Autowaves in the moving excitable media
Within the framework of kinematic theory of autowaves we suggest a method for analytic description of stationary autowave structures appearing at the boundary between the moving and fixed excitable media. The front breakdown phenomenon is predicted for such structures. Autowave refraction and, p...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Autowaves in the moving excitable media / V.A. Davydov, N.V. Davydov, V.G. Morozov, M.N. Stolyarov, T.Y amaguchi // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 565–578. — Бібліогр.: 16 назв. — англ. |
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Davydov, V.A. Davydov, N.V. Morozov, V.G. Stolyarov, M.N. Yamaguchi, T. 2017-06-03T04:19:18Z 2017-06-03T04:19:18Z 2004 Autowaves in the moving excitable media / V.A. Davydov, N.V. Davydov, V.G. Morozov, M.N. Stolyarov, T.Y amaguchi // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 565–578. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 05.50.+q, 05.70.Ln, 82.40.-g, 87.10.+e DOI:10.5488/CMP.7.3.565 https://nasplib.isofts.kiev.ua/handle/123456789/119027 Within the framework of kinematic theory of autowaves we suggest a method for analytic description of stationary autowave structures appearing at the boundary between the moving and fixed excitable media. The front breakdown phenomenon is predicted for such structures. Autowave refraction and, particulary, one-side “total reflection” at the boundary is considered. The obtained analytical results are confirmed by computer simulations. Prospects of the proposed method for further studies of autowave dynamics in the moving excitable media are discussed. В рамках кінематичної теорії автохвиль запропоновано метод для аналітичного опису стаціонарних автохвильових структур, що виникають на межі між рухомим та нерухомим збудженими середовищами. Передбачено явище розриву фронту автохвилі в таких структурах. Розглядається заломлення автохвилі та одностороннє “повне відбиття”на границі. Отримані аналітичні результати знайшли своє підтвердження при комп’ютерному моделюванні. Обговорюються перспективи застосування запропонованого методу для подальшого дослідження автохвильової динаміки в рухомих середовищах, що збуджуються. This work was partially supported by U.S.CRDF-RF Ministry of Education Award VZ–010–0. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Autowaves in the moving excitable media Автохвилі в рухомих збуджених середовищах Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Autowaves in the moving excitable media |
| spellingShingle |
Autowaves in the moving excitable media Davydov, V.A. Davydov, N.V. Morozov, V.G. Stolyarov, M.N. Yamaguchi, T. |
| title_short |
Autowaves in the moving excitable media |
| title_full |
Autowaves in the moving excitable media |
| title_fullStr |
Autowaves in the moving excitable media |
| title_full_unstemmed |
Autowaves in the moving excitable media |
| title_sort |
autowaves in the moving excitable media |
| author |
Davydov, V.A. Davydov, N.V. Morozov, V.G. Stolyarov, M.N. Yamaguchi, T. |
| author_facet |
Davydov, V.A. Davydov, N.V. Morozov, V.G. Stolyarov, M.N. Yamaguchi, T. |
| publishDate |
2004 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Автохвилі в рухомих збуджених середовищах |
| description |
Within the framework of kinematic theory of autowaves we suggest a method
for analytic description of stationary autowave structures appearing at the
boundary between the moving and fixed excitable media. The front breakdown
phenomenon is predicted for such structures. Autowave refraction
and, particulary, one-side “total reflection” at the boundary is considered.
The obtained analytical results are confirmed by computer simulations.
Prospects of the proposed method for further studies of autowave dynamics
in the moving excitable media are discussed.
