influence of injection and dissipation en-
ergy, momenta, or matter usually exhibit coexistence of different states—this
feature is usually denominated multistability [1,2]. Inhomogeneous initial
conditions, usually caused by the inherent fluctuations, generate domains
that are separated by their respective interfaces. These interfaces are known
as front solutions, interfaces, domain walls or wavefronts [2,3,4], depending
on the physical context where they are considered. Interfaces between these
∗Corresponding author
Email address: kapaalbi@gmail.com (K. Alfaro-Bittner)
Preprint submitted to Commun Nonlinear Sci Numer Simulat May 3, 2016
arXiv:1605.00556v1 [nlin.PS] 2 May 2016

metastable states appear in the form of propagating fronts and give rise to
rich spatiotemporal dynamics [5,6,7]. Front dynamics occurs in systems
as different as walls separating magnetic domains [8], directed solidification
processes [6], nonlinear optical systems [9,10,11,12], oscillating chemical
reactions [13], fluidized granular media [14,15,16,17], and population dy-
namics [18,19,20], to mention a few.
In one spatial dimension—from the point of view of dynamical systems
theory—a front is a nonlinear solution that is identified in the co-moving
frame system as a heteroclinic orbit linking two steady states [21,22]. The
evolution of front solutions can be regarded as a particle-type one, i.e., they
can be characterized by a set of continuous parameters such as position,
core width and so forth. The front dynamics depends on the nature of the
steady states that are connected. In the case of a front connecting two stable
uniform states, a variational system tends to minimize its energy or Lyapunov
functional. Thus, the front solution always propagates with a well defined
unique speed towards the less energetically favorable steady state. There
is only one point in the parameter space for which the front is motionless,
the Maxwell point. In systems with only local interaction between adjacent
neighbors—a discrete medium—the front solutions persist [23,24]. Once
more, the most favorable state invades the less favorable one, being now the
speed oscillatory. Actually, there is a region of the parameter space, close
to the Maxwell point—the pinning range—where the fronts are motionless
[24]. These properties can be explained by considering a potential for the
front position, the Peierls-Nabarro barrier [25], which it is a result of the
discreteness of the system.
The former scenario changes drastically for a front connecting a stable
and an unstable state, usually called Fisher-Kolmogorov-Petrosvky-Piskunov
(FKPP) front [18,26,27]. FKPP fronts have been observed in auto-catalytic
chemical reaction [28], Taylor-Couette instability [29], Rayleigh-Benard ex-
periments [30], pearling and pinching on the propagating Rayleigh instability
[31], spinodal decomposition in polymer mixtures [32], and liquid crystal light
valves with optical feedback [11]. One of the features of these fronts is that
their speed is determined by initial conditions. When initial conditions are
bounded, after a transient period, two counter propagative fronts emerge
with the minimum asymptotic speed [18,26,27]. In discrete media, FKPP
fronts also persist exhibiting an oscillatory behavior, however, the pinning
phenomenon no longer exists. Beyond that, there is few understanding about
their general behavior in discrete systems in our knowledge.
2

The aim of this article is to investigate, theoretically and numerically,
the FKPP front propagation in discrete media. Based on paradigmatic mod-
els: the dissipative Frenkel-Kontorova and the discrete Fisher-Kolmogorov-
Petrosvky-Piskunov equation. We determine analytically the minimum mean
speed of FKPP fronts as a function of the medium discreteness. Numerically
we characterize the oscillatory front propagation. We reveal that the front
behaves as an extended object. Different parts of the front oscillate with the
same frequency but with different amplitude. To describe this phenomenon
we generalize the notion of the Peierls-Nabarro potential, which allows us to
have an effective continuous description of the discreteness effect.
2. Chain of dissipative coupled pendula
Let us consider a chain of dissipative coupled pendula, known as the
dissipative Frenkel-Kontorova model,
¨
θi=−ω2sin θi−µ˙
θi+θi+1 −2θi+θi−1
dx2,(1)
where θi(t) is the angle formed by the pendulum and the vertical axis in
the i-position at time t, iis the index label the i-th pendulum, ωis the
pendulum natural frequency, µaccounts for the damping coefficient, and dx
stands for the interaction between adjacent pendulums. This last parameter
controls the degree of discreteness of the system. When dx →0 the system
describes front propagation in a continuous medium, the dissipative sine-
Gordon equation. In the conservative or Hamiltonian limit, µ= 0, the
above equation is known as the Frenkel-Kontorova model, which describes
the dynamics of a chain of particles interacting with the nearest neighbors
in the presence of an external periodic potential. The Frenkel-Kontorova
model, Eq. (1), is a paradigmatic model with application to several physical
contexts. It has been used to describe the dynamics of atoms and atom layers
adsorbed on crystals surfaces, incommensurate phase in dielectric, domain
wall in magnetic domain, fluxon in Josephson transmission lines, rotational
motion of the DNA bases, and plastic deformations in metals (see textbook
[25] and references therein).
Note that equation (1) can rewrite in the following manner
µ˙
θi=−δF
δθi
,(2)
3

