J. Phys. Chem. 1994, 98, 9676-9681
9676
Dynamics of the Oscillation Phase Distribution in the BZ Reaction Rubin R. Aliev*J and Vadim N. Biktashev* Department of Theoretical Biology, University of Utrecht, Padualaan 8, 3584 CH Utrecht, The Netherlands, and Institute of Mathematical Problems of Biology, Puschino, Moscow region, 142292, Russia Received: March 16, 1994; In Final Form: July 12, 1994@
Oscillation phase dynamics in the homogeneous autooscillating Belousov-Zhabotinsky (BZ) reaction have been studied by computer simulations. The wave dynamics in autooscillating media are qualitatively different from those in excitable media for sufficiently long waves. We propose a classification of chemical waves which is based on the analysis of the shape of the dispersion curve. This delimits conventionally the waves with properties analogous to those in excitable media, “trigger waves”, from “phase waves”. In particular, we study the phenomenon of the “dephasing wave”, recently observed experimentally (Aliev, R. R. J . Phys. Chem. 1994, 98, 3999). This is a special form of phase wave dynamics, which looks like a propagating wave of phase triggering toward a uniform value. We show that phase wave dynamics, including the dephasing waves, can be described in terms of the phase diffusion equation, the well-known Burgers equation. This description remains qualitatively valid for a wide range of wavelengths.
1. Introduction
coefficients W O , w1, and P will be discussed below. Other approaches to studying phase dynamics are natural and computational experiments, carried out with the BZ reaction. Reusser and Field26described the transition of phase waves to trigger waves, when a spatially nonuniform phase distribution was imposed on the system. Similar observations were recently reported by Su et al.,35who studied the dynamics of phase waves arising from oscillation frequency gradients in an inhomogeneously cooled BZ reaction. The effect of phase gradient on the type of waves was also studied experimentally by Bodet and Ross.” Particular interest was attracted to the role of phase waves on the spontaneous formation of wave sources in an oscillating medium.11-13J9-21 In this article, we report the results of computational experiments carried out with a two-variable model of the BZ reaction proposed in refs 28 and 29 on the dynamics of phaseshifted oscillations. We describe the transition of trigger waves to phase waves, using the dispersion relation that covers all types of chemical waves in the BZ reaction. The transition to a uniform phase of oscillations and the dynamics of phase waves in the BZ reaction are measured and interpreted in terms of the Burgers equation (eq 1.1). The correspondence of the computational data obtained to the recently observed experimental results27 is discussed.
During the last decades, the family of excitable and autooscillating chemical reactions has been discovered.’ Among such reactions, the Belousov-Zhabotinsky (BZ) r e a ~ t i o nhas ~ , ~been studied most intensively, and it shows the behaviors also observed in other systems. This makes the BZ reaction a useful experimental model for studying nonlinear phenomena in distributed systems. Following the observation of bulk oscillations, two basic types of propagating chemical waves were found to occur in distributed These are slow waves of constant shape with sharp fronts, the so called trigger waves, and high-velocity waves that can penetrate through a barrier impermeable to diffusion. These waves resemble phase shifted oscillations in a chain of weakly coupled pendula and are known as phase waves, also called pseudowaves7 or kinematic waves.” In addition, “phase diffusion waves” were introduced in refs 11-13, as an intermediate class. In the present paper the term “phase waves” stands for both “phase” and “phase diffusion” waves. Since the discovery of the trigger waves, their dynamics have been studied in detail (see ref 1 for refs), while less attention was paid to the phase waves. Meanwhile, nonuniform phase distribution and phase waves can play a significant role in formation of various wave patterns, such as focal sources (“target vortices,14 and spatio-temporal 2. Model and Methods of Calculation chaos.14-16 Phase storing and dynamics in a light sensitive chemical reaction can be used for image p r o ~ e s s i n g . ’ ~ ~ ’ ~ ~ ~ ~We use a two-variable model of the ferroin-catalyzed BZ reaction developed by Rovinsky and Zhabotinsky2* with rate Ortoleva and Ross were the first to describe phase wave constants evaluated in ref 29. This model was experimentally dynamics as a result of limit cycle perturbation^.'^ This and verified to describe adequately spatio-temporal phenomena in following p a p e r ~ ~ Oconsider -~~ the dynamics of the phase in the BZ r e a c t i ~ n . ~ ~ . ~ ~ the case of small perturbations of the limit cycle. As is shown, The dynamics in a perfectly stirred vessel are described by e.g. in ref 23, these dynamics can be described by the Burgers the differential equations for bromous acid (2.la) and ferroin equations: (2.ib) 3 2 9 drpldt = wo 0 ~ l V r p 1 ~ P V 2 q (1.1) 1 - x ) - 2qa- Z (2.1a) 1-2 x+p Here p is the phase of oscillations and vp and Vrp are the Laplacian and the gradient of the phase; the meaning of the Z d d d t = x - a(2.lb) 1-2 t University of Utrecht.