В рамках кінематичної теорії автохвиль запропоновано метод для аналітичного опису стаціонарних автохвильових структур, що виникають на межі між рухомим та нерухомим збудженими середовищами. Передбачено явище розриву фронту автохвилі в таких структурах. Розглядається заломлення автохвилі та одностороннє “повне відбиття”на границі. Отримані аналітичні результати знайшли своє підтвердження при комп’ютерному моделюванні. Обговорюються перспективи застосування запропонованого методу для подальшого дослідження автохвильової динаміки в рухомих середовищах, що збуджуються.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119027 |
| citation_txt |
Autowaves in the moving excitable media / V.A. Davydov, N.V. Davydov, V.G. Morozov, M.N. Stolyarov, T.Y amaguchi // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 565–578. — Бібліогр.: 16 назв. — англ. |
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2025-11-24T02:23:59Z |
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| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 3(39), pp. 565–578
Autowaves in the moving excitable
media
V.A.Davydov 1 , N.V.Davydov 1 , V.G.Morozov 1 ,
M.N.Stolyarov 2 , T.Yamaguchi 3
1 Moscow State Institute of Radioengineering, Electronics and Automation,
117454, Vernadsky Prospect 78, Moscow, Russia
2 Department of theoretical physics, Physical Institute of Russian Academy
of Sciences, 117924, Leninsky Prospect 53, Moscow, Russia
3 National Institute of Advanced Industrial Science and Technology (AIST),
Higashi, 1–1–1, Tsukuba, Ibaraki 305–8565, Japan
Received April 5, 2004
Within the framework of kinematic theory of autowaves we suggest a method
for analytic description of stationary autowave structures appearing at the
boundary between the moving and fixed excitable media. The front break-
down phenomenon is predicted for such structures. Autowave refraction
and, particulary, one-side “total reflection” at the boundary is considered.
The obtained analytical results are confirmed by computer simulations.
Prospects of the proposed method for further studies of autowave dynam-
ics in the moving excitable media are discussed.
Key words: nonlinear dynamics and nonlinear dynamical systems,
reaction-diffusion media, autowave refraction
PACS: 05.50.+q, 05.70.Ln, 82.40.-g, 87.10.+e
1. Introduction
There are many physical, chemical, and biological systems which may be re-
garded as excitable media consisting of locally connected active elements capable of
forming pulses [1]. The most intriguing property of excitable media is the existence of
autowaves, i.e., nonlinear spatiotemporal structures propagating through the medi-
um. Autowave patterns is one of the most dramatic examples of self-organization in
macroscopic systems [1,2].
Over many years, most attention has been concentrated on the study of au-
towave processes in plane homogeneous media. More recently, however, the trend
has been toward the study of autowaves in anisotropic, curved, inhomogeneous, and
non-stationary excitable media. This range of problems is of great importance in
practice. Using feed-back techniques, such as enhancing catalytic efficiencies, fo-
c© V.A.Davydov, N.V.Davydov, V.G.Morozov, M.N.Stolyarov, T.Yamaguchi 565
V.A.Davydov et al.
cused laser light, imposing gradients of excitability, etc., it is possible to control
autowave propagation. Another promising way of controlling the autowave patterns
is based on the fact that, in the moving excitable media, gradients of concentration
of different species, velocity gradients, and temperature gradients can dramatically
modify the behavior of autowaves. Our aim in the present paper is to consider some
new effects in the moving excitable media.
Distributed excitable systems are usually described by a set of reaction-diffusion
equations
∂U
∂t
= F (U) + D̂ · ∆U , (1.1)
where U(r, t) is a vector whose components U
n
give spatial concentrations of re-
acting species. The source term F specifies the local reactions in small volumes of
the medium, and D̂ is the matrix of transport (diffusion or thermoconductivity)
coefficients. Equations (1.1) may be viewed as a “microscopic” model of excitable
media.
Unfortunately, general methods for the analytic solution of equations (1.1) are
not available. Therefore, these equations serve as a basis for numerical and other ap-
proximate methods. In many realistic reaction-diffusion systems, the local curvature
k of wave fronts is so small that the curvature radius R = 1/k is much larger than
the wave width a. In such cases a wave can be represented by a smooth infinitely
thin oriented curve. This approximation is fundamental for the so-called kinematic
approach (see, e.g., [3,4]) which is successfully employed in the studies of large-scale
phenomena in 2D and 3D excitable media. In the region of its applicability the kine-
matic approach is proved to give results which are in good agreement with those
obtained by numerical calculations of reaction-diffusion equations (1.1). It should
be noted that the kinematic approach offers some advantages over direct numerical
schemes. First, in many cases the results can be obtained in analytic form. This pro-
vides new and interesting insights into the behavior of autowave structures. Second,
qualitative predictions of the kinematic approach are “universal” in a sense that
they refer to any reaction-diffusion excitable medium.
