refraction of active waves in reaction—diffusion media

$: Refraction of active waves in reaction—diffusion media$
cm _- __ l!iB 8 January 1996

PHYSICS LETTERS A

ELSEVIER Physics Letters A 210 (1996) 189-194

Refraction of active waves in reaction-diffusion media Hiroyasu Yamada a, Chihiro Matsuoka b, Akira Yoshimori a

a Department of Physics, NAROYA University, NAGOYA 444-01, Japan

b Department of’ Physics, Ehime University, Ehime 790, JApAn

Received 2 June 1995; revised manuscript received 23 October 1995; accepted for publication 7 November 1995 Communicated by A.P. Fordy

Abstract

The refraction of active waves is analyzed for a stable-metastable reaction-diffusion system consisting of two regions with different diffusion coefficients. The equations governing the evolution of wavefronts are derived by means of an asymptotic perturbation method for boundary layers. These equations describe non-stationary refraction near the steady state regime. It is shown that the dynamics of wavefronts separates into that in the region near the boundary and that far from the boundary.

Non-uniformity or inhomogeneity of system parameters are often observed in biological tissues and ecological populations. The inhomogeneity of these systems is due to gradients of concentrations, spatial dependence of local kinetic parameters, complicated structure and formation of media. Wave propagation in inhomogeneous reaction-diffusion media is different from that in homogeneous media [ I-31. Wave patterns are deformed, broken and pinned. Inhomogene- ity sometimes generates wave sources/sinks.

Refraction of propagating active waves is one of the characteristic phenomena in two-dimensional inhomogeneous reaction-diffusion media. Wave refraction in reaction-diffusion systems appears at a boundary between two media with different diffusivities and local kinetics. Momev [4] has theoretically studied refraction in a bistable reaction-diffusion system with a piecewise constant profile of the diffusion coefficient and identical local kinetics. He found solutions describing the steady state regime of refraction. Agladze and de Kepper [ 51 have observed refraction of chemical waves in the ferroin-catalyzed Belousov-

Zhabotinsky (BZ) reaction-diffusion system at the boundary between the solution and the polyacrylamide gel. Zhabotinsky, Eager and Epstein [6] have studied refraction and reflection of chemical waves in the ferroin-catalyzed BZ reaction-diffusion medium. They created a sharp boundary between two regions with different wave velocities by controlling the local chemical parameters (the oxygen concentration). Their measurements of the refraction angles and velocities showed that refraction of a chemical wave obeys Snell’s law.

In the present paper, we discuss the refraction of active waves in a stable-metastable reaction-diffusion system consisting of two regions with different diffusion coefficients and identical local kinetic parameters. We treat the dynamics of refraction near the steady state regime, and derive a system of equations governing the propagation of wavefronts in this refraction regime. First, we comment on wavefront solutions and the steady state refraction in a reaction- diffusion medium. Next, we introduce a geometrical view of wavefronts in refraction, and then analyze non-

0375-9601/96/$12.00 0 1996 Ekvier Science B.V. All rights reserved SSDI 0375-9601(95)00885-3

190 H. Yamada et al./Physics Letters A 210 (1996) 189-194

stationary refraction in the medium. For refraction phenomena in reaction-diffusion

media, we use a simple model: a stable-metastable reaction-diffusion equation of the form

l ($+V-j) = f(u), j = +DVu, (1)

where E is a small parameter, u is the concentration of the reaction species, j is the diffusion flux of u, D is the diffusion coefficient depending on the space points, V is the nabla operator in two spatial dimensions, and f(u) is an N-shaped function which represents the local kinetics of u. The function f(u) has three zeros, u_, uc and u+ (u_ < uc < u+): u_ and u+ give the metastable and stable state of the medium, respectively (Fig. la). Here we assume

When D is homogeneous, Eq. (1) is known to have stationary solutions of the one-dimensional travelling wavefront type. The wavefront is propagated into the region with the metastable state u_ at fixed velocity, and the back of the wave is in the stable state u+ (Fig. 1 b) . For some polynomial nonlinearities of f(u), the wavefront solution can be given explicitly with a unique wave velocity [7]. For a more gen- eral f(u) as shown in Fig. 1 a, the wavefront solutions to Eq. (1) are obtained from a nonlinear eigenvalue problem,

d2u -@ + A$ + f(u) = 0,

with the eigenvalue A = c/D, here 5 = (X - cr) /~a and c is the wave velocity [ 8,9]. The wavefront solution of Eq. ( 1) given by the eigenfunction C/(l) for Eq. (2) has a thin layer with a width O(E) at 5 = 0 where the gradient of u is not negligible.

