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Steady-state autowave patterns in a two-dimensional excitable medium with a band of different excitability
PHYSICA ELS EVIER
Physica D 102(1997)300-312
Steady-state autowave patterns in a two-dimensional excitable medium with a band of different excitability Pavel K. Brazhnik *, John J. Tyson Department of Biology, Virginia Polytechnic Institute and State Universi~, Blacksburg, VA 24061-0406, USA
Received 4 July 1995; revised 20 June 1996; accepted 20 June 1996 Communicated by C.K.R.T. Jones
Abstract
A kinematic approach is used to construct steady-state traveling wave patterns in a two-dimensional excitable medium with a band of different excitability. New stationary autowave structures of bell-like shape are shown to exist when the excitability in a band is decreased compared to surrounding areas. The front profiles of the patterns and their propagation velocities are evaluated. Total internal reflection of waves from a band of higher excitability is predicted. The failure of steady-state propagation because of the critical curvature effect is discussed.
I. Introduction
An excitable medium (EM) is a spatially distributed system of excitable elements that can be triggered by their (excited) neighbors to become excited themselves and then relax back to the resting state. Waves of excitation (autowaves) propagate through neural networks, through muscular tissue [1], in certain chemical reactions (e.g. Belousov-Zhabotinsky (BZ) reaction) [2], in non-equilibrium plasma and in some kinds of solid state systems [3]. In two- and three-dimensional (2D and 3D) media excitation fronts exhibit complicated shapes and movements whose description presents challenging mathematical problems. The shapes of autowave patterns (spirals, vortex rings, etc.) refltct their unusual properties compared to acoustic or electromagnetic waves. For instance, autowaves annihilate upon collision, and autowave velocity depends on local wave front curvature and on the period of wave trains. While much is known about the properties of autowaves in homogeneous systems, our understanding of inhomogeneous EM is still primitive. Real systems, however, are often inhomogeneous and anisotropic. For example, catalytic surfaces show different speeds of propagation in different crystallographic directions, muscles like the heart are built of many tissue layers, chemical reactions may occur in systems with concentration gradients. Inhomogeneity * Corresponding author. E-mail: [email protected].
0167-2789/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S01 67-2789(96)001 82-0
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301
can be created artificially, e.g. by visible light [4] or electrical fields [5] in chemical EM. Therefore, in recent years, autowaves in heterogeneous systems have received increasing attention. In this paper we explore autowave propagation in stratified EM, that is, inhomogeneous media whose properties vary in space in only one direction. The solution of nonlinear models governing excitation spreading is a challenging problem by itself which becomes even more difficult when complicated by inhomogeneities. To simplify the problem for stratified media we approximate continuous variation of properties by step-like variations, that is, we consider media consisting of homogeneous layers with different properties abutting each other along semi-penetrable boundaries. Propagating wave solutions for such a medium can be constructed from the solutions for homogeneous media with an appropriate concatenation of the functions (and their derivatives) along the boundary lines. In this way the solution of inhomogeneous differential equations is reduced to the solution of algebraic equations at the semi-penetrable boundaries. The properties of autowaves propagating in stratified media are relevant to cardiac physiology, marine and lake ecology, and multilayer solid state devices (consider, for instance, the practical importance of electromagnetic waves in optical layers). Wave refraction and reflection [6,7] are, of course, of fundamental importance to wave motion of any nature. But the study of autowaves in layered EM poses, in addition, conceptual questions which have not yet been appropriately addressed. Consider for instance the stationary-wave patterns observed recently in chemical EM with two parallel semi-penetrable borders: the patterns propagate along a band with higher excitability placed in a homogeneous EM with lower excitability, Fig. 1 [4]. Boundaries between areas with different excitability were negligibly thin and therefore the medium can be considered as being piecewise constant. In the extensive literature of EM, two kinds of steady-state structures dominate: moving plane fronts and rotating spiral waves. No combination of pieces of these two structures can give the patterns in Fig. 1. Some help comes from existence of V-shaped steady-state solution for EM [8,9], which provides building blocks for the external wings of the pattern in Fig. 1(a). But what about the front inside the band? Also, consider the opposite situation when the excitability is lower in the band than in the bulk medium. The steady-state patterns one may intuitively expect to form in such a medium are depicted in Fig. 2(a) for bounded, and Figs. 2(b) and 3 for unbounded medium. This time the front inside the band is reminiscent of the V-shaped wave but what are the external wings? The lack of appropriate building blocks indicates that, apart from plane, spiral, and V-shaped waves, there are other steady-state solutions of EM which, in spite of extensive numerical exploration of PDE models, have not yet been identified and characterized. In this paper we predict bell-like autowave patterns which may exist in an EM with a band of lower excitability (Figs. 2 and 3) and characterize their properties. The effects of total internal reflection from a band of higher excitability and steady-state propagation failure because of the critical curvature are also discussed. Our study is based on a kinematic theory of curvature-driven wavefronts, which allows us to make exhaustive predictions about the number of patterns possible in a given medium and their properties without invoking intuitive arguments, and to include a consideration of critical curvature effects.
2. Mathematical model
A comprehensive study of autowaves propagating in piecewise media has to be based on corresponding PDE models. Unfortunately the nonlinearity of reaction~liffusion systems describing EM prohibits their exact solution. A coarse-scale geometrical model, the so-called kinematic approach, which is with respect to the underlying reactiondiffusion equations like the geometric optics approximation in the theory of electromagnetic waves, often provides reasonable insight into the behavior of solitary waves. The approach is especially suitable for the problem under consideration because for non-circulating solutions the steady-state kinematic model can be integrated exactly [9], thereby providing all necessary building blocks for steady-state patterns in layered EM. A complete description
302
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t
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.
.
.
(~)~, ~
~
~ -
-
_~4. ,-,-, .~_ - - , - , - , - , ~
x (mm)
,-O!l'
/
.1---.<2.. v (ran)
-'-'-
,i f
5. '- :;"
,(°)
Fig. 1. Stationary-wave patterns which develop in an EM with a band of higher excitability from an initially planar wave non-parallel to the boundaries. Excitability is higher within the band between the dashed lines, while the diffusion coefficient is constant throughout the EM. Planar front velocity (V0)in is larger in the band than in surrounding areas (V0)out. The steady-state propagating wavefront (thick solid line) is depicted for: (a) an EM of infinite width, and (b) an unbounded EM. In Figs. 1-3, the curves are computed by the kinematic approach. Parameters of the medium are taken as those in experiments with light-sensitive BZ system [4]: (V0)out = 77.9 ~m/s, (V0)in -----97.9 Izm/s, V(0) = 83.9 Izm/s, D = 4500 ~m2/s, L = 1.44 mm. W = 0.1 mm.
o f k i n e m a t i c t h e o r y o f a u t o w a v e s c a n b e f o u n d e l s e w h e r e [ 10]. H e r e w e m e n t i o n o n l y those details n e c e s s a r y for u n d e r s t a n d i n g o u r results. T h e k i n e m a t i c d e s c r i p t i o n r e d u c e s an e x c i t a t i o n p u l s e s t r u c t u r e in 1D to a single point, so that a n e x c i t e d region in a 2 D E M s h r i n k s to a t h i n c u r v e d line w i t h a n o r m a l v e c t o r p o i n t i n g in the p r o p a g a t i o n direction. E a c h section o f s u c h a f r o n t m o v e s in the n o r m a l d i r e c t i o n w i t h v e l o c i t y V (k) d e t e r m i n e d b y the local w a v e f r o n t c u r v a t u r e k. F o r relatively s m a l l k [11], V(k)
= Vo -
Dk.
(1)
303
P.K. Brazhnik, J.J. T y s o n / P h y s i c a D 102 (1997) 3 0 0 - 3 1 2
L--
Y (ram)
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i
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.
.
.