Autowaves in the moving excitable media, essentially under shear flow, were
studied by many authors (for references see, e.g., recent papers [5,6]). The most
common theoretical approach in all these studies is based on numerical solution of
reaction-diffusion equations (1.1) with some specific expressions for generic reactivity
functions in the source term F . Within the kinematic approach, the problem of
autowave structures in the moving media was posted for the first time in [7]. In
the present paper the method outlined in [7] will be used to describe stationary
autowave structures appearing at a boundary between the moving and fixed 2D
excitable media.
The paper is organized as follows. Section 2 reviews stationary autowave struc-
tures in a homogeneous excitable medium. Section 3 turns to stationary structures
appearing at a boundary between the moving and fixed media with tangential veloc-
ity discontinuity. We show that such structures can be represented in terms of the
stationary excitation fronts in an infinite homogeneous medium. Critical properties
566
Autowaves in the moving media
of autowave structures at the boundary are discussed. Section 4 deals with autowave
refraction at the boundary between semi-infinite moving and fixed media. Section 5
presents the results of computer simulations confirming theoretical predictions. Sec-
tion 6 concludes with remarks on further applications of the kinematic method to
the studies of autowave structures in excitable media with penetrable boundaries.
2. Non-spiral autowave structures in homogeneous media
We start with basic kinematic equations that are necessary for our discussion.
As already noted, the kinematic description reduces an autowave structure to
an infinitely thin excitation front. At any time t this front can be specified by its
natural equation k(l, t) which gives the curvature k as a function of the path length
l measured from a certain point on the front. Assuming that each small segment
of the front moves in its normal direction with some local velocity V (l, t), simple
considerations lead to the evolution equation [3,4]
∂k
∂t
+
∂k
∂l
l
∫
0
kV dξ + C
+ k2V +
∂2V
∂l2
= 0, (2.1)
where C is the growth velocity of free tips of the front. The kinematic approach
supposes that V and C depend on l through the local curvature. For k considerably
smaller than the inverse width of the excitation pulse, this dependence can be taken
as linear. The relation for the local velocity has the form
V = V
0
− Dk, (2.2)
where V
0
is the velocity of the plane front, and D is the diffusion coefficient. Equa-
tion (2.2) is known in the literature as the eikonal equation [8].
In this paper we will be interested in steady-state autowave structures moving
with constant shape and velocity. In such cases ∂k/∂t = 0 and C = 0. Then, after
one integration over l, equation (2.1) becomes
k
l
∫
0
kV dξ +
dV
dl
= 0. (2.3)
Generally speaking, the right-hand side of this equation is equal to an integration
constant ω which has the meaning of the angular speed of rotation of the front [4].
In what follows we will consider only non-spiral autowave structures, so that we put
ω = 0.
Before proceeding to a study of the moving excitable media with boundaries
we shall touch briefly on stationary autowave structures in an infinite homogeneous
medium. In this case equation (2.3) is exactly integrable if the local velocity is given
by (2.2). The solutions were fully considered in [9,10]. Here we run through the
structures available for different values of the curvature k(0) at the point l = 0.
567
V.A.Davydov et al.
Figure 1. Non-spiral stationary autowave structures in a homogeneous excitable
media: a) V -shaped pattern; b) self-intersecting loop; c) separatrix; d) multi-loop.
First we note that equation (2.3) has two trivial solutions: k = 0 (a plane front)
and k = V
0
/D (a stationary circle). Let us list the non-trivial steady-state configu-
rations:
(i) For k(0) < 0, equation (2.3) describes the so-called V-shaped patterns (figu-
re 1a). The explicit expression for k(l) is given by
k(l) = −
1
D
V 2(0) − V 2
0
V
0
+ V (0) cosh
[(
√
V 2(0) − V 2
0
/D
)
l
] , (2.4)
where V (0) = V
0
− D k(0). It is notable that the stable V-shaped patterns in
excitable media were first predicted within the kinematic approach [9]. Short-
ly thereafter they were observed experimentally in the Belousov-Zhabotinsky
(BZ) reaction [11].
(ii) For k(0) > 2V
0
/D, the stationary solution of equation (2.3) is the self-intersecting
loop (figure 1b):
k(l) =
1
D
V 2(0) − V 2
0
V (0) cosh
[(
√
V 2(0) − V 2
0
/D
)
l
]
− V
0
. (2.5)
(iii) In the case that k(0) = 2V
0
/D, equation (2.3) gives
k(l) =
2V
0
/D
1 + (V
0
l/D)2
. (2.6)
Following [10], we shall call this configuration (figure 1c) the separatrix.