c)
b) Front position
x0
20 60 140 180
×104
1.590
1.591
1.592
time
v
time
0 100 200 300
0.4
0.5
0.6
Speed
dx=10
dx=7
dx=3
θ
g
a)
v
x0
100
dx=5
dx=2
dx=0.1
Figure 1: (color online) Chain of dissipative coupled pendula. a) Schematic
representation of a chain of dissipative coupled of pendula. b) Temporal
evolution of the front position x0(t) with ω= 1 and µ= 20. The upper
(yellow), middle (orange) and lower (blue) lines correspond to dx = 10,
dx = 7, and dx = 3, respectively. c) Temporal evolution of the front speed
˙x0(t) with ω= 1 and µ= 6 . The upper (yellow), middle (orange) and lower
(blue) lines correspond to dx = 5, dx = 2, and dx = 0.1, respectively.
where the Lyapunov functional Fhas the form
F≡
N
X
i=0 "˙
θ2
i
2−ω2cos θi+(θi+1 −θi+1)2
2dx2#.(3)
Hence, the dynamics of Eq. (1) is characterized by the minimization of func-
tional Fwhen µ6= 0.
2.1. Propagation of a π-kink in a dissipative chain of pendula
In the range {0,2π}, Eq. (1) has steady states θ0= 0 and θ1=π,
which corresponds to the upright and upside-down position of pendulum,
respectively. The upright (upside-down) position of the pendulum is a stable
(unstable) equilibrium. Hence, the chain of dissipative coupled pendula can
exhibits domains of upright or upside-down pendula as an extended state
4

and a domain wall or connective front between both domains. This solution
is usually denominated as a π-kink. Figure 1a illustrates a π-kink solution
of this chain. The position of the domain wall, x0(t), is defined by a spa-
tial location that interpolates a horizontal pendulum (cf. Fig. 1a). Due to
coupling between pendulums, the domain wall propagates into the unstable
state. Figures 1b and 1c show, respectively, front position and speed for dif-
ferent values of discreteness obtained from numerical simulations of equation
(1). Numerical simulations were conducted using finite differences method
with Runge-Kutta order-4 algorithm and specular boundary conditions. In-
deed, the speed of propagation of the π-kink is oscillatory with a well defined
average speed, hvi. Unexpectedly, when discreteness dx increases, the mean
speed, amplitude and frequency of oscillation increases. Note that the os-
cillations exhibited by the speed are non-harmonic. Figure 2a shows the
mean speed as function of the discreteness. For large discreteness, the speed
increases linearly.
×10
4
2.651 2.6518 2.6526
-1
0
1
X0
x0
F
dx
<v>
a) b)
-8.115
-8.125
2.659 2.661
×104
×104
0 2 4 6 8 10
0.6
1.0
1.4
Figure 2: Front propagation into an unstable state in discrete chain of dis-
sipative coupled pendula. a) Mean front speed as a function of the coupling
parameter. Dots (red) are obtained by means of numerical simulations of
Eq. (1) with ω= 1.0, dx = 5.0 and µ= 2.0. The solid line is obtained
by using the formulas (25) and (26). b) Lyapunov functional as function
of front position obtained by numerical simulations of Eq. (1) for the same
parameters. Inset: Lyapunov functional computed in the co-mobile system.
From a numerical solution of a chain of dissipative coupled pendula, we
have computed the Lyapunov functional. Figure 2b shows evolution of the
Lyapunov functional as a function of the front position x0. As a matter
of fact, the Lyapunov functional decreases with time in an oscillatory man-
5