.
.
.
.
.
_
~ Y (rnm)
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.
-
_
.
.
.
v(o~
Fig. 2. As in Fig. 1, except excitability is smaller within the central band: (V0)in < (V0)out.
Here V0 is the velocity of the planar front and D is the diffusion coefficient of the excitable variable. A front shape is specified then by the intrinsic equation k = k(l, t) that at each moment of time t relates curvature of the front-line with arclength 1. In this paper we consider only steady-state configurations, k = k(l). The profile of a stationary propagating front obeys the equation /
k(l) f k(~)V[k(~)] d~ +
dV[k(t)] d-----[~- -
w.
(2)
o
Limiting ourselves to non-circulating solutions, we set ~o = 0. The intrinsic equation k(l) determines the front curve up to its position on the plane. A parametric representation of the actual front-line in the Cartesian frame of reference can be constructed from the intrinsic equation following a standard procedure:
304
P.K. Brazhnik, J.J. Tyson/Physica D 102 (1997)300-312
3 2
/
1 /
o: =
xc,.,.)
,4,~°
' I ' ' ' ' I ' ' ' ' I ....
-2i
(vo)~
-3
I
N~v(o)
Fig. 3. A stationary-wave pattern which develops in an EM with a band of lower excitability, as in Fig. 2, in the process of oblique collision of two plane waves. l
x(l)
l
= f sin[0(~)]d~,
y(l)----
0
f cos[0(¢)]d~,
(3)
0
where x and y are Cartesian coordinates of the front-line and l 0(l) = - /
k(~) dse
(4)
o
is the angle b e t w e e n the tangent to the wavefront at the point 1 and the axis O Y taken as positive if measured clockwise from the positive direction of the axis OY. It has been shown recently that, for w ---- 0, Eq. (2), with V (k) given by (1), can be integrated exactly [9] providing, d e p e n d i n g on the value of the 'initial' condition k ( l = 0) - k(0), three one-parameter families of propagating solutions plus two specific cases: the plane wave and a separatrix solution. For k(0) < 0, the fronts propagation in the form of V-shaped waves: 1
k ( l ) --
[v(o)]
2 -
Vo2
D V(O) + V ( O ) c o s h ( 1 / l o ) '
-~
< 1 < o0,
(5)
with characteristic length l0 = D - (I IV(0)] 2 - V021)-°-5. The corresponding front-line is depicted in the Cartesian frame of reference in Fig. 4. The curvature of this front is always negative and decreases exponentially to zero as II] goes to infinity. For all solutions of Eq. (2) the tangential angle 0 satisfies the condition V = V(0) cos 0,
(6)
where V(0) denotes the normal velocity of the wave at 0 ---- 1 = 0 and is to be distinguished from V0, the velocity at k ----0. F r o m here, the asymptotic angle between wings of the V pattern, ~, is o~ =- Jr - 20(1 ~
±cx~) = 2 arcsin [ V o / V ( 0 ) ] .
(7)
PK. Brazhnik, J.J. Tyson/Physica D 102 (1997) 300-312 V-skaped froml
plamar
osdllmtimg
fromt
fromt
separatrlz
305
ome Loop fr~!
< Fig. 4. Shapes of stationary propagating fronts corresponding to solutions of the kinematic model in the eikonal approximation. Solutions are parametrized by the 'initial' value k(0). Properties of each pattern are discussed in the text. The V-shaped wave moves uniformly, with velocity V(0) > V0, i.e. faster than a planar front. As k(0) ~ 0, the V-wave straightens, c~ increases to zr, and the propagation velocity decreases to V0, that is the pattern converts into a plane wave (k -- 0). Positive k(0) produces space oscillating fronts described by
/c(l)
l -
IV(0)] 2 -
v02
D VO+ V(O) cos(I/lo)"
(8)
As k(0) increases from zero up to k(0) "~ Vo/D, both the amplitude of the front curvature oscillations and the period decrease: the amplitude from Vo/D to zero, and period from infinity to 2rr D~ Vo. The front propagates with a speed V(0) < V0. The corresponding front-line is shown in Fig. 4. The value k(0) = 2Vo/D corresponds to a separatrix solution, with algebraic soliton-type shape k(/) -
2Vo/D
(9)
In the Cartesian frame of reference this front exhibits a loop (see Fig. 4) and has asymptotically flat wings, k(I --~ +o~) --~ 0, separated by an angle of rr. The front propagates with the velocity of a plane front. For k(0) > 2Vo/D the solution in Eq. (5) gives a one-loop front but the asymptotic angle c~ between the wings determined by (6) is smaller than zr, going to zero as k(0) ~ oo (see Fig. 4). The front propagates with velocity V(0) > V0.
3. B e l l - l i k e w a v e s
The motion of an autowave front in a chemical EM with a band of higher excitability was investigated experimentally in [4] and theoreitically in [9]. The steady-state patterns occurring in such a medium are depicted in Fig. 1. Inside the band wavefront is positively curved and constitutes a piece of the oscillating solution, Eq. (8). The wings of the pattern in a bounded EM (Fig. 1(a)) are fragments of V-shaped waves (Eq. (5)) matching smoothly the front-line inside the band and approaching the impenetrable boundaries perpendicularly. For an unbounded EM (Fig. 1(b)) the wings degenerate into straight lines (pieces of a plane wave) tilted away from the direction of pattern propagation. The velocity of the pattern, V (0), in a bounded EM of fixed bandwidth (W), is a monotonically decreasing function of the medium width (L) running from the velocity of a planar front in the band (when L = W) to the velocity of the pattern in an unbounded medium (L ~ cx~). The latter, in turn, is a monotonically increasing function of the bandwidth approaching the velocity of a planar front in the band when the bandwidth goes to infinity.
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Next we consider the opposite situation, when excitability is higher in the bulk medium than in the band. We characterize the medium by its macroscopic parameters: the planar front velocity V0 and the diffusion coefficient D. We suppose that in a band of half width W (W < L) located along the symmetry axis of the medium (the axis OX in Figs. 2 and 3) excitability and hence planar front velocity are smaller than in surrounding areas: (V0)in < (V0)out. The diffusion coefficient is considered the same everywhere. Such conditions can be generated, for instance, in a light-sensitive BZ system by the technique developed in [4]. If we start with a planar autowave perpendicular to the OX axis, how will it evolve in time? Obviously, the shape of the front must change in the course of its evolution because local velocity is different for different parts for the front-line: the part in the middle band will lag compared to parts outside, and, as a result, the front-line will become negatively curved in the direction of propagation in the middle band and positively curved outside. Negative curvature of the front in the middle will increase its velocity (due to the curvature effect, see Eq. (l)) compared to (V0)in, that its, it will tend to compensate for the velocity difference. Positive curvature of the external wings also contributes to this compensation. The final steady-state configuration depicted in Fig. 2(a) we call a bell-like wave. As the width of the EM (2L) increases, the pattern becomes more extended and as L -~ cc turns into a structure with asymptotically planar wings (Fig. 2(b)). In Section 3.1 we construct bell-like wave explicitly for the kinematic model. We suppose that the transition region between areas with different excitability is infinitely thin and the medium can be treated as piecewise continuous. Also the size of the band and the EM itself are supposed to be quite large compared to the width of excitation pulse so that the kinematic approach is applicable. At the semi-penetrable boundary we impose geometric constraints on the wavefront line: the front-line must be a continuous function across the boundary, that is, 0 must be a continuous function of l. 3.1. A bell-like wave in a bounded excitable medium
We describe patterns in terms of their intrinsic equations, measuring arclength l from the point of symmetry, y = 0, hence V ( y = 0) = V(0). In this case, l spans the interval from --IL to IL, where 21L is the total length of the front. We start with the bell-like wave in a bounded EM, Fig. 2(a). Among the solutions in Fig. 4, the only appropriate candidate for the negatively curved section inside the central band is a V-shaped wave. Thus the central part of the pattern is described by the expression 1 IV(0)] 2 - [(V0)in] 2 D (V0)in + V(0)cosh[l/(lO)in]'
kin(l) =
l • ( - l * , l*),
(10)
where +l* are the values of l at the semi-penetrable boundaries (y(+l*) = +W), and the characteristic length within the band is: (/0)in = D/v/[V(O)] 2 - [(V0)in] 2. Fragments of the front adjoined to impenetrable boundaries are positively curved in the direction of propagation and approach the impenetrable borders at right angles. For these parts, an appropriate solution among those in Fig. 4 is the oscillating one (Eq. 8) which for our purpose has to be written in the form 1 [V(0)] 2 - [(V0)out] 2 kout(l) = - - ~ (V0)out q- V(0) cos[(/-4- lL)/(lo)out]'
1 • (+l*, i l L ) ,
(1 1)
where the sign ' + ' or ' - ' holds for y > 0 or y < 0, respectively, and we have taken into account the fact that V ( + I L ) = V(O) since the front is orthogonal to impenetrable borders (Oout(ilL) = 0) as it is to the line of its s y m m e t r y (0in(0) = 0). This relates also, through Eq. (1), the curvature of the front at impenetrable boundaries, kout(+lL) =- k(IL), to the curvature in the middle of the pattern, kin(0) -~ k(O), as (V0)out
-
-
D . k(IL) = (V0)in -- D . k(0).