568
Autowaves in the moving media
(iv) Finally, for 0 < k(0) < V
0
/D and V
0
/D < k(0) < 2V
0
/D, the solution of
equation (2.3),
k(l) = −
1
D
V 2(0) − V 2
0
V
0
+ V (0) cos
[(
√
V 2(0) − V 2
0
/D
)
l
] , (2.7)
represents the multi-loop shown in figure 1d.
A few comments should be made here about the above-listed autowave struc-
tures. Using the method developed in [4], it can be shown that all these structures
are stable in a sense that their profiles are recovered after small perturbations.
Note, however, that structures with self-crossing cannot propagate in homogeneous
infinitely extended media because autowave pulses annihilate after collision. This ef-
fect is not described by the kinematic equation (2.1). Thus, the only stable non-spiral
autowave structures in infinitely extended media are plane fronts and V-shaped pat-
terns. Nevertheless, in the next section we will show that non-self-crossing branches
of the other structures can exist near a boundary between the moving and fixed
excitable media.
3. Stationary autowave structures at a boundary between the
moving and fixed media
In order to gain insight into characteristic properties of autowave structures in
the moving media, we will consider a homogeneous medium in a strip of width 2h
which is infinitely extended along the X-axis. Zero-flux conditions will be used for
both boundaries of the strip (i. e., there is no diffusion of the reacting substances
through the boundaries). Suppose also that the strip was cut into two strips each of
width h, and the upper one moves with velocity w along the X-axis (see figure 2).
This system with the tangential velocity discontinuity can easily be realized exper-
imentally and, besides, it may serve as a basis for the study of autowave structures
in more complicated systems, say, in excitable media with shear flow.
Figure 2. A stationary autowave structure at the boundary between moving and
fixed excitable media. (A) is a branch of the separatrix; (B) is a branch of the
V -shaped pattern. The wave moves along the relative velocity of the media w.
569
V.A.Davydov et al.
Our aim now is to search for steady-state autowave structures which can be gen-
erated in the system described above. It is clear that an excitation front propagating
along the X-axis with constant velocity must be constructed from branches of the
stationary solutions of equation (2.3) listed in the previous section. This alleviates
the problem.
We shall restrict our consideration to the case where the width of the strip h
is very much larger than the characteristic length l
0
= D/V
0
which determines the
size of the separatrix loop, equation (2.6). Taking the values D = 2 · 10−5 cm2/c
and V
0
= 3 mm/min, which are typical of the BZ reaction, we obtain l
0
= 0.04 mm.
This estimate shows that the case where h ≈ l
0
is very unlikely to be experimentally
realizable.
Since the upper and lower boundaries of the medium are assumed to be imper-
vious to diffusion, an autowave front must be orthogonal to them (see, e.g., [4]).
Then it can be easily verified that in the case h � l
0
the only steady-state autowave
which can propagate along the X-axis is the solution of equation (2.3) constructed
from the branch of the separatrix (in the moving medium) and the “half” of the
V -shaped pattern (in the fixed medium). This solution is pictured in figure 2.
Note that the excitation front shown in figure 2 moves as a whole along the
X-axis with velocity V
0
+ w because its part in the upper strip is the branch of
the separatrix (see figure 1c) and h � l
0
. Taking l = 0 at the tip of the structure
on the lower boundary of the fixed medium, we find that for the V -shaped pattern
V (0) = V
0
+ w. This condition, together with equation (2.2), determines uniquely
the excitation front in the fixed medium. The essential point is that the negative
curvature of almost all front line of a V -shaped pattern, equation (2.4), is close to
zero and only near the vertex it becomes very large (see, e.g., [9,11]). Thus, due to
the condition h � l
0
, the front of the V -pattern near the boundary between the
media in figure 2 is nearly a straight line. The angle β between this front and the
boundary can be easily determined by the parameter V (0) as [11]
sin β = V
0
/V (0) = V
0
/ (V
0
+ w) . (3.1)
The curvature kb of the front in the moving medium near the boundary differs
from zero and, as shown below, essentially depends on the velocity w. To calculate
k
b
, we note that the sewing condition requires that the velocities of the V -pattern
and the separatrix along the X-axis should be the same at the boundary between the
media. Let V
b
be the corresponding normal velocity of the front in the upper strip
(in the co-moving frame of reference). Then, taking into account that the asymptotic
normal velocity of the V -pattern must be equal to the velocity of the plane front,
the sewing condition can be written as
V
0
/ sin β = V
b
/ sin β + w. (3.2)
Since V
b
and k
b
satisfy the eikonal equation (2.2), from (3.1) and (3.2) it follows
that
k
b
=
1
D
V
0
w
V
0
+ w
. (3.3)
570
Autowaves in the moving media
This result, together with equation (2.6), determines the value l = l
b
on the bound-
ary and, consequently, the shape of the front in the moving medium.