To examine refraction of active waves, we make the profile of the diffusion coefficient in Eq. ( 1) piecewise homogeneous with a jump at the line y = 0,

D= D; forrE II; (i= 1,2). (3)

f

-c

Fig. I. (a) The function f(u). The three zeros of f(u), L(_, UC) and u+, indicate the metastable, unstable and stable state of the

medium, respectively. (b) A profile of the travelling wavefront.

The wavefront propagates into the medium with the metastable

state and turns the medium into the stable state.

Here DI and D2 are positive constants, II, = {(x, y) : y > 0} is the upper half plane, and II2 = {(x, y) : y < 0) is the lower half plane. From now on, we will call the medium with the diffusion coefficient DI medium (I) and the medium with the diffusion coefficient 02 medium (II). At the boundary y = 0 between the two regions II, and II2, we require the continuities of u and j,

uI?;=+o = uIp-o, 3. &+0 = 3. j&-0, (4)

where 3 is a unit vector in the direction of the y-axis. Eq. ( 1) with continuities (4) has some remarkable

stationary solutions [ 41. One type of stationary solutions describes the steady state regime of refraction.

H. Yamadaetal./PhysicsLettersA2l0f1996) I89-194 191

(a> 09

Y t

I-II s-0

t

s

+=

n z s=o

SC0 y=o

u=U(O) I-I2

x- x- Fig. 2. The curvilinear coordinate system (s, z) in the half planes fl I and fI2 with the boundary ?: - 0. A thick line denotes the level

curve (wavefront) u = U(0). (a) The coordinates s and r are the arclength along the curve and distance from the curve, respectively.

The set of tangential and normal vectors (t.n) is an orthogonal frame. (b) The two coordinate systems for media (I) and (II) overlap

in the region near the boundary (y - 0). A thick dotted line denotes the wavefront in the fictitious medium. A thin dotted line is the

distance coordinate from the fictitious wavefront.

When the incident wave propagates in II 1, a solution of this type is

(5)

where ki = (sinGi,-costi;) and ci = Afi for r E Iii (i- 1,2). This solution is a travelling plane wave refracted at the boundary y = 0 with the incidence angle $1 and the refraction angle $2. The diffusion coefficients Dt and 02 uniquely determine the refraction angles $1 and & with the continuity condition (4). When the incident wave propagates in the positive direction of the x-axis, the refraction angles become

+; = sin-’ (/z) (i= 1,2). (6)

In experiments of chemical reaction-diffusion media consisting of two regions with different wave velocities, the incident wave arrives at a boundary between two regions at various angles. If the incidence angle does not satisfy Eq. (6), the refraction of the active wave is no longer in the steady state regime: the wavefront has a non-zero curvature and its profile is deformed. We will now discuss propagation of wavefronts in such a non-stationary refraction regime by means of an asymptotic perturbation method for boundary layers.

To the derive kinetics for the geometrical evolution of wavefronts, we introduce a curvilinear coordinate system moving with the wavefront: (s, z, t). Here s is the arclength from the boundary y = 0 along a level curve of the wavefront u( X, y, t) = U( 0) , and z is the distance from this curve (Fig. 2a). Let the position vector x( S, t) denote the curve at time t, t( s, t) be the tangent vector and n( S, t) be the normal vector to the curve x( S, t). Then the two-dimensional space is represented as

r(s, z, t) = x(s, t) + zn(s, t). (7)

When the curve (wavefront) x( S, t) is propagating in each of the regions II, and II2, the integrability conditions for the curve yield a geometrical relation of the form [lo-131

(8)

where 19 is the angle between the tangential vector t and the x-axis; K = CM/& is the curvature of the wavefront; CY and /? are the tangential velocity and the normal velocity.