(12)
P.K. Brazhnik, J.J. Tyson/Physica D 102 (1997) 300-312
307
5-
I] ! ! !
~11
i
4-,
3--
! I t t
2£ -
!' ! #
1
i
....
- 2.0
, ....
....
1.0
- 1.O ~ g
2.0
I" "1"
2
Length of the front I, mm Fig. 5, Dependence of the curvature of bell-like waves on front arclength l: the solid line corresponds to the pattern in a bounded medium (Fig. 2(a)); the dashed line corresponds to the pattern in an unbounded medium (Fig. 2(b)). Though fronts cross lines of excitation-jump smoothly, their curvatures undergo jumps determined by the difference of planar front velocities in areas with different excitability, Eq. (1.4).
Fronts in inner and outer areas must join smoothly at the semi-penetrable boundaries, that is, 0out(-4-1*) = 0in(+l*),
(13)
and this leads to relationship between curvatures on different sides of the semi-penetrable border (V0)out --
D- kout(+l*) = (V0)in - D . k i n ( + / * ) .
(14)
The angle between the tangent to the front and the axis O Y, 0, is also related to the half-widths of the band and EM through Eqs. (3) as l* W = /
cos[0in(~; k(0))] d~,
(15)
,/
0 IL
/i
L - W = / cos[0out(~; k(lL))] d~,
(16)
J
/"
where the parametric dependence of 0in (l) o n k ( 0 ) is indicated explicitly, as well as 0out(l) on k(IL). For given (V0)in, (V0)out, D, L and W, the four equations (12), (14)-(16) can be solved for the four unknown constants k (0), k (IL), l*, lL. Since solution (11) is an oscillating function, the solution of this problem is not unique. The solution set {k (0), k (IL), l*, 1L} with minimum value of IL, which also has minimal curvature k (1L), corresponds to the bell-like wave in Fig. 2(a). Other solutions, with larger lL and larger k(1L), introduce one or more periods of the oscillatory solutions into the lip of the bell. We will have more to say about these oscillatory solutions in the discussion. The intrinsic equation of the pattern (Fig. 2(a)) we have just constructed is depicted in Fig. 5 by a solid line. Note that, though the front-line of the pattern is smooth, its curvature undergoes a jump (given by Eq. (14)) at the
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P.K. Brazhnik, J.J. Tyson/Physica D 102 (1997) 300-312
semi-penetrable boundaries. This jump is equal to the difference between curvatures of the front in the middle of the pattern and at the impenetrable border (compare Eqs. (14) and (17)): kin(l*) - kout(l*) = k(0) - k(IL) =
(W0)in - (Vo)out D
(17)
If l* turns out to be much greater than (/0)in, as it is in Fig. 2, the front is almost flat as it approaches the semipenetrable boundary from the inner side and, hence, tilted to the boundary line at the angle ½or determined by the asymptotic angle of the V-shaped wave (see Eq. (7)). Eqs. (15) and (16) written in an explicit form can be used to relate propagation velocity of the pattern V(0) to the width of the band and the EM: W-
V(0)D [ 0*
L - W --
--[(1D/(V0)in2(10)in_ Uintan(½0*))/(l+Uintan(½0*))]}
°/°"
V(O)
D/(Vo)outarctan [Uout tan(½0*)l
I
,
(18)
where 0 * --= 0 (l*) is the angle the front makes with the band boundaries and Uin, out --- ~/( V (0) + (V0)in,out)/( V (0) (V0)in,out). Excluding 0* from here we get the desired expression: V(0) = V(0; L, W). Sets of curves for V(0; L, W) at fixed L and at fixed W are depicted in Figs. 6(a) and (b), respectively. For fixed L the velocity of the pattern decreases monotonically from (V0)out when W = 0 to (VO)in when W = L. In both limiting cases, the front degenerates to a planar wave. For fixed W, the velocity of the pattern is a monotonic function of L increasing from (V0)in when L = W to (V0)out when L ~ oo. 3.2. A bell-like wave in an unbounded medium We expect that for the bell-like wave, when L ~ oo, the front-line becomes asymptotically (at l --+ -1-o~) flat and orthogonal to the direction of front propagation (Fig. 2(b)), therefore V(0) = (V0)out. For the latter condition, the oscillating solution (1 1) does not work any more, and the appropriate front-line is formed by the separatix solution (9), with V0 = (V0)out and l replaced by l + l*, l • (+l*, -4-~). The central part of the bell-like wave in an unbounded medium is still described by Eq. (10) with V(0) replaced by (V0)out. The intrinsic equation of this bell-like wave is shown by a dashed line in Fig. 5. Since kout(l --+ ±oo) = 0, the curvature of the bell-like wave in the middle is given by (see Eq. (12)) k(0) =
(V0)in
(V0)out
-
D
(19)
and does not depend on the bandwidth. 3.3. A bell-like wave with tilted wings Now suppose that instead of launching a plane wave, as we did in Section 3.2, we collide two plane waves (at oblique angle or) in a way that the angular corner they form at the point of collision is located inside the band. The evolution of such a sharp wedge into a V-shaped wave in homogeneous EM has been studied in [8]. In unbounded EM with a band of reduced excitability, the corner will become a fragment of a V-shaped wave only within the band, while the wavefront outside the band will gain some positive curvature in order to equalize velocity differences along the front, see Fig. 3. The intrinsic equation of the front fragment of this pattern inside the band is given by Eq. (10), while the external wings are described by the one-loop solution (Eq. (5), for positive k(0)). The pattern propagates with the velocity (V0)out/cos(½ot) which does not depend on the bandwidth.
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_lO; 90--[
I
N~
.
.
.
.
.
\ L •0.1;
0.5;
/mf ......................
70 l 0
.
N
: 1.o1 (V°)
.
'
I
' I ' I ' half-width W (ram)
0.9
I
'
I 1
I
'
I 1
100-
[[
90
-~~'
so (v)~
w=o.1; o~;
70
~ 0
I
0.4
' I ' I ' half-width L ( m m )
Fig. 6. Propagation velocity of bell-like waves in bounded medium (Fig. 2): (a) dependence of V(0) on W for fixed values of L; (b) dependence of V(0) on L for fixed values of W. Conclusions about the stability of bell-like waves can be made on the basis of the stability of the solutions from which the patterns are constructed. Linear stability analysis shows that the latter are stable with respect to small localized perturbations which disappear diffusively, traveling along the front towards the region of maximum curvature ([k[) [9]. Hence, the bell-like patterns we predict are stable with respect to small perturbations which travel from the periphery to the center of the patterns and disappear diffusively.