+
+
* Institute of Mathematical Problems of Biology.
@
Abstract published in Advance ACS Abstracts, September 1, 1994.
0022-365419412098-9676$04.50/0
+,BY>]
where 1994 American Chemical Society
Oscillation Phase Distribution in the BZ Reaction
J. Phys. Chem., Vol. 98, No. 38, 1994 9617 bromous acid
ferriin
a a=- k4K8B (k,AhfJ2’
v=-
k4k7
k4C
t=-
z,
p=-
(k,A),h,,
2k4k13B
0.9
(k,A)2h,
0.1
+
C = Fe(~hen)~+3 Fe(~hen)~+3, A = NaBrO3, B = CH2-
(COOH)2, ho = acidity function, q = 0.6 is a stoichiometric factor, and ki are rate constants. All the calculations were carried out for the following parameters: A = 0.33, B = 0.167, C = 0.00167, = 0.35. This composition mimics the BZ reaction recipe used for the experimental study of similar processes.27 In the present work rate constants are rescaled to 25 “C using the temperature dependence described in ref 29:
0‘ 0
K8 = 3.56 x
M
s-l,
k,, = 1.78 x
+ AQX Wdz = G(x,z) + 6A,z ‘
s-l
‘0.7
(2.2)
(2.3)
,,p%L]
0.5
k4C 0,
where F(x,z) and G(x,z) are the right-hand sides of eqs 2.la and 2.lb, respectively, i i are spatial coordinates, Q i are scaled spatial coordinates, AB is the Laplacian operator with respect to coordinates e, and 6 = DJD,is the ratio of the diffusion cm2 s-l. coefficients. Here we assume Dx = D,= 2 x The computations were carried out in a one-dimensional array of 600 elements using the explicit Euler’s method of integration and Neumann’s (“no flux”) boundary conditions. Space and time steps, h, = 0.005 cm and ht = 0.0083 s, were chosen so that a further increase in resolution could improve the accuracy by no more than a few percent. Prior to one-dimensional calculations, we integrated the zerodimensional system (eqs 2.1) and measured the period of bulk oscillations, TO,with a precision of about 0.5%. This period was determined as the time interval between two successive excitations. An excitation was assumed to occur at those time instants t where x ( t - h,) < xth, and x(t) > xth, with xth, = 20p. The transients were considered to have died out when the difference between two successive periods was less than 0.5%. The amplitudes of the x and z variables (eqs 2.1) were stored during an oscillation cycle through equal time intervals to form phase-calibrated arrays x,d[q] and z , d [ q ] , 0 cp 2 n (see Figure 1). So, the index of thus filled arrays corresponds to the phase of bulk oscillations. These arrays were used to set phase-shifted oscillation distributions as the initial conditions for computations. The initial conditions were imposed with a piece-wise constant phase of oscillations, which equals 90in the left part of the medium (from 0 to 1.4 cm in Figure 3), equals 90 A 9 in the right part (from 1.6 to 3 cm), and linearly increases in the middle (see curve 0 in Figure 8). Several values of A q from n to 3 n were tested in calculations and gave a similar effect. Dispersion relation was measured in a one-dimensional array of length L with periodic boundary conditions. We used a linear phase distribution as initial conditions: q(l)= 2nlIL, 0 < I L. This allowed us to create a solitary pulse enclosed on a ring
+
’
50
k7 = 2.67 M-’s-l,
dxldz = F(x,z)
=
40
0.95
Adding diffusion terms to eqs 2.1 yields the reaction-diffusion system
@,
20 30 time (s)
b
k , = 17.8 M - 2 s-l, k4 = 3023 M-’ s-l, k5 = 1.78 x lo6M-’s-’,