With respect to this coordinate system, Eqs. ( 1) and (4) become

192 H. Yamada et al./Physics Letters A 210 (1996) 189-194

1 a 1 au ____

1 - KZ as 1 - KZ aS

cost?? -sine: 2

aZ >I, =O, (10)

where [ 1: is the difference between the boundary values of the terms in brackets for media (I) and (II). We require that the projected velocities of the wavefronts on the boundary are the same in each medium,

[ 1 P 2_o. sin0 ,

(11)

This relation is Snell’s law. We use notations [ ] I and [ 12 to represent the boundary values for media (I) and (II). With respect to these notations, [ 1: = [ ] 2 - [ ] 1. The spatial coordinates (s, z ) at the boundary satisfy

s s

[I

2

cost’ds+tantJ J I, sitreds =0,

0 0

s

[z]i = [--& /sinBdsIi (i = 1,2),

0

(12)

from the geometrical point of view. We consider that medium (I) is spread on the whole

plane but fictitiously on the lower half plane II2 when we treat the boundary conditions. We similarly consider medium (II). The wavefront in the fictitious media on the half planes is extended from that in the (real) media on the other half planes. We introduce the curvilinear coordinate for the fictitious media in the same way as for the real media. Two coordinate systems for each medium overlap in the region near the boundary (Fig. 2b). The boundary conditions ( 10) and ( 11) are applied to the inner region of the wavefront, i.e. [z] i N 0( l ) (i = 1,2), which implies [S]i -O(E) (i= 132).

We see refraction phenomena near the steady state regime on the following assumptions: the differences of the refraction angles from the steady state refraction angles are small; the radius of curvature l/~ is

much larger than the width of the wavefronts; the profile of the wavefronts varies slightly from the one- dimensional wavefront solutions.

We now introduce a stretched coordinate transfor- mation and perturbation expansions,

(s,z,t) + (s,5,t), t= -&

u=u(“)+EU(‘)+...,

a=d”)+Ea(‘)+ . . . . p=/?(O)+,/?(‘)+ . . . .

K=K(‘)+EK(‘)+ . . . . ,=,co’+E6”)+ . . . .

(13)

where the zero-order expansion variables are

U(O) = U(5), p(O) = c (= AJD),

lim 8”’ = $, .s+O

l@o~o”(o) = --&. (14)

Substitution of Eq. (13) into Eqs. (8)-( 11) yields equations describing the refraction dynamics of wavefronts for each order in E. We discuss the dynamics of wavefronts derived from the geometrical evolution equation (8) and the constraints near the boundary (lo), (11).

By the perturbation expansions, the zero-order variables give the steady state regime of refraction near the boundary. In terms of expansion variables ( 13)) Eqs. (8) -( 11) describe the propagation of wavefronts,

de(O) _ = K(o) [,(O) at o +~(e(~)-e~))], (15)

with constraints at s N 0,

[dBcOse(“)]; = 0, 1 1 C * = o, sin@@ ,

(16)

where the subscript 0 denotes a value on the boundary s = 0. Near the boundary, Eq. ( 15) has a stationary solution which is consistent with the steady state regime of refraction (6). Far from the boundary, the wavefronts in media (I) and (II) obey Eq. ( 15) al- most independently of each other. The effects of the boundary come from the boundary condition of e(O).

Now we derive the equations governing the first- order variables in the expansion ( 13)) and discuss the non-stationary refraction near the steady state regime.

H, Yamada et al./Physics Letters A 210 (1996) 189-194 193

With the expansion ( 13), the geometrical evolution equation (8) becomes

a?(” a/3(” - = at

as + K(“[(Yp + c(B(O’ - go’) ]

s

+ K(O) + c(B”’ -e;“) +/p"'Kio'dS),

0 (17)

at first order in E, where K( ’ ) = ~%(‘)/ds and

&” = c ( P (1)

tan@O’ c @I) 1 - tan* /3(O)

) tan8(O’ ’

Using Eqs. (8)-( 11) and the solvability conditions for u(I) near the boundary, we obtain the following constraints at s N 0,

1 P (1) e(l) * ~ =o,

C tan e(o) I’

&jet’) sine(0) utauO *

1 at ’

= cos@O’ (P”’ [

_DK’O’) /- ut!$d(],

case(o) (p(‘) - DK”’ ,c*$$‘d[,;,

%C (18)

where ut is a null eigenfunction for the adjoint differential equation of the linearized operator

a* L = - + *a + -(p) af

at* at au ’

on the assumption that ,t satisfies the same continuities as u at the boundary. Eq. (18) gives PC” as a function of e(“, 8”’ and K(O) for media (I) and (II), which makes Eq. (17) a closed system.