4. D i s c u s s i o n
In Section 3 we have shown that in EM with a band of reduced excitability three steady-state patterns may exist, one for a bounded media (Fig. 2(a)) and two for an unbounded (Figs. 2(b) and 3). In [4,9] two steady-state patterns were studied in EM with a band of increased excitability (Fig. 1). Our theory indicates though that one more
P.K. Brazhnik, J.J. Tyson / Physica D 102 (1997) 300-312
310
W I
,,,
l,
....
,,
-I -W -2
(vo)~ ~
v(o)
Fig. 7. The stationary pattern which develops in an EM with a band of higher excitability from an initial front (two colliding plane waves) forming a sharp corner inside the band. The difference of plane front velocities inside the band and in the rest of the EM is supposed to be sufficiently small.
steady-state wavefront configuration is possible in this case if the EM is not confined. The pattern develops from two colliding plane waves when the ratio (VO)out/(VO)in is close enough to unity. The resulting steady-state wavefront is shown in Fig. 7. Both wings and the portion inside the band are now pieces of V-shaped waves matching smoothly across the semi-penetrable boundaries. The 'elbow' of this pattern has smaller curvature than would be necessary to connect the same wings in a homogeneous medium. The pattern propagation velocity does not depend on the bandwidth. When the difference of velocities increases (say the velocity inside the band increases), the portion of the steady-state pattern inside the band flattens and finally, when the initial inclination angle 0out(l ~ o0) satisfies the condition COS 0out(/ ~
(X)) = (Vo)out/(Vo)in,
(20)
the inner portion becomes a piece of a plane wave oriented perpendicular to the band boundaries, and the wings become exactly halves of V-shaped waves. For given (V0)in and (V0)out the angle determined from the ratio (20) may be called the critical angle. For the conditions of the experiments with BZ reported in [4] (see also the caption for Fig. 1) this angle is around 40 °. Further increase of (V0)in will flip the inclination of the external wings in the neighborhood of boundaries and this disturbance will propagate with time away from the boundary forming finally the steady-state pattern with opposite to initial inclination, that is, one depicted in Fig. l(b). In this regime the two initially colliding waves undergo a total internal reflection from the band [7]. In the theory proposed so far, the concatenation of V-wave solutions (5) with oscillating solutions (8) can be made at any angle from 0 to ½zr. Therefore the patterns we have described should exist for an arbitrarily large difference of velocities in the band and in the bulk medium. However, some of the elementary solutions we have used to build the more complicated patterns exhibit non-physical self-crossings of the front-lines, which is an artifact of extrapolating the linear dependence, Eq. (1), beyond the critical curvature Kcr, beyond which continuous propagating fronts do not exist. Accounting for critical curvature replaces loops by cusps [ 12] so that self-crossings disappear. Corresponding front-lines in this case permit only a restricted variation of the angle between the tangent to the front and the direction of the pattern propagation. For non-oscillating solutions 0 runs from 0 (k = 0) up to 0 (k = kcr) and for the oscillating solution, from O(k = kmin) to 0(k = kcr). This, in turn, puts the restrictions on possible concatenation angles for
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311
our patterns. If the difference in properties of the EM across the semi-penetrable boundary is so large that it would require a concatenation angle outside the permissible regions, the steady-state patterns become impossible. The wavefront will break at the boundary. The condition for failure of stationary propagation can be easily derived, for example, for the pattern in Fig. 1(b). The maximum angle the oscillating solution fragment inside the band can make with the axis O Y is l:r + lot 2 cusp, where the cusp angle of the oscillating pattern otcusp = 2 arcsin[( Vcr)in/ V (0)] and ( Vcr)in is the critical velocity for the EM inside the band [ 12]. On the other hand, the angle between the external wing of the pattern and axis O Y is given by the expression 0out = - 2 arccos[(V0)out/V(0)]. Outside and inside 1 portions of the front can be matched smoothly only if 0out > ½zr + ~otcusp, what gives the desirable condition (V0)out > (Vcr)in. Since, generally speaking, the angle a stationary wave makes with the semi-penetrable boundary depends on the bandwidth, the critical curvature effect may also introduce a critical bandwidth beyond which the EM with a band cannot support stationary patterns. Only a small portion of the oscillating solution was used for constructing patterns in Figs. 1 and 2. Formally, exploiting our algorithm, one could introduce several periods of the oscillating solution in the outside regions. Such configurations, though, would include cusps which lead to front instability [ 12], and therefore the whole pattern would be unstable. If the properties of the EM are different on both sides of the band (three layers of different properties), nonsymmetrical steady-state patterns arise. The technique for their construction should, by now, be clear.
5. Conclusions
We have predicted three new steady-state autowave patterns in EM with a band of reduced excitability. The patterns have a bell-like shape, that is, their front-lines are negatively curved in the middle being concave in the direction of the pattern propagation and become positively curved (convex) outside the band, and finally orthogonal to the direction of the pattern propagation or achieving an asymptotic angle. We have described one new stationary pattern in unbounded EM with a band of increased excitability (Fig. 7), similar to the V-shaped wave observed in homogeneous EM but slightly disturbed by the band of increased excitability. The six patterns illustrated in Figs. 1-3 and 7 represent a complete set of stable stationary patterns which can form in EM with a band of different excitability, according to kinematic theory. Propagation velocities of the patterns in unbounded media do not depend on the bandwidth while in the bounded EM patterns propagate with velocities which are functions of the bandwidth. The V-shaped pattern in Fig. 7 exhibits a notable behaviour when (go)out/(go)in exceeds a critical value: the V-shaped wave disintegrates and the front evolves to another stionary patterns. Patterns similar to those we have just described should also appear in EM with a band of different diffusivity (but uniform excitability). This situation can be realized experimentally using either silica gel or porous glasses [ 13].
Acknowledgements
This work was supported by NSF Grant No. CHE 9500763.
References
[ 11 A.T. Winfree, When Time Breaks Down (Princeton UniversityPress, Princeton, NJ, 1987). [2] Oscillations and Traveling Waves in Chemical Systems, eds. R.J. Field and M. Burger (Wiley, New York, 1985); Chemical Waves and Patterns, eds. R. Kapral and K. Showalter (Kluwer Academic Publishers, Dordrecht, 1995). [3] B.S. Kerner and V.V. Osipov, Sov. Phys. Usp. 32 (1989) 101.
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[4] O. Steinbock, V.S. Zykov and S.C. Muller, Phys. Rev. E 48 (1993) 3295. [5] J. Schutze, O. Steinbock and S.C. Muller, Phys. Rev. Lett. 68 (1992) 248; V. Perez-Munuzuri, R. Aliev, B. Vasiev and V.I. Krinsky, Physica D 56 (1992) 229. [6] L.M. Brekhovskikh, Waves in Layered Media (Academic Press, New York, 1980); A.M. Zhabotinsky, M.D. Eager and I.R. Epstein, Phys. Rev. Lett. 71 (1993) 1526. [7] P.K. Brazhnik and J.J. Tyson, Phys. Rev. E 54 (1996) 205. [8] P.K. Brazhnik and V.A. Davydov, Physics Lett. A 199 (1995) 40. [9] P.K. Brazhnik, Physica D 94 (1996), in press. [10] A.S. Mikhailov, V.A. Davydov and V.S. Zykov, Physica D 70 (1994) 1. [11] P. Foerster, S. Muller and B. Hess, Science 241 (1988) 685; Development 109 (1990) 11. [12] P.K. Brazhnik, J.J. Tyson, Phys. Rev. E 54 (1996), in press. [ 13] T. Yamaguchi, L. Kuhnert, Z.S. Nagy-Ungvarai, S.C. Muller and B. Hess, J. Phys. Chem. 96 (1991) 5831 ; T. Amemiya, M. Nakaiwa, T. Ohmoi and T. Yamaguchi, Physica D 84 (1995) 103.