’‘
J l 10
0.8 0.000
0.001
0,010 0.100 bromous acid
1.000
Figure 1. Bulk oscillations in the BZ reaction: (a) Ferriin concentration, z (upper curve), and bromous acid concentration, x (lower curve), vs time. To determine the phases of the oscillations, the full cycle, Le. the interval between successive excitations (the points A and B ) , was scaled to 2n radians. (b) The limit cycle in the (x,z) coordinates. The numbers near the curve denote the phases of the oscillations measured in radians. 0.5
$ 2;
v1
0.4
0.3
0.2 3 CT
-P
L
1.4
a,
>
x
0.1
0 0 0
1
’
20
40
;
0.8 t
69
Wavenumber, r a d / c m
1
,
10
-1 15
20
25
3C
Period s
d
x
~
Wavenumber
Period
Figure 2. Numerically obtained dispersion relations in the oscillating BZ reaction (a and b) and a sketch of typical dispersion relations in an excitable medium (c and d). Special points on the curves: 0, bulk oscillations; Z, point of inflection; C,critical point, corresponding to the shortest possible waves. A point of inflection Z separates trigger
waves (CZ arc) from phase waves (0Z arc). and to measure the established period T of its rotation. Gradually diminishing the circle size L, we measured the dispersion relation T(L), which was then rescaled to appropriate coordinates (Figure 2). These measurements were performed under the same computation parameters as described above, except that space and time steps were reduced to improve the accuracy at small L. Spatial and temporal phase distributions were measured during computations by the following procedure. First, we stored the moments of excitation at every point of the medium.
Aliev and Biktashev
9678 J. Phys. Chem., Vol. 98, No. 38, 1994 The phase at these moments was set to be equal to 2nn, with n being the index of excitation at this point, and was linearly interpolated between these moments. This procedure yields the distribution of the phase vs space and time, q(x,t). On the basis of the thus determined phase distributions, we calculated local wave characteristics: =2 n 1= ~ aviat
v =- d k
(2.4a)
(2.4~)
Here u) is the frequency of oscillations, T is the period of oscillations, k is the wavenumber, ;1is the wavelength, and v is the visible propagation (“phase”) velocity. All these quantities are functions of space and time. To evaluate the coefficients of Burgers equation (eq 1. l), we calculated spatial and temporal derivatives of the phase distribution obtained in the computations. Then we plotted the graph of dqldt vs IVqI2 and Vq. The Burgers equation (eq 1.1) requires this graph to lie in a plane in space (Vq, lVqI2, dqldt). The parameters of this plane, which are the coefficients of the Burgers equation, were fitted by the standard least-squares technique. By this technique, we fit the coefficients u)l and P, while the frequency of bulk oscillations, UO, was taken from a direct zero-dimensional experiment as 2n/To.
Figure 3. Spatio-temporal distribution of the bromous acid (left) and ferriin (right) concentrations, in a simulation with stepwise initial conditions, with phase shift A 9 = 37c rad. Large concentrations are shown with black. The horizontal axis stands for space in the range 0-3 cm; the vertical axis stands for time in the range 0-803 s from bottom to top that corresponds to 30 periods of bulk oscillations.