Near the boundary s - 0, the wavefront dynamics is in the stationary refraction state and thus K(O) - 0. Then Eq. ( 17) turns into first order linear partial differential equations for [ 8’ “1 I and [ 0’ ’ ) ] *. In these differential equations, we do not find terms which force B to become the stationary refraction angle + in Eq. (6). This result agrees with the experiments by

Zhabotinsky, Eager and Epstein [ 61 in which refraction with different incidence angles is observed but obeying Snell’s law.

For a large value of 1.s (this implies 6 is much larger than the width of wavefronts), the second equation of Eq. (18) gives p (‘I = DK(‘) because

lim ~4~ alJo - =o,

&-+*oo a[

which is consistent with PC” in homogeneous media. When K(O) is small, the first term ~Yp(“/as and third term (proportional to K(O)) in the right hand side of Eq. (17) are small. Then disturbances of 8(” are propagated along s with velocities --cx~’ - c( /3(” - $). The value of the propagating velocities is nega- tive. Thus the disturbances come into the region near the boundary in medium (I), pass the boundary with deformation, and move away from the boundary in medium (II).

In summary, we have analyzed wavefront refraction in a stable-metastable reaction-diffusion system. By means of an asymptotic perturbation method for boundary layers, we derived the system of equations ( 17)) ( 18) governing the deformation kinetics for the geometrical evolution of wavefronts. We were able to study non-stationary refraction near the steady state regime. When the incident wave is deformed from the incidence angle in the steady state regime of refraction, a small perturbation goes through the boundary, and the refraction dynamics is not in the steady state regime.

We would like to thank Professor Nozaki for help- ful discussions and advice. We also thank Professor Konishi for useful comments and encouragement.

References

[ I ] A.M. Zhabotinsky, in: Chemical waves and patterns, eds. R.

Kapral and K. Showalter (Kluwer, Dordrecht, 1995) p. 401.

[2] H.L. Swinney and V.I. Krinsky, eds., Waves and patterns in

chemical and biological media, Physica D 49 ( 1991 ).

[3] D.P. Zipes and J. Jalife, eds., Cardiac electrophysiology:

From cell to bedside (Saunders, Philadelphia, 1990).

[ 41 O.A. Momev, in: Self-organization, autowaves and structures

far from equilibrium, ed. V.I. Krinsky (Springer, Berlin,

1984) p. III.

194 H. Yamada et al. /Physics Letters A 210 (1996) 189-194

[5] K.I. Agladze and F? de Kepper, in: Spatio-temporal

organization in nonequilibrium systems, eds. S.C. Miiller and

T. Plesser (Projekt Verlag, Dortmund, 1992) p. I.

[6] A.M. Zhabotinsky, M.D. Eager and I.R. Epstein, Phys. Rev.

Lett. 71 (1993) 1526.

[7] M. Otwinowski, R. Paul and W.G. Laidlaw, Phys. Lett. A

128 (1988) 483.

[S] A. Scott, Active and nonlinear wave propagation in

electronics ( Wiley-Interscience, New York, 1970).

[9] A.C. Scott, Rev. Mod. Phys. 47 ( 1975) 487.

[IO] R.C. Brower, D.A. Kessler, J. Koplik and H. Levine, Phys.

Rev. A 29 ( 1984) 1335.

[ I I ] A.S. Mikhailov and V.S. Zykov, Physica D 52 ( 1991) 379.

[ 121 H. Yamada and K. Nozaki, Physica D 64 ( 1993) 153.

[ 131 K. Nakayama and M. Wadati, J. Phys. Sot. Japan 62 ( 1993)

1895.

refraction of active waves in reaction—diffusion media

Documents