Figure 3. The same as in figure 2, but the wave moves in the direction opposite
to the relative velocity of the media w.
Equation (3.3) shows that the curvature k
b
of the front in the moving media near
the boundary is a monotonically increasing function of w. This leads to a remarkable
effect which does not occur when linear waves (for instance, electromagnetic waves)
are refracted on a boundary between the moving and fixed media. It is known that,
for autowave fronts, there exists a critical curvature k∗ (k∗ < V
0
/D), above which
stable propagation becomes impossible [12,13]. Turning to the situation shown in
figure 2, we may state that for some critical velocity w∗ the steady-state structure
described above becomes unstable. The value of w∗ follows from (3.3) if we set there
k
b
= k∗:
w∗ =
V
0
V
0
/(Dk∗) − 1
. (3.4)
For w > w∗, the excitation front will be broken at the boundary, so that the au-
towaves in the moving and fixed media will propagate independently of one another.
We have considered the case where the autowave structure moves in the direction
of the relative velocity w of the media. If the excitation front moves in the opposite
direction, then the only possible steady-state configuration is constructed from the
“half” of a V -shaped pattern in the moving medium and the branch of the separatrix
in the fixed medium (see figure 3). Following the same line of reasoning as before, it is
easy to derive the stationary front shape and the critical velocity w∗. It is interesting
to note that the corresponding expressions coincide with those cited above.
4. Refraction of autowaves
In the limit that h → ∞ in figure 2, we deal with a plane boundary between the
moving and fixed semi-infinite excitable media. Stationary solutions of equation (2.3)
will now describe refraction of autowaves at the boundary.
Clearly the structures considered in the previous section (figure 2 and figure 3)
are stable as before if the curvature (3.3) does not exceed the critical value k∗. The
only new point is that now the vertex of the V -pattern goes to infinity and the
corresponding branch of the steady-state structure becomes a real plane front.
571
V.A.Davydov et al.
Note, however, that in the case of semi-infinite media new stable stationary
structures are possible. They are constructed from the plane front and the non-self-
crossing branch of the loop (see figure 4).
Figure 4. Autowave refraction at the boundary between the moving and fixed
excitable media.
Using the natural equation for the loop, equation (2.5), and recalling the argu-
ments from the previous section, it is an easy matter to give an exhaustive analysis
of autowave refraction. For the structure shown in figure 4a, as an example, we
obtain the refraction law
sin β =
sin α
1 + w sin α/V
0
, (4.1)
where α is the angle between the boundary and the asymptote of the loop front in
the moving medium (“incidence angle”), while β is the angle between the boundary
and the plane front in the fixed medium (“angle of refraction”). The curvature of
the loop branch at the boundary is derived from equations (2.5) and (4.1):
k
b
=
1
D
V
0
w sin α
V
0
+ w sin α
. (4.2)
Finally, for a given α, the continuous excitation front shown in figure 4a breaks at
the boundary when the relative velocity of the media w exceeds the critical value
w∗(α) =
w∗
sin α
, (4.3)
Figure 5. A schematic picture of the one-side “total reflection” of autowaves at
the boundary between the excitable media.
572
Autowaves in the moving media
where w∗ is given by equation (3.4) and has the meaning of the critical velocity for
α = π/2. The steady-state structure shown in figure 4b can be analyzed in a similar
fashion.
In closing, we briefly describe an interesting phenomenon which can be observed
on a boundary between the moving and fixed excitable media. In a sense this phe-
nomenon may be called the one-side “total reflection” of autowaves. Suppose that a
circular autowave was excited at some point “a” in the fixed medium (figure 5). For
simplicity, the centre of the wave is assumed to be far removed from the boundary
so that the curvature dependence of the front velocity near the boundary can be
neglected. When the front reaches the boundary it generates the wave in the mov-
ing medium. The sources of this secondary wave are two intersection points of the
incident wave and the boundary. One point is to the left, and the other is to the
right of a vertical axis Y passing through the centre of the incident wave (figure 5).