3. Dispersion Relation One of the most important characteristics of nonlinear waves is the dispersion relation, Le. the relation between the fundamental wave parameters. We have measured the dispersion relation frequently used in the analysis of nonlinear wave function^:^'.^^ frequency vs wavenumber, ~ ( k )and , velocity vs period, ~(7‘).The dependencies obtained are shown in Figure 2, together with typical dependencies for excitable media. Analysis of the dispersion relation shows that in both excitable and self-oscillating chemical media there are critical values kc and Tc such that stable propagation of periodic waves with k > kc and T -= Tc is impossible (points C on Figure 2). In selfoscillating media, waves with k and T close to k, and Tc exhibit the properties of trigger waves; in excitable media, trigger waves are the only class of large amplitude nonlinear traveling waves. In contrast to excitable media, a self-oscillating medium has a qualitatively different regime, occurring as the period of the wave train, T, is close to the period of bulk oscillations, TO, and the wavelength is sufficiently large (points 0 on Figure 2). Waves with T = TO and small k exhibit characteristics qualitatively different from those of trigger waves-e.g. the visible velocity of propagation (the phase velocity) is significantly higher than the velocity of information transfer (the group velocity). These waves are usually referred to as phase waves. Another characteristic feature of the dispersion curves of a self-oscillating medium is that they both are not convex, unlike those of excitable media. This results from the different behavior of the dispersion relation at large wavelengths. Therefore, we can conventionally separate the dispersion curves into two branches, corresponding to slow trigger waves (CZ branch) and rapid phase waves (IO branch). A natural boundary between the branches may be chosen at the point of inflection of either curve (points Z on Figure 2). We use this classification below for the analysis of phase-shifted oscillations. Naturally, this separation is conventional, and in fact the trigger waves and phase waves are limit cases. They are separated by a zone which is referred to as “phase diffusion waves” by some authors.’
Figure 4. Spatio-temporal distribution of the local period of oscillations. The black denotes the smallest period. The region of faster oscillations forms a dephasing wave moving leftward. The horizontal axis stands for space in the range 0-3 cm; the vertical axis stands for time in the range 0-803 s from bottom to top that corresponds to 30 periods of bulk oscillations.
Strictly speaking, the dispersion relation (Figure 2 ) cannot be applied to nonstationary processes (nonperiodic waves). Nevertheless, we will use it below, because we deal with approximately periodic waves, also known as “slowly varying
4. Dynamics of Phase-Shifted Oscillations Figure 3 illustrates the dynamics of bromous acid and ferriin in the system with phase-shifted oscillations. A stepwise phase distribution, initially sharp, gets smoother during its evolution. Note also that the phase step initially placed in the center drifts to the left. The mechanism of this effect will be discussed below. Figure 4 shows the space-time distribution of the period of oscillations. The gray color in the left and right parts, where unperturbed oscillations occur, corresponds to the period of bulk oscillations, TO. A phase gradient applied to the system resulted in the decrease of the period of oscillations, seen as a darker region in the figure. The black color in this figure marks the fastest oscillations with a period of about 0.7T0, which occur at the place of the steepest gradient. It is seen that the region of high-frequency oscillations gets smoother during the evolution
J. Phys. Chem., Vol. 98, No. 38, 1994 9679
Oscillation Phase Distribution in the BZ Reaction
a
“T~-
120
l/velocity (slcm)
i/v
100-
1
C I
“I, 1
I
!