Physica D 102(1997)300-312
Steady-state autowave patterns in a two-dimensional excitable medium with a band of different excitability Pavel K. Brazhnik *, John J. Tyson Department of Biology, Virginia Polytechnic Institute and State Universi~, Blacksburg, VA 24061-0406, USA
Received 4 July 1995; revised 20 June 1996; accepted 20 June 1996 Communicated by C.K.R.T. Jones
Abstract
A kinematic approach is used to construct steady-state traveling wave patterns in a two-dimensional excitable medium with a band of different excitability. New stationary autowave structures of bell-like shape are shown to exist when the excitability in a band is decreased compared to surrounding areas. The front profiles of the patterns and their propagation velocities are evaluated. Total internal reflection of waves from a band of higher excitability is predicted. The failure of steady-state propagation because of the critical curvature effect is discussed.
I. Introduction
An excitable medium (EM) is a spatially distributed system of excitable elements that can be triggered by their (excited) neighbors to become excited themselves and then relax back to the resting state. Waves of excitation (autowaves) propagate through neural networks, through muscular tissue [1], in certain chemical reactions (e.g. Belousov-Zhabotinsky (BZ) reaction) [2], in non-equilibrium plasma and in some kinds of solid state systems [3]. In two- and three-dimensional (2D and 3D) media excitation fronts exhibit complicated shapes and movements whose description presents challenging mathematical problems. The shapes of autowave patterns (spirals, vortex rings, etc.) refltct their unusual properties compared to acoustic or electromagnetic waves. For instance, autowaves annihilate upon collision, and autowave velocity depends on local wave front curvature and on the period of wave trains. While much is known about the properties of autowaves in homogeneous systems, our understanding of inhomogeneous EM is still primitive. Real systems, however, are often inhomogeneous and anisotropic. For example, catalytic surfaces show different speeds of propagation in different crystallographic directions, muscles like the heart are built of many tissue layers, chemical reactions may occur in systems with concentration gradients. Inhomogeneity * Corresponding author. E-mail: [email protected].
0167-2789/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S01 67-2789(96)001 82-0
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301
can be created artificially, e.g. by visible light [4] or electrical fields [5] in chemical EM. Therefore, in recent years, autowaves in heterogeneous systems have received increasing attention. In this paper we explore autowave propagation in stratified EM, that is, inhomogeneous media whose properties vary in space in only one direction. The solution of nonlinear models governing excitation spreading is a challenging problem by itself which becomes even more difficult when complicated by inhomogeneities. To simplify the problem for stratified media we approximate continuous variation of properties by step-like variations, that is, we consider media consisting of homogeneous layers with different properties abutting each other along semi-penetrable boundaries. Propagating wave solutions for such a medium can be constructed from the solutions for homogeneous media with an appropriate concatenation of the functions (and their derivatives) along the boundary lines. In this way the solution of inhomogeneous differential equations is reduced to the solution of algebraic equations at the semi-penetrable boundaries. The properties of autowaves propagating in stratified media are relevant to cardiac physiology, marine and lake ecology, and multilayer solid state devices (consider, for instance, the practical importance of electromagnetic waves in optical layers). Wave refraction and reflection [6,7] are, of course, of fundamental importance to wave motion of any nature. But the study of autowaves in layered EM poses, in addition, conceptual questions which have not yet been appropriately addressed. Consider for instance the stationary-wave patterns observed recently in chemical EM with two parallel semi-penetrable borders: the patterns propagate along a band with higher excitability placed in a homogeneous EM with lower excitability, Fig. 1 [4]. Boundaries between areas with different excitability were negligibly thin and therefore the medium can be considered as being piecewise constant. In the extensive literature of EM, two kinds of steady-state structures dominate: moving plane fronts and rotating spiral waves. No combination of pieces of these two structures can give the patterns in Fig. 1. Some help comes from existence of V-shaped steady-state solution for EM [8,9], which provides building blocks for the external wings of the pattern in Fig. 1(a). But what about the front inside the band? Also, consider the opposite situation when the excitability is lower in the band than in the bulk medium. The steady-state patterns one may intuitively expect to form in such a medium are depicted in Fig. 2(a) for bounded, and Figs. 2(b) and 3 for unbounded medium. This time the front inside the band is reminiscent of the V-shaped wave but what are the external wings? The lack of appropriate building blocks indicates that, apart from plane, spiral, and V-shaped waves, there are other steady-state solutions of EM which, in spite of extensive numerical exploration of PDE models, have not yet been identified and characterized. In this paper we predict bell-like autowave patterns which may exist in an EM with a band of lower excitability (Figs. 2 and 3) and characterize their properties. The effects of total internal reflection from a band of higher excitability and steady-state propagation failure because of the critical curvature are also discussed. Our study is based on a kinematic theory of curvature-driven wavefronts, which allows us to make exhaustive predictions about the number of patterns possible in a given medium and their properties without invoking intuitive arguments, and to include a consideration of critical curvature effects.
2. Mathematical model
A comprehensive study of autowaves propagating in piecewise media has to be based on corresponding PDE models. Unfortunately the nonlinearity of reaction~liffusion systems describing EM prohibits their exact solution. A coarse-scale geometrical model, the so-called kinematic approach, which is with respect to the underlying reactiondiffusion equations like the geometric optics approximation in the theory of electromagnetic waves, often provides reasonable insight into the behavior of solitary waves. The approach is especially suitable for the problem under consideration because for non-circulating solutions the steady-state kinematic model can be integrated exactly [9], thereby providing all necessary building blocks for steady-state patterns in layered EM. A complete description
302
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t
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.
.
.
(~)~, ~
~
~ -
-
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x (mm)
,-O!l'
/
.1---.<2.. v (ran)
-'-'-
,i f
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,(°)
Fig. 1. Stationary-wave patterns which develop in an EM with a band of higher excitability from an initially planar wave non-parallel to the boundaries. Excitability is higher within the band between the dashed lines, while the diffusion coefficient is constant throughout the EM. Planar front velocity (V0)in is larger in the band than in surrounding areas (V0)out. The steady-state propagating wavefront (thick solid line) is depicted for: (a) an EM of infinite width, and (b) an unbounded EM. In Figs. 1-3, the curves are computed by the kinematic approach. Parameters of the medium are taken as those in experiments with light-sensitive BZ system [4]: (V0)out = 77.9 ~m/s, (V0)in -----97.9 Izm/s, V(0) = 83.9 Izm/s, D = 4500 ~m2/s, L = 1.44 mm. W = 0.1 mm.
o f k i n e m a t i c t h e o r y o f a u t o w a v e s c a n b e f o u n d e l s e w h e r e [ 10]. H e r e w e m e n t i o n o n l y those details n e c e s s a r y for u n d e r s t a n d i n g o u r results. T h e k i n e m a t i c d e s c r i p t i o n r e d u c e s an e x c i t a t i o n p u l s e s t r u c t u r e in 1D to a single point, so that a n e x c i t e d region in a 2 D E M s h r i n k s to a t h i n c u r v e d line w i t h a n o r m a l v e c t o r p o i n t i n g in the p r o p a g a t i o n direction. E a c h section o f s u c h a f r o n t m o v e s in the n o r m a l d i r e c t i o n w i t h v e l o c i t y V (k) d e t e r m i n e d b y the local w a v e f r o n t c u r v a t u r e k. F o r relatively s m a l l k [11], V(k)
= Vo -
Dk.