ner. It is clear from this results that the observed dynamical behavior is a
consequence of the discreteness of the system.
To assess the above proposition we shall consider first a minimal theoret-
ical model—the discrete Fisher-Kolmogorov-Petrosvky-Piskunov equation—
that contains the main ingredients: coexistence between a stable and unstable
state in a discrete medium.
3. The Discrete Fisher-Kolmogorov-Petrosvky-Piskunov model
Before study the effects of the discreteness in the FKPP-front, we shall
establish some well-known facts about the front propagation into unstable
states in the continuous case.
3.1. The Fisher-Kolmogorov-Petrosvky-Piskunov model: continuos medium
The most simple model that present front propagation into an unstable
state is the Fisher-Kolmogorov-Petrosvky-Piskunov equation,
∂tu=u(1 −u) + ∂xxu, (4)
where u(x, t) is an order parameter that accounts for an extended transcritical
bifurcation. The above model was used to study the populations dynamics in
several contexts [18], where the main ingredients are linear growth, nonlinear
saturation (logistic nonlinearity), and Fickian transport process.
In this model, u= 0 is an unstable fixed point and u= 1 is a stable one.
If the initial conditions have compact support, i.e., u(x, 0) = u0(x) with
u0(x) = 


0, x < x1
f(x), x1≤x≤x2
0, x < x2
(5)
where f(x) is a positive and bounded function, the solution u(x, t) will evolve
to a two counter propagating front solutions, which propagate to speed v=
2. In other words, the fronts move with minimum speed vmin = 2 [26,
27]. This speed was determined by considering traveling solutions in the
co-mobile dynamical system. After, we perform a linear analysis around
the unstable state, imposing that the front solutions do not exhibit damped
spatial oscillations [26]. Henceforth, we shall denominate this method to
determine the minimum speed as FKPP procedure. This result was obtained
in the pioneering work of Luther [28], in the context of wave propagation in
6

catalytic chemical reaction. As this minimum speed is determined by means
of a linear analysis it is usually called linear criterion. This linear criterion for
the determine the minimum speed of front propagation into unstable states is
valid for weakly nonlinearity, as it is established in the work of Kolmogorov
et al. [26] (for more details see Rev. [27] and references therein). For
strong nonlinearity, nonlinear criterion, the front propagation into unstable
states also have a minimum speed, however there is no general formulation
to determine the value of such speed. For gradients equations have been
developed a variational method to determine an adequate approximation
to the minimum speed [33]. Fronts whose minimum speed is determined
by the linear or nonlinear criteria are usually called pulled or pushed front,
respectively [27].
As we have mention for different initial conditions, the front solution
will strongly depend of the asymptotic behavior of u(x, 0) for x→ ±∞.
Considering an initial conditions of the form u(x, 0) ∼Ae−kx for x→ ∞
where {k, A}are positive constants. The front propagates as a wave of the
form u(x, t) = u(k(x−vt)). Linearizing Eq. (4) and considering u(x, t)∼
Ae−k(x−vt)for x→ ∞, after straightforward calculations one can obtain the
following relation [39],
v=1
k+k. (6)
Thus, vas function of kis a convex function. Minimizing the previous curve
with respect to k, we obtain the critical steepness k≡kc= 1, for which we
obtain the minimal speed v(kc)≡vmin = 2. This method is the asymptotic
process. For any other value of the steepness kthe front propagates with a
speed larger than vmin (v(k)>vmin). Hence, using the asymptotic shape of
the front solution one can determine the minimum speed, when the linear
criterion is valid. Note that this procedure can only determine the minimum
speed of a pulled front. It is noteworthy to mention that the asymptotic
solution for all co-mobile reference systems, i.e., z=x−vt is [38]
u(z) = 1
(1 + ez/v )+1
v2
ez/v
(1 + ez/v )2ln "4ez/v
(1 + ez/v )2#+O1
v4(7)
where v≥vmin = 2.
7