It is easy to verify that the intersection points move relative the fixed medium with
velocity
V
c
= V
0
/ sinα, (4.4)
where α is the angle between the Y -axis and the ray traced from the centre “a”
to the intersection point. Since the angle α increases with time, the velocity V
c
monotonically decreases.
The evolution of a wave front in the moving medium essentially depends on
whether its points are to the left or to the right of the Y -axis. First we consider
the front to the right of the Y -axis. Sooner or later, the velocity of the intersection
point of the incident wave becomes equal to V
0
+ w. As follows from (4.4), it occurs
for α = α
0
, where
sin α
0
=
V
0
V
0
+ w
. (4.5)
From this time on the right intersection point will not generate the wave in the
moving medium. In contrast, a “reflected” wave will be generated in the fixed medi-
um. As a result, the wave front arranges itself into the form shown in figure 5. It
should be noted that the angle between the “reflected” wave and the Y -axis at the
boundary will not vary with time and will be equal to α
0
given by equation (4.5).
Clearly the velocity of the left intersection point in figure 5 will never become
smaller than the velocity of the wave front in the moving medium. Therefore no
“reflected” wave is generated at this point and, as time elapses, the wave front
near the boundary takes the shape shown in figure 4b. The resulting structure is
illustrated in figure 5.
By analogy with optics, the above effect may be called the one-side “total re-
flection” of autowaves at a boundary between moving and fixed excitable media.
It is interesting to note that a similar effect was observed at a boundary between
excitable media with different diffusion coefficients [14].
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V.A.Davydov et al.
5. Computer simulations
The above considerations were based on the kinematic equation (2.1) and the
eikonal equation (2.2), which may be regarded as a phenomenological approach
describing the main features of the “microscopic” model of excitable media, equa-
tions (1.1). It is of interest to check our analytic results by numerical solution of
reaction-diffusion equations.
We have performed computer simulations of equations (1.1) for a two-variable
excitable medium [3]:
∂u
∂t
= F
u
(u, v) + D
u
∆u,
∂v
∂t
= εF
v
(u, v) + D
v
∆u, (5.1)
where u and v are the activator and inhibitor variables respectively, D
u
and D
v
are the diffusion coefficients of both species. The formal parameter ε determines
the time scale for the inhibitor relaxation. We used the model with piecewise linear
generic reactivity functions F
u
and F
v
(see, e.g., [13]):
F
u
(u, v) = f(u) + v, F
v
(u, v) =
{
k
v
u − v, k
v
u − v > 0,
k
ε
(k
v
u − v) , k
v
u − v < 0 ,
(5.2)
where
f(u) =
−k
1
u, u < σ
k
u
(u − a) , σ 6 u 6 1 − σ
k
2
(1 − u), 1 − σ < u ,
(5.3)
k
1
=
a − σ
σ
k
u
, k
2
=
1 − σ − a
σ
k
u
. (5.4)
Constants a, σ, k
u
, k
v
, k
ε
are free parameters of the model. In all simulations the
values of these parameters were
a = 0.1, σ = 0.01, k
u
= 1.7, k
v
= 2, k
ε
= 6. (5.5)
The dimensionless activator diffusion coefficient was D
u
= 1.5 ·10−4. Since the main
objective of computer simulations was to test the qualitative properties of autowaves
discussed in the previous sections, we have restricted ourselves to the simplest model
with D
v
= 0.
Equations (5.1) were integrated in a 2D array of 128× 128 elements using the
Runge-Kutta method with the following boundary conditions:
(i) Zero-flux boundary conditions for the reagents at the top and bottom of the
medium;
(ii) Periodic boundary conditions at the left and right boundaries of the domain;
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Autowaves in the moving media
Figure 6. Formation of a stationary wave structure at the boundary between the
moving and fixed excitable media. The velocity of the plane front V
0
= 0.023, the
velocity of the medium w = 0.008; a) t = 0, b) t = 18.4, c) t = 70.