(1)
303
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Fig. 2. As in Fig. 1, except excitability is smaller within the central band: (V0)in < (V0)out.
Here V0 is the velocity of the planar front and D is the diffusion coefficient of the excitable variable. A front shape is specified then by the intrinsic equation k = k(l, t) that at each moment of time t relates curvature of the front-line with arclength 1. In this paper we consider only steady-state configurations, k = k(l). The profile of a stationary propagating front obeys the equation /
k(l) f k(~)V[k(~)] d~ +
dV[k(t)] d-----[~- -
w.
(2)
o
Limiting ourselves to non-circulating solutions, we set ~o = 0. The intrinsic equation k(l) determines the front curve up to its position on the plane. A parametric representation of the actual front-line in the Cartesian frame of reference can be constructed from the intrinsic equation following a standard procedure:
304
P.K. Brazhnik, J.J. Tyson/Physica D 102 (1997)300-312
3 2
/
1 /
o: =
xc,.,.)
,4,~°
' I ' ' ' ' I ' ' ' ' I ....
-2i
(vo)~
-3
I
N~v(o)
Fig. 3. A stationary-wave pattern which develops in an EM with a band of lower excitability, as in Fig. 2, in the process of oblique collision of two plane waves. l
x(l)
l
= f sin[0(~)]d~,
y(l)----
0
f cos[0(¢)]d~,
(3)
0
where x and y are Cartesian coordinates of the front-line and l 0(l) = - /
k(~) dse
(4)
o
is the angle b e t w e e n the tangent to the wavefront at the point 1 and the axis O Y taken as positive if measured clockwise from the positive direction of the axis OY. It has been shown recently that, for w ---- 0, Eq. (2), with V (k) given by (1), can be integrated exactly [9] providing, d e p e n d i n g on the value of the 'initial' condition k ( l = 0) - k(0), three one-parameter families of propagating solutions plus two specific cases: the plane wave and a separatrix solution. For k(0) < 0, the fronts propagation in the form of V-shaped waves: 1
k ( l ) --
[v(o)]
2 -
Vo2
D V(O) + V ( O ) c o s h ( 1 / l o ) '
-~
< 1 < o0,
(5)
with characteristic length l0 = D - (I IV(0)] 2 - V021)-°-5. The corresponding front-line is depicted in the Cartesian frame of reference in Fig. 4. The curvature of this front is always negative and decreases exponentially to zero as II] goes to infinity. For all solutions of Eq. (2) the tangential angle 0 satisfies the condition V = V(0) cos 0,
(6)
where V(0) denotes the normal velocity of the wave at 0 ---- 1 = 0 and is to be distinguished from V0, the velocity at k ----0. F r o m here, the asymptotic angle between wings of the V pattern, ~, is o~ =- Jr - 20(1 ~
±cx~) = 2 arcsin [ V o / V ( 0 ) ] .
(7)
PK. Brazhnik, J.J. Tyson/Physica D 102 (1997) 300-312 V-skaped froml
plamar
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fromt
fromt
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305
ome Loop fr~!
< Fig. 4. Shapes of stationary propagating fronts corresponding to solutions of the kinematic model in the eikonal approximation. Solutions are parametrized by the 'initial' value k(0). Properties of each pattern are discussed in the text. The V-shaped wave moves uniformly, with velocity V(0) > V0, i.e. faster than a planar front. As k(0) ~ 0, the V-wave straightens, c~ increases to zr, and the propagation velocity decreases to V0, that is the pattern converts into a plane wave (k -- 0). Positive k(0) produces space oscillating fronts described by
/c(l)
l -
IV(0)] 2 -
v02
D VO+ V(O) cos(I/lo)"
(8)
As k(0) increases from zero up to k(0) "~ Vo/D, both the amplitude of the front curvature oscillations and the period decrease: the amplitude from Vo/D to zero, and period from infinity to 2rr D~ Vo. The front propagates with a speed V(0) < V0. The corresponding front-line is shown in Fig. 4. The value k(0) = 2Vo/D corresponds to a separatrix solution, with algebraic soliton-type shape k(/) -
2Vo/D
(9)
In the Cartesian frame of reference this front exhibits a loop (see Fig. 4) and has asymptotically flat wings, k(I --~ +o~) --~ 0, separated by an angle of rr. The front propagates with the velocity of a plane front. For k(0) > 2Vo/D the solution in Eq. (5) gives a one-loop front but the asymptotic angle c~ between the wings determined by (6) is smaller than zr, going to zero as k(0) ~ oo (see Fig. 4). The front propagates with velocity V(0) > V0.
3. B e l l - l i k e w a v e s
The motion of an autowave front in a chemical EM with a band of higher excitability was investigated experimentally in [4] and theoreitically in [9]. The steady-state patterns occurring in such a medium are depicted in Fig. 1. Inside the band wavefront is positively curved and constitutes a piece of the oscillating solution, Eq. (8). The wings of the pattern in a bounded EM (Fig. 1(a)) are fragments of V-shaped waves (Eq. (5)) matching smoothly the front-line inside the band and approaching the impenetrable boundaries perpendicularly. For an unbounded EM (Fig. 1(b)) the wings degenerate into straight lines (pieces of a plane wave) tilted away from the direction of pattern propagation. The velocity of the pattern, V (0), in a bounded EM of fixed bandwidth (W), is a monotonically decreasing function of the medium width (L) running from the velocity of a planar front in the band (when L = W) to the velocity of the pattern in an unbounded medium (L ~ cx~). The latter, in turn, is a monotonically increasing function of the bandwidth approaching the velocity of a planar front in the band when the bandwidth goes to infinity.
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P.K. Brazhnik, J.J. Tyson/Physica D 102 (1997) 300-312
Next we consider the opposite situation, when excitability is higher in the bulk medium than in the band. We characterize the medium by its macroscopic parameters: the planar front velocity V0 and the diffusion coefficient D. We suppose that in a band of half width W (W < L) located along the symmetry axis of the medium (the axis OX in Figs. 2 and 3) excitability and hence planar front velocity are smaller than in surrounding areas: (V0)in < (V0)out. The diffusion coefficient is considered the same everywhere. Such conditions can be generated, for instance, in a light-sensitive BZ system by the technique developed in [4]. If we start with a planar autowave perpendicular to the OX axis, how will it evolve in time? Obviously, the shape of the front must change in the course of its evolution because local velocity is different for different parts for the front-line: the part in the middle band will lag compared to parts outside, and, as a result, the front-line will become negatively curved in the direction of propagation in the middle band and positively curved outside. Negative curvature of the front in the middle will increase its velocity (due to the curvature effect, see Eq. (l)) compared to (V0)in, that its, it will tend to compensate for the velocity difference. Positive curvature of the external wings also contributes to this compensation. The final steady-state configuration depicted in Fig. 2(a) we call a bell-like wave. As the width of the EM (2L) increases, the pattern becomes more extended and as L -~ cc turns into a structure with asymptotically planar wings (Fig. 2(b)). In Section 3.1 we construct bell-like wave explicitly for the kinematic model. We suppose that the transition region between areas with different excitability is infinitely thin and the medium can be treated as piecewise continuous. Also the size of the band and the EM itself are supposed to be quite large compared to the width of excitation pulse so that the kinematic approach is applicable. At the semi-penetrable boundary we impose geometric constraints on the wavefront line: the front-line must be a continuous function across the boundary, that is, 0 must be a continuous function of l. 3.1. A bell-like wave in a bounded excitable medium
We describe patterns in terms of their intrinsic equations, measuring arclength l from the point of symmetry, y = 0, hence V ( y = 0) = V(0). In this case, l spans the interval from --IL to IL, where 21L is the total length of the front. We start with the bell-like wave in a bounded EM, Fig. 2(a). Among the solutions in Fig. 4, the only appropriate candidate for the negatively curved section inside the central band is a V-shaped wave. Thus the central part of the pattern is described by the expression 1 IV(0)] 2 - [(V0)in] 2 D (V0)in + V(0)cosh[l/(lO)in]'
kin(l) =
l • ( - l * , l*),
(10)
where +l* are the values of l at the semi-penetrable boundaries (y(+l*) = +W), and the characteristic length within the band is: (/0)in = D/v/[V(O)] 2 - [(V0)in] 2. Fragments of the front adjoined to impenetrable boundaries are positively curved in the direction of propagation and approach the impenetrable borders at right angles. For these parts, an appropriate solution among those in Fig. 4 is the oscillating one (Eq. 8) which for our purpose has to be written in the form 1 [V(0)] 2 - [(V0)out] 2 kout(l) = - - ~ (V0)out q- V(0) cos[(/-4- lL)/(lo)out]'
1 • (+l*, i l L ) ,
(1 1)
where the sign ' + ' or ' - ' holds for y > 0 or y < 0, respectively, and we have taken into account the fact that V ( + I L ) = V(O) since the front is orthogonal to impenetrable borders (Oout(ilL) = 0) as it is to the line of its s y m m e t r y (0in(0) = 0). This relates also, through Eq. (1), the curvature of the front at impenetrable boundaries, kout(+lL) =- k(IL), to the curvature in the middle of the pattern, kin(0) -~ k(O), as (V0)out
-
-
D . k(IL) = (V0)in -- D . k(0).