3.2. Front propagation in discrete FKPP model
Let us consider a simple discrete version of the Fisher-Kolmogorov-Petrosvky-
Piskunov model, Eq.(4),
˙ui=ui(1 −ui) + ui+1 −2ui+ui−1
dx2,(8)
where ui(t) stands for the population in i-th position. It is assumed that
locally the growth is linear, the saturation is nonlinear, and that the popula-
tion flow is proportional to the population difference of near neighbors. Note
that the dynamics of discrete FKPP model can rewrite in the following form
∂tui=−∂F
∂ui
(9)
where the Lyapunov function is defined as
F=X
i−u2
i
2+u3
i
3+(ui+1 −ui)2
2dx2=X
i
Vi+(ui+1 −ui)2
2dx2,(10)
where Viis the potential. Hence, the dynamics of Eq. (8) is characterized by
the minimization of functional F. Indeed, using Eq. (8), one obtains
dF
dt =X
i
∂F
∂ui
∂ui
∂t =−X
i∂F
∂ui2
.(11)
The discrete Fisher-Kolmogorov-Petrosvky-Piskunov Eq. (8) exhibits front
propagation into an unstable state. In Ref.[40], it has been established the
existence of these solutions, however their oscillatory propagation has not
been yet characterized. Figure 3shows a schematic representation of po-
tential Viand the front solution. From this figure, one trivially deduces the
energy source of the front propagation.
Defining the front position as the spatial position that interpolate the
maximum spatial gradient, u(x0)=1/2 (cf. Fig. 3b), one can study the front
propagation. Figure 4shows, respectively, the front position and minimum
speed for different values of discreteness obtained from numerical simulations
of model Eq. (8). We can observe similar dynamical behavior that those
exhibit by a chain of dissipative coupled pendula (cf. Fig. 1), that is, the
front propagates with an oscillatory speed with a given mean speed. When
coupling parameter dx increases, the mean speed, amplitude and frequency
8

a) b)
ui
i
Vi
i
ui
x0
1.0
0.8
0.4
0.2
0.0
210 230 250 270
Figure 3: (color online) Front solution of the FKKP model Eq. (8). a)
Schematic representation of potential Vi. b) Front solution obtained numeri-
cally from Eq. (8) (blue dots) and the asymptotic solution Eq.(7) (solid line);
x0accounts for the front position.
of oscillations increases. Moreover, the oscillations exhibited by the speed
are non-harmonic type. Figure 5a shows the mean speed as a function of the
coupling parameter. For large dx the speed increases linearly.
From a numerical solution of FKPP model Eq. (8), we have computed the
Lyapunov functional (10). Figure 5b shows evolution of Lyapunov functional
as a function of the front position. In the inset of Fig. 5b, we show the
Lyapunov functional in the co-mobile system. As we can see, Lyapunov
functional decreases with time in a oscillatory manner.
Usually, the study of the front dynamics is reduced to the front position,
i.e., the dynamical tracking of point x0, where u(x0) = 1/2 is the maximum
of the spatial gradient. This is based on the assumption that the front be-
haves as a point-like particle. Thus, point x0will gives us enough information
about the whole structure dynamics. Surprisingly, the FKPP front exhibits
an extended object behavior: each point of the front shows an oscillation
dynamics with the same frequency but different amplitude. Figure 6a shows
the spatiotemporal diagram of the front. From this figure, is easy to infer
that the front propagates as an extended object. Moreover, the oscillation
with respect to front position x0are in anti-phase (see Fig. 6b). That is,
the maximum oscillation of a point to the left of the front position coin-
cides with the minimum oscillation of a point to the right. To explore the
structure of the potential over which the front propagates, we have followed
different points or ”cuts” along the front profile, studying each one sepa-
rately. Figure 6c displays the amplitude of oscillation of the front speed for
9

time
300 600 900
2
2.6
3.2
3.8
Speed
v
b)
0
0
5
10
15
20
25
Front Position
x0
a)
dx=5
dx=2
dx=0.1
time
6004002000
dx=9.5
dx=5
dx=2
Figure 4: (color online) Front propagation into an unstable state in discrete
FKPP Eq. (8). a) Temporal evolution of front position x0(t). The upper
(yellow), middle (orange) and lower (blue) lines correspond to dx = 9.5,
dx = 5, and dx = 2, respectively. b) Temporal evolution of front speed ˙x0(t).
The upper (yellow), middle (orange) and lower (blue) lines correspond to
dx = 5, dx = 2, and dx = 0.1, respectively.
different cuts. From this figure, we conclude that this amplitude is minimal
at the front position, increases as one moves away from the front position
and decays to zero abruptly in the front tails.
3.3. Theoretical description of the mean speed for the discrete FKPP model
For discrete media, the FKPP procedure is unsuitable to determine the
minimum speed. Due to there is not a continuos dynamical system associated
to the co-mobile system inferred for traveling wave solutions. To compute
the minimal front speed, we generalize the asymptotic procedure ansatz for
the front tail [39],
ui(t) = e(αt−2iβ)1 + fω
dx;i(t), i 1,(12)
with α≡khviand β≡k dx/2 are parameters. The index i≥0 is a
positive and large integer number, dx is the discretization parameter, hvi
is the mean speed of the front, and fω
dx,i(t) is a time periodic function with
period T≡2π/ω, i.e., fω
dx;i(t) = fω
dx;i(t+T), which accounts for the oscillation
of the front speed at the i-th position (cf. Fig. 6). In addition, fω
dx;i(t)→0
when i→ ∞. Hence, function fω
dx,i(t) takes into account the periodicity
introduced by the discreteness. Linearizing discrete FKKP model (8) and
10