Figure 7. Excitation fronts for different relative velocities of the media: a) The
value of w is changed from w = 0.008 to w = 0.04 (subcritical behavior); b) The
stable structure for w = 0.04; c), d) The value of w is changed from w = 0.04 to
w = 0.05 (w∗ ≈ 0.045).
(iii) Continuity conditions for the concentrations u, v and their gradients at the
boundary between the moving and fixed media.
Note also that the drift terms w ·∇u and w ·∇v must be added in the left-hand
sides of equations (5.1) for the moving medium.
The results of computer simulations are shown in figures 6–8. Figure 6 illustrates
the formation of a stationary structure at the boundary between the moving and
fixed media. Initially the plane excitation front was generated in the homogeneous
medium with w = 0 (figure 6a). Then, at time t = 0 the upper half of the medium
begins to move with velocity w. After a lapse of time the stationary structure is
formed (figure 6c). For w < w∗, the increase of the relative velocity w of the media
leads only to distortions of the structure which move along the V -shaped branch
in the fixed medium and a new stable wave front is formed (figures 7a and 7b). If
575
V.A.Davydov et al.
Figure 8. Computer simulation of the “total reflection” of autowaves: a) t = 1.4;
b) t = 5.5; c) t = 7.9; d) t = 13.9. V
0
= 0.023, w = 0.03.
w exceeds the critical value w∗, the wave front breaks at the boundary (figures 7c
and 7d) in accordance with the prediction of the kinematic theory.
Finally, the “total reflection” of autowaves is shown in figure 8. The resulting
structure coincides with that predicted by the kinematic theory (see figure 5).
6. Conclusion
In this paper we have discussed some effects related to autowave dynamics in the
moving excitable media. An important point is that the qualitative features of these
effects are common to a variety of reaction-diffusion excitable media with different
generic reactivity terms F in equations (1.1). Considerable attention has been re-
cently focussed on mesoporous glasses as a support for the BZ reaction [15,16]. These
“solid” excitable media appear to be best suited for experimental observation of the
critical properties and refraction of autowave discussed in section 3 and section 4.
It should be noted that the method for studying stationary autowave structures
proposed in this paper applies to any piecewise homogeneous 2D excitable media
with plane boundaries between the parts having different properties. For instance,
it is not difficult to consider the refraction of autowaves at a boundary between
excitable media with different diffusion coefficients and velocities of the plane front.
Within the last decade, there has been a marked interest in the study of autowave
patterns in fluid media. Recently a simple kinematic model was proposed to explain
some features of the spiral wave meandering induced by fluid convection [6], but a
methodical kinematic theory of autowaves in reaction-diffusion-convection media is
still lacking. We intend to consider this important problem in future publications.
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Autowaves in the moving media
Acknowledgement
This work was partially supported by U.S.CRDF-RF Ministry of Education
Award VZ–010–0.
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V.A.Davydov et al.
Автохвилі в рухомих збуджених середовищах
В.А.Давидов 1 , М.В.Давидов 1 , В.Г.Морозов 1 ,
М.Н.Столяров 2 , Т.Ямагучі 3
1 Московський державний інститут радіотехніки, електроніки та
автоматики, 117454, проспект Вернадського 78, Москва, Росія
2 Відділення теоретичної фізики Інституту фізики РАН, 117924,
Ленінський проспект 53, Москва, Росія
3 Національний інститут сучасної науки та техніки, Хігаші, 1–1–1,
Цукуба, Ібаракі 305–8565, Японія
Отримано 5 квітня 2004 р.
В рамках кінематичної теорії автохвиль запропоновано метод для
аналітичного опису стаціонарних автохвильових структур, що вини-
кають на межі між рухомим та нерухомим збудженими середовища-
ми. Передбачено явище розриву фронту автохвилі в таких структу-
рах. Розглядається заломлення автохвилі та одностороннє “повне
відбиття”на границі. Отримані аналітичні результати знайшли своє
підтвердження при комп’ютерному моделюванні. Обговорюються
перспективи застосування запропонованого методу для подальшо-
го дослідження автохвильової динаміки в рухомих середовищах, що
збуджуються.
Ключові слова: нелінійна динаміка та нелінійні динамічні системи,
реакційно-дифузійні середовища, заломлення автохвиль
PACS: 05.50.+q, 05.70.Ln, 82.40.-g, 87.10.+e
578
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