(12)
P.K. Brazhnik, J.J. Tyson/Physica D 102 (1997) 300-312
307
5-
I] ! ! !
~11
i
4-,
3--
! I t t
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!' ! #
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i
....
- 2.0
, ....
....
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2
Length of the front I, mm Fig. 5, Dependence of the curvature of bell-like waves on front arclength l: the solid line corresponds to the pattern in a bounded medium (Fig. 2(a)); the dashed line corresponds to the pattern in an unbounded medium (Fig. 2(b)). Though fronts cross lines of excitation-jump smoothly, their curvatures undergo jumps determined by the difference of planar front velocities in areas with different excitability, Eq. (1.4).
Fronts in inner and outer areas must join smoothly at the semi-penetrable boundaries, that is, 0out(-4-1*) = 0in(+l*),
(13)
and this leads to relationship between curvatures on different sides of the semi-penetrable border (V0)out --
D- kout(+l*) = (V0)in - D . k i n ( + / * ) .
(14)
The angle between the tangent to the front and the axis O Y, 0, is also related to the half-widths of the band and EM through Eqs. (3) as l* W = /
cos[0in(~; k(0))] d~,
(15)
,/
0 IL
/i
L - W = / cos[0out(~; k(lL))] d~,
(16)
J
/"
where the parametric dependence of 0in (l) o n k ( 0 ) is indicated explicitly, as well as 0out(l) on k(IL). For given (V0)in, (V0)out, D, L and W, the four equations (12), (14)-(16) can be solved for the four unknown constants k (0), k (IL), l*, lL. Since solution (11) is an oscillating function, the solution of this problem is not unique. The solution set {k (0), k (IL), l*, 1L} with minimum value of IL, which also has minimal curvature k (1L), corresponds to the bell-like wave in Fig. 2(a). Other solutions, with larger lL and larger k(1L), introduce one or more periods of the oscillatory solutions into the lip of the bell. We will have more to say about these oscillatory solutions in the discussion. The intrinsic equation of the pattern (Fig. 2(a)) we have just constructed is depicted in Fig. 5 by a solid line. Note that, though the front-line of the pattern is smooth, its curvature undergoes a jump (given by Eq. (14)) at the
308
P.K. Brazhnik, J.J. Tyson/Physica D 102 (1997) 300-312
semi-penetrable boundaries. This jump is equal to the difference between curvatures of the front in the middle of the pattern and at the impenetrable border (compare Eqs. (14) and (17)): kin(l*) - kout(l*) = k(0) - k(IL) =
(W0)in - (Vo)out D
(17)
If l* turns out to be much greater than (/0)in, as it is in Fig. 2, the front is almost flat as it approaches the semipenetrable boundary from the inner side and, hence, tilted to the boundary line at the angle ½or determined by the asymptotic angle of the V-shaped wave (see Eq. (7)). Eqs. (15) and (16) written in an explicit form can be used to relate propagation velocity of the pattern V(0) to the width of the band and the EM: W-
V(0)D [ 0*
L - W --
--[(1D/(V0)in2(10)in_ Uintan(½0*))/(l+Uintan(½0*))]}
°/°"
V(O)
D/(Vo)outarctan [Uout tan(½0*)l
I
,
(18)
where 0 * --= 0 (l*) is the angle the front makes with the band boundaries and Uin, out --- ~/( V (0) + (V0)in,out)/( V (0) (V0)in,out). Excluding 0* from here we get the desired expression: V(0) = V(0; L, W). Sets of curves for V(0; L, W) at fixed L and at fixed W are depicted in Figs. 6(a) and (b), respectively. For fixed L the velocity of the pattern decreases monotonically from (V0)out when W = 0 to (VO)in when W = L. In both limiting cases, the front degenerates to a planar wave. For fixed W, the velocity of the pattern is a monotonic function of L increasing from (V0)in when L = W to (V0)out when L ~ oo. 3.2. A bell-like wave in an unbounded medium We expect that for the bell-like wave, when L ~ oo, the front-line becomes asymptotically (at l --+ -1-o~) flat and orthogonal to the direction of front propagation (Fig. 2(b)), therefore V(0) = (V0)out. For the latter condition, the oscillating solution (1 1) does not work any more, and the appropriate front-line is formed by the separatix solution (9), with V0 = (V0)out and l replaced by l + l*, l • (+l*, -4-~). The central part of the bell-like wave in an unbounded medium is still described by Eq. (10) with V(0) replaced by (V0)out. The intrinsic equation of this bell-like wave is shown by a dashed line in Fig. 5. Since kout(l --+ ±oo) = 0, the curvature of the bell-like wave in the middle is given by (see Eq. (12)) k(0) =
(V0)in
(V0)out
-
D
(19)
and does not depend on the bandwidth. 3.3. A bell-like wave with tilted wings Now suppose that instead of launching a plane wave, as we did in Section 3.2, we collide two plane waves (at oblique angle or) in a way that the angular corner they form at the point of collision is located inside the band. The evolution of such a sharp wedge into a V-shaped wave in homogeneous EM has been studied in [8]. In unbounded EM with a band of reduced excitability, the corner will become a fragment of a V-shaped wave only within the band, while the wavefront outside the band will gain some positive curvature in order to equalize velocity differences along the front, see Fig. 3. The intrinsic equation of the front fragment of this pattern inside the band is given by Eq. (10), while the external wings are described by the one-loop solution (Eq. (5), for positive k(0)). The pattern propagates with the velocity (V0)out/cos(½ot) which does not depend on the bandwidth.
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PK, Brazhnik, J.J. Tyson/Physica D 102 (1997) 300-312
_lO; 90--[
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-~~'
so (v)~
w=o.1; o~;
70
~ 0
I
0.4
' I ' I ' half-width L ( m m )
Fig. 6. Propagation velocity of bell-like waves in bounded medium (Fig. 2): (a) dependence of V(0) on W for fixed values of L; (b) dependence of V(0) on L for fixed values of W. Conclusions about the stability of bell-like waves can be made on the basis of the stability of the solutions from which the patterns are constructed. Linear stability analysis shows that the latter are stable with respect to small localized perturbations which disappear diffusively, traveling along the front towards the region of maximum curvature ([k[) [9]. Hence, the bell-like patterns we predict are stable with respect to small perturbations which travel from the periphery to the center of the patterns and disappear diffusively.