×104
x0
F
10-3
10
4
x0
dx
<V>
-919.4
-919.0
-918.6
-918.2
b)
×
×
5.510 5.512 5.514
a)
4.0
3.0
2.0
0 2 4 6 8 10
-4
4
0
5.5285 5.5305 5.5325
Figure 5: (color online) Front propagation into an unstable state in FKPP
model, equation (8). a) Minimum mean speed as a function of the discrete-
ness dx. Dots (blue) correspond to numerical simulations of model (8). The
solid (red) and dashed (yellow) lines are the exact and approximative curve
obtained form expressions (18) and (21), respectively. b) Lyapunov func-
tional as a function of front position obtained from numerical simulations of
Eq. (8) with dx = 10. Inset: Lyapunov functional computed in the co-mobile
system.
replacing ansatz (12), we get
˙ui=˙
fω
dx;i+α1 + fω
dx;i
=1 + fω
dx;i+k2
β2sinh2(β)+sinh2(β)fω
dx;i.
Integrating this expression in a normalized period T
h˙
fω
dx;i+α1 + fω
dx;ii=h1 + fω
dx;i+k2
β2sinh2β+sinh2βfω
dx;ii,(13)
where
hg(t)i ≡ 1
TZT
0
g(t)dt, (14)
we obtain an expression for the mean speed hvi
hvi=1
k+ksinh β
β2
.(15)
with hfω
dx;ii=h˙
fω
dx;ii= 0, due to fω
dx;i(t) periodicity. This expression accounts
for the mean speed as a function of steepness and discreteness parameters.
11

t
x
n
Speed Amplitude
time
time
b)
c)
n
50 150 250 350
2.7495
2.7502
2.7509
2.7514
2.5
1
0
01
180 200 220
0
5
10
15
20
a)
0.2
0
0.4
0.6
0.8
1
120 140 160 180 200 220
0.2 0.4 0.6 0.8
0.5
1.5
2
0.8
0.2
0.5
Figure 6: Front propagation into an unstable state in discrete FKPP
model (8). a) Spatiotemporal evolution of the front propagation into an
unstable state in discrete FKPP model (8) with dx = 7.5. b) Trajectory of
three different points or cuts: Above (upper yellow line), in (middle red line),
and below (lower blue line) the front position. Inset illustrates different cuts
under consideration. c) Oscillation amplitude of the front speed in different
points.
Note that hvitends to expression (6) when dx →0 (β→0), which corre-
sponds to the continuous limit. Figure 7shows the mean speed as a function
of the parameter kfor different values of the discretization parameter dx.
For different values of discretization parameter dx,hviis a concave function.
We can observe, that the minimum speed hvimin increases as the discretiza-
tion parameter dx grows. Meanwhile, the critical steepness kcdecreases. By
differentiating the mean speed relation (15) and equating to zero, we obtain
an expression for the discretization parameter,
dx2= 4 sinh βc(2βccosh β−sinh βc),(16)
where βc=kcdx/2 and kcis the critical steepness to obtain the minimum
speed. Replacing the definition of βcin above expression, we get
dx2= 4 sinh kcdx
2kcdx cosh kcdx
2−sinh kcdx
2.(17)
One can not explicitly determine the critical steepness as fa unction of dis-
cretization parameter dx,kc(dx). Hence, minimum speed as a function of dx
12

k
<V>
<V>min
0
0
21 3 4 5 6 7
8
6
4
2
Figure 7: Mean speed hvias a function of the steepness parameter kfor
different values of the discretization parameter dx, formula (15). From the
lower to upper curve we consider dx = 0,2,5 and 9, respectively.
is a implicit formula
hvimim =1
kc(dx)+kc(dx)sinh β(dx)
β(dx)2
.(18)
The continuos curve in Fig. 5a is the minimal mean speed as a function of
the discretization, expression (18). Numerical simulations show quite good
agreement with this expression (cf. Fig. 5a).
To have an explicit analytical expression we consider the limit β→0,
thus expression (19) can be simplified to
dx ≈2βcp1 + β2
c.(19)
From here, we can write the parameter βc
βc≈dx
q21 + √1 + dx2
and the critical steepness
kc≈s2
1 + √1 + dx2(20)
as a function of the discretization parameter dx. Figure 8shows the dis-
cretization parameter dx as a function of β(Eq. (19)). The shadow area
13