4. D i s c u s s i o n
In Section 3 we have shown that in EM with a band of reduced excitability three steady-state patterns may exist, one for a bounded media (Fig. 2(a)) and two for an unbounded (Figs. 2(b) and 3). In [4,9] two steady-state patterns were studied in EM with a band of increased excitability (Fig. 1). Our theory indicates though that one more
P.K. Brazhnik, J.J. Tyson / Physica D 102 (1997) 300-312
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W I
,,,
l,
....
,,
-I -W -2
(vo)~ ~
v(o)
Fig. 7. The stationary pattern which develops in an EM with a band of higher excitability from an initial front (two colliding plane waves) forming a sharp corner inside the band. The difference of plane front velocities inside the band and in the rest of the EM is supposed to be sufficiently small.
steady-state wavefront configuration is possible in this case if the EM is not confined. The pattern develops from two colliding plane waves when the ratio (VO)out/(VO)in is close enough to unity. The resulting steady-state wavefront is shown in Fig. 7. Both wings and the portion inside the band are now pieces of V-shaped waves matching smoothly across the semi-penetrable boundaries. The 'elbow' of this pattern has smaller curvature than would be necessary to connect the same wings in a homogeneous medium. The pattern propagation velocity does not depend on the bandwidth. When the difference of velocities increases (say the velocity inside the band increases), the portion of the steady-state pattern inside the band flattens and finally, when the initial inclination angle 0out(l ~ o0) satisfies the condition COS 0out(/ ~
(X)) = (Vo)out/(Vo)in,
(20)
the inner portion becomes a piece of a plane wave oriented perpendicular to the band boundaries, and the wings become exactly halves of V-shaped waves. For given (V0)in and (V0)out the angle determined from the ratio (20) may be called the critical angle. For the conditions of the experiments with BZ reported in [4] (see also the caption for Fig. 1) this angle is around 40 °. Further increase of (V0)in will flip the inclination of the external wings in the neighborhood of boundaries and this disturbance will propagate with time away from the boundary forming finally the steady-state pattern with opposite to initial inclination, that is, one depicted in Fig. l(b). In this regime the two initially colliding waves undergo a total internal reflection from the band [7]. In the theory proposed so far, the concatenation of V-wave solutions (5) with oscillating solutions (8) can be made at any angle from 0 to ½zr. Therefore the patterns we have described should exist for an arbitrarily large difference of velocities in the band and in the bulk medium. However, some of the elementary solutions we have used to build the more complicated patterns exhibit non-physical self-crossings of the front-lines, which is an artifact of extrapolating the linear dependence, Eq. (1), beyond the critical curvature Kcr, beyond which continuous propagating fronts do not exist. Accounting for critical curvature replaces loops by cusps [ 12] so that self-crossings disappear. Corresponding front-lines in this case permit only a restricted variation of the angle between the tangent to the front and the direction of the pattern propagation. For non-oscillating solutions 0 runs from 0 (k = 0) up to 0 (k = kcr) and for the oscillating solution, from O(k = kmin) to 0(k = kcr). This, in turn, puts the restrictions on possible concatenation angles for
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our patterns. If the difference in properties of the EM across the semi-penetrable boundary is so large that it would require a concatenation angle outside the permissible regions, the steady-state patterns become impossible. The wavefront will break at the boundary. The condition for failure of stationary propagation can be easily derived, for example, for the pattern in Fig. 1(b). The maximum angle the oscillating solution fragment inside the band can make with the axis O Y is l:r + lot 2 cusp, where the cusp angle of the oscillating pattern otcusp = 2 arcsin[( Vcr)in/ V (0)] and ( Vcr)in is the critical velocity for the EM inside the band [ 12]. On the other hand, the angle between the external wing of the pattern and axis O Y is given by the expression 0out = - 2 arccos[(V0)out/V(0)]. Outside and inside 1 portions of the front can be matched smoothly only if 0out > ½zr + ~otcusp, what gives the desirable condition (V0)out > (Vcr)in. Since, generally speaking, the angle a stationary wave makes with the semi-penetrable boundary depends on the bandwidth, the critical curvature effect may also introduce a critical bandwidth beyond which the EM with a band cannot support stationary patterns. Only a small portion of the oscillating solution was used for constructing patterns in Figs. 1 and 2. Formally, exploiting our algorithm, one could introduce several periods of the oscillating solution in the outside regions. Such configurations, though, would include cusps which lead to front instability [ 12], and therefore the whole pattern would be unstable. If the properties of the EM are different on both sides of the band (three layers of different properties), nonsymmetrical steady-state patterns arise. The technique for their construction should, by now, be clear.
5. Conclusions
We have predicted three new steady-state autowave patterns in EM with a band of reduced excitability. The patterns have a bell-like shape, that is, their front-lines are negatively curved in the middle being concave in the direction of the pattern propagation and become positively curved (convex) outside the band, and finally orthogonal to the direction of the pattern propagation or achieving an asymptotic angle. We have described one new stationary pattern in unbounded EM with a band of increased excitability (Fig. 7), similar to the V-shaped wave observed in homogeneous EM but slightly disturbed by the band of increased excitability. The six patterns illustrated in Figs. 1-3 and 7 represent a complete set of stable stationary patterns which can form in EM with a band of different excitability, according to kinematic theory. Propagation velocities of the patterns in unbounded media do not depend on the bandwidth while in the bounded EM patterns propagate with velocities which are functions of the bandwidth. The V-shaped pattern in Fig. 7 exhibits a notable behaviour when (go)out/(go)in exceeds a critical value: the V-shaped wave disintegrates and the front evolves to another stionary patterns. Patterns similar to those we have just described should also appear in EM with a band of different diffusivity (but uniform excitability). This situation can be realized experimentally using either silica gel or porous glasses [ 13].
Acknowledgements
This work was supported by NSF Grant No. CHE 9500763.
References
[ 11 A.T. Winfree, When Time Breaks Down (Princeton UniversityPress, Princeton, NJ, 1987). [2] Oscillations and Traveling Waves in Chemical Systems, eds. R.J. Field and M. Burger (Wiley, New York, 1985); Chemical Waves and Patterns, eds. R. Kapral and K. Showalter (Kluwer Academic Publishers, Dordrecht, 1995). [3] B.S. Kerner and V.V. Osipov, Sov. Phys. Usp. 32 (1989) 101.
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[4] O. Steinbock, V.S. Zykov and S.C. Muller, Phys. Rev. E 48 (1993) 3295. [5] J. Schutze, O. Steinbock and S.C. Muller, Phys. Rev. Lett. 68 (1992) 248; V. Perez-Munuzuri, R. Aliev, B. Vasiev and V.I. Krinsky, Physica D 56 (1992) 229. [6] L.M. Brekhovskikh, Waves in Layered Media (Academic Press, New York, 1980); A.M. Zhabotinsky, M.D. Eager and I.R. Epstein, Phys. Rev. Lett. 71 (1993) 1526. [7] P.K. Brazhnik and J.J. Tyson, Phys. Rev. E 54 (1996) 205. [8] P.K. Brazhnik and V.A. Davydov, Physics Lett. A 199 (1995) 40. [9] P.K. Brazhnik, Physica D 94 (1996), in press. [10] A.S. Mikhailov, V.A. Davydov and V.S. Zykov, Physica D 70 (1994) 1. [11] P. Foerster, S. Muller and B. Hess, Science 241 (1988) 685; Development 109 (1990) 11. [12] P.K. Brazhnik, J.J. Tyson, Phys. Rev. E 54 (1996), in press. [ 13] T. Yamaguchi, L. Kuhnert, Z.S. Nagy-Ungvarai, S.C. Muller and B. Hess, J. Phys. Chem. 96 (1991) 5831 ; T. Amemiya, M. Nakaiwa, T. Ohmoi and T. Yamaguchi, Physica D 84 (1995) 103.