dx
β
Steepness
dx
b)
a)
0
0
4
8
0
0.5
12108642
0.3
0.7
0.9
1.1
12
21.51.00.5
Figure 8: (color online) Parameter as function of discretization. a) Dis-
cretization as a function of βparameter. Solid (blue) and dashed (yellow)
lines are the exact and approximative analytic curves (17) and (19), respec-
tively. b) Steepness kas a function of the discretization parameter, dx. Solid
and dashed lines are the exact and approximative analytic curves (17) and
(20), respectively. Dots (blue) are obtained by numerical simulations.
illustrates the limit where the approximation β→0 is valid. Likewise, fig-
ure (8) shows the steepness kas a function of the parameter dx. From both,
we can infer that expressions (19) and (20) are valid in a wide range of the
parameter dx. Therefore, in a good approximation the mean speed hvican
take the form,
v≈s1 + √1 + dx2
2
1 + 4
dx2sinh2

dx
q21 + √1 + dx2

.(21)
Figure 5shows the mean speed as a function of the discretization param-
eter dx. Up to a value of dx = 6.0, expression (21) is an adequate approxi-
mation. We observe a good accordance between the analytic expression and
the mean speed obtained by numerical simulations.
In brief, the asymptotic procedure allows an adequate characterization
of the average features of front propagation into unstable states in discrete
media. In the next section, we shall apply this procedure to characterize
the mean properties of front propagation into unstable state in a chain of
dissipative coupled pendula.
14

3.4. Theoretical description of the mean speed for the Chain of dissipative
coupled pendula
For the chain of dissipative coupled pendula, Eq. (1), the unstable state
correspond to θi=π/2. Considering the asymptotic ansatz for the front tail
around this state, we get,
θi(t) = π
2+A0e(αt−2iβ)1 + fω
dx;i(t),(22)
where A0is a constant that characterizes the shape of the front tail, α≡khvi
and β≡k dx/2 are parameters. fω
dx;i(t) is a periodic function of frequency
ωin i-th position of the chain that describes the oscillatory behavior of the
speed. Introducing the above ansatz in Eq. (1) and taking into account only
the linear leading terms, we obtain,
¨
θi=α21 + fω
dx;i(t)+ 2α˙
fω
dx;i(t) + ¨
fω
dx;i(t)
=ω21 + fω
dx;i(t)−µhα1 + fω
dx;i(t)+˙
fω
dx;i(t)i+4
dx2sinh2(β)1 + fω
dx;i(t)
Integrating this expression in a normalized period T= 2π/ω, and considering
hfω
dx;i(t)i=h˙
fω
dx;i(t)i=h¨
fω
dx;i(t)i= 0, after straightforward calculations, we
obtain
α2=ω2−µα +4
dx2sinh2(β).(23)
Substituting the definition of α, the mean speed reads
hvi=−µ
k+1
ksµ2+ω2+k2
β2sinh2(β),(24)
and replacing k= 2β/dx,
hvi=−µ
2βdx +1
2βqdx2(µ2+ω2) + 4 sinh2(β) (25)
The above expression accounts for front speed as a function of the steepness.
In order to deduce the minimal front speed, we differentiate the above speed
with respect to β
ω2µ2+ω2dx4+ 4 µ2+ 2ω2sinh2(β)−2µ2+ω2βsinh(β) cosh(β)dx2
+ 16 sinh2(β) [sinh(β)−βcosh(β)]2= 0.
(26)
15

This expression gives us a relation between the critical steepness kcand the
coupling parameter dx. An explicit expression kc(dx) cannot be derived.
Using expression (26) in formula (25), we obtain the minimal front speed for
the chain of dissipative coupled pendula, Eq (1). Note that this analytical
results has quite fair agreement with the numerical simulations as it is shown
in Fig. 2a. Therefore, the asymptotic procedure is a suitable method to
characterize the mean properties of front propagation.
4. Effective continuous model: oscillatory properties of front pro-
pagation
Due to the complexity of discrete dissipative systems, to obtain analytical
results is a daunting task. In order to figure out the oscillatory behavior of
the front, we shall consider a similar strategy to that used in Ref. [24], which
is based on considering an effective continuous equation that accounts for
the dynamics of the discrete system. The benefit of this approach is that
analytical calculations are accessible.
4.1. Generalized Peierls-Nabarro potential
Let us consider the continuos order parameter u(x, t), which satisfies
∂tu=−δF
δu ,(27)
where the Lyapunov functional has the form
F=Z−u2
2+u3
3+(∂xu)2
2+ (∂xu)2Γdx(x)dx, (28)
Γdx(x) is a spatial periodic function with dx period, Γdx(x+dx)=Γdx(x).
This function accounts for the discreteness of the system. The last term
of the free energy is a generalization of the Peierls-Nabarro potential. An
effective potential has been used to explain the dynamics of defects position
such as dislocations in condensed matter physics or dynamics of the position
of kink or fronts (see textbook [25] and reference therein). Here, we consider
an effective equation for the entire field u(x, t), which reads
∂tu=u(1 −u) + D∂xxu+ 2Γdx(x)∂xxu+ 2Γ0
dx(x)∂xu. (29)
This equation is a populations dynamical model with linear growth, nonlinear
saturation, inhomogeneous diffusion and drift force. Numerical simulations
16

v
time
c)
0 4000 8000
530
540
550
560
x0
04000 8000
1
3
5
7
time
b)
a)
40 60 80 100
0
0.2
0.4
0.6
0.8
1
0
2
4
6
8
10
12
14
time
Figure 9: (color online) Front propagation into an unstable state in FKPP
Eq. (29) with a harmonic generalized Peierls-Nabarro potential, Γdx(x) =
Acos(2πx/dx) with A= 0.06, and dx = 5.0. The numerical discretization
parameter of the finite differences method is 0.1. a) Spatiotemporal evolution
of the front propagation into unstable state. b) Temporal evolution of the
front position and c) minimum speed.
with a harmonic potential Γdx exhibit front solutions. It is important to
note that for these numerical simulations, we have discretized the Laplacian
and gradient of uto first neighbors considering a small dx. For which the
discreteness effects are negligible. Figura 9shows the spatiotemporal diagram
of the front into an unstable state of the effective FKPP, Eq. (29), with a
harmonic generalized Peierls-Nabarro potential. Its trajectory and speed
are also illustrated. We can observe that the numerical simulations of the
effective FKPP, Eq. (29), and discrete FKPP, Eq. (8), have similar qualitative
dynamical behaviors.
To understand better the generalized Peierls-Nabarro potential, Figure 11a
shows the effective force for the harmonic case and the amplitude speed for
the effective FKPP Eq. (29). From this figure, we infer that the effective
force, f≡2Γdx(x)∂xxu+ 2Γ0
dx(x)∂xu, has an oscillatory structure concen-
trated in the region where the front displays larger spatial variations. More-
over, we observe that the structure of the amplitude of the speed is similar
17

to that observed in the discrete case (cf. Figs. 11b and 6c).
0.12
0.13
0.11
0.1
10.80.60.40
-2
-1
0
1
2
u
Speed Amplitude
x
u(x,t)
f(x,t)
b)
630600570
a)
0.2
Figure 10: The generalized Peierls-Nabarro force for a harmonic case,
Γdx(x) = Acos(2πx/dx). a) Front solution and effective force f≡
2Γdx(x)∂xxu+2Γ0
dx(x)∂xu. b) Amplitude of the speed for the effective FKPP
Eq. (29) with D= 1.97, A= 0.03, and dx = 5.0. The numerical discretiza-
tion parameter of the finite differences method is 0.1.
4.2. Dynamics of front position
The equilibria are not affected by the presence of the periodical extra
terms, effective force. In the continuous limit, dx →0 and Γdx(x)→0 one
recovers the Fisher-Kolmogorov-Petrosvky-Piskunov model Eq. (4). Then
for dx 1, the last two terms of Eq. (29) are perturbative. We shall analyze
this region of parameters, where we can obtain analytical results.
The Fisher-Kolmogorov-Petrosvky-Piskunov model Eq. (4) has front so-
lutions of the form uF KP P (x−vt −p), where pis a constant that accounts
for the front position and vthe front speed. Analytical expressions of this
solution are unknown, however solutions in the form of perturbative series
are available [18]. Considering the following ansatz for small discreteness
(dx 1)
u(x, t) = uF KP P (x−vt −p(t)) + w(x−vt −p(t), p(t)),(30)
where front position is promoted to a temporal function, p(t) and wis a cor-
rective function on the order of the perturbative force. Introducing the above
ansatz in Eq. (29) and linearizing in w, after straightforward calculations, we
obtain
Lw=−˙p(t)∂ξuFK P P −2Γdx(x)∂ξξuF K P P −2Γ0
dx(x)∂ξuF K P P ,(31)
18

where L ≡ ∂ξξ +v∂ξ+ 1 −2uF K P P (ξ) is a linear operator and ξ=x−vt −p
is the coordinate in the co-mobile system. Considering the inner product
hf|gi=ZL
−L
f(ξ)g(ξ)dξ, (32)
where 2Lis the system size. In order to solve the linear Eq. (31), we apply
the Fredholm alternative or solvability condition [2], and obtain
˙p(t) = −2hΓdx(ξ+vt +p)∂ξξ uF KP P |ψi
h∂ξuF KP P |ψi−2hΓ0
dx(ξ+vt +p)∂ξuFK P P |ψi
h∂ξuF KP P |ψi,
(33)
where ψ(ξ) is an element of kernel of adjoint of L,L†≡∂ξξ −v∂ξ+ 1 −
2uF KP P (ξ), that is L†ψ= 0. The ψfunction is unknown analytically, how-
ever the asymptotic behavior of this function are characterized to diverges
exponentially with the the same exponent that uF KP P converges to their
equilibria. Therefore the above integrals diverge proportional to L, however
the ratio is well defined.
To understand the dynamics described by the above equation for simplic-
ity we shall consider the generalized Peierls-Nabarro potential for a harmonic
case, that is,
Γdx(x) = γ(x)≡Acos 2πx
dx ,(34)
Replacing this expression in Eq. (33), after straightforward calculations, we
obtain
˙p(t) = qK2
1+K2
2cos 2π
dx(p−vt) + φ0,(35)
with
K1=Acos 2πξ
dx ∂ξξ uF KP P (ξ)−2π ξ
dx sin 2πξ
dx ∂ξuF KP P |ψ(ξ)
h∂ξuF KP P |ψ(ξ)i,
K2=−Asin 2πξ
dx ∂ξξ uF KP P (ξ) + 2πξ
dx cos 2πξ
dx ∂ξuF KP P |ψ(ξ)
h∂ξuF KP P |ψ(ξ)i,
tan(φ0) = K1
K2
.(36)
Therefore, the front position propagates in a oscillatory manner. Notice that
the Peierls-Nabarro potential propagates together with the front. Figure 11
19

shows fitting curve for ˙pusing solution of (35) if Γdx(x) is given by (34).
We can see that the analytical result is in good agreement with the observed
dynamics.
1100 1200 1300
0.8
1
1.2
1.4
1.6
p
.
time
Figure 11: Fitting curve for ˙pgiven by the expression ˙p=asec2(bx+c)
1+(dtan(bx+c)−e)2+f
with a= 0.57, b= 0.0682, c= 0.6157, d= 1, e= 0.8, and f= 0.5
5. Conclusions and remarks
We have studied, theoretically and numerically, the front propagation
into unstable states in discrete dissipative systems. Based on a paradig-
matic model of coupled chain of oscillators (the dissipative Frenkel-Kontorova
model) and population dynamic model (the discrete and effective Fisher-
Kolmogorov-Petrosvky-Piskunov model), we have determined analytically
the mean speed of FKPP fronts when nonlinearities are weak. Numerically
we have characterized the oscillatory front propagation. Likewise, we have
revealed that different parts of the front oscillate with the same frequency
but with different amplitude. To describe this phenomenon, we have gener-
alized the notion of the Peierls-Nabarro potential, which allows us to have
an effective continuous description of discreteness effect.
The analysis presented only is valid for weak nonlinearity, where linear
criterium is valid. The characterization of pushed front in local coupling
dissipative systems is an open and relevant question. Propagation fronts
in two dimensions is affected by the curvature of the interface, which can
increase o decrease the speed of the propagating interface. Study of front
propagation into unstable states in these contexts are in progress.
20

Acknowledgments
M.G.C., M.A.G-N., and R.G.R thank for the financial support of FONDE-
CYT projects 1150507, 11130450, and 1130622, respectively. K.A-B. was
supported by CONICYT, scholarship Beca de Doctorado Nacional No.21140668.
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