0.3
0.2‘ 0
500
1000 time ( 8 )
1500
2000
0
0.5
1 1.5 2 distance (cm)
2.5
3
minimal velocity (cm/s)
position (cm) 1.51,
I
b
0.7‘ 0
500
1000 time (SI
1500
I
2000
Figure 5. (a) Time dependence of the maximal frequency of oscillations in the system, i.e. the amplitude of the dephasing wave (cf. 3). (b) Position where the maximal frequency was observed (the crest of the dephasing wave). This plot shows the motion of the dephasing wave. The vertical dotted line shows the moment f* = 70 s corresponding to the qualitative change in the wave dynamics.
and drifts leftward. This process, called the dephasing wave,27 leads to the adjusting of the out-of-phase oscillations to a uniform phase. In other words, the dephasing wave moves from the region where phase-lead points occur (right wall) to phaselag regions (left wall) and syncronizes oscillations, establishing unifonnly the most advanced phase while passing. The amplitude of the dephasing wave can be defined as the difference between the frequency inside the phase jump and the frequency of bulk oscillations. Figure 5a presents a plot of the maximal frequency of oscillations, on the crest of the dephasing wave, vs time. The maximal frequency decreases with time and remains larger than the frequency of bulk oscillations. Figure 5b illustrates the motion of the dephasing wave, showing the position of its crest vs time. After the time t* x 70 s, the velocity of the drift is approximately constant v c d s . This value corresponds to the rate of the = 1.9 x phase synchronization in the system and is significantly lower than the velocity of the chemical waves that constitute the background for the dephasing wave. Similar results were observed in the natural experiments with the light sensitive BZ reaction.27 Let us note that at the moment t* the local frequency and local phase gradient at the crest of the dephasing wave are close to the coordinates of the inflection point on the dispersion curve in Figure 2a. This substantiates the introduction of the inflection point on the dispersion curve as a boundary between phase and trigger waves. Figure 6a presents the distribution of local wave velocity vs space at several time moments. The minimal velocity occurs at the phase jump, at the place of the maximum phase gradient (see eq 2.4). The velocity tends to infinity far ahead and back of the phase jump, Le. in the regions of bulk oscillations. The shift of the profiles in the left direction, seen in this figure, is caused by the motion of the dephasing wave. The time dependence of the value of minimal velocity (Figure 6b) illustrates the transition of trigger waves to phase waves. The initially large phase gradient, corresponding to the CZ branch
0.008
0
500
1000 time (SI
1500
2000
Figure 6. (a) Velocity profiles in the system. The curves 1-8 show an instantaneous velocity distribution measured at moments 16.8 s (curve 1) through 1835 s (curve 8) with the interval of 259.7 s. The
velocity profile pulse drifts in the left direction, getting smoother with time. (b) Minimal propagation velocity vs time. Initially, the propagation velocity corresponds to that inherent to trigger waves (CZinterval) but during evolution becomes significantly larger, indicating the transition of trigger waves to phase waves. of the dispersion curve (see Figure 2), results in low-velocity trigger waves. The trigger waves have turned to phase waves after approximately 70 s, due to the decrease of the phase gradient. It is interesting that the opposite transition was observed by Reusser and Field,26and Su et al.35in the system with the imposed phase gradient.
5. Phase Wave Dynamics: The Burgers Equation Approach We have estimated the coefficients of the Burgers equation (1.1) in application to the BZ reaction:
wo = 0.2346 (rad/s)
w1= 1.139 x
(cm2/(s rad))
P = 2.277 x
(5.1)
(cm2/s)
These values were estimated by fitting of the dependence dpl dt vs the Laplacian and squared gradient of p (Figure 7) by a plane using least-squares techniques (see Section 2). In addition to the above procedure, some a priori estimations can be made reasoning from the physical meaning of the coefficients in eqs 5.1. Indeed, wo (the frequency of bulk oscillations) and 01 are just the coefficients of the Taylor expansion for the dispersion curve w(k):
+
w(k) = w o 01k2
+ ...
(5.2)
We can evaluate the coefficients, approximating the dispersion curve (Figure 2a) by a parabola. With k ranged from 0 to 17 radcm, we obtained w1 = 1.037 x c d s . The coefficient P is determined by the diffusion coefficient of the reagents. In the case of equal diffusion coefficients, P is equal to them,19-25 cm2/s. A minor divergence of the P = D1 = DZ = 2 x thus evaluated coefficients in comparison to the data set in eqs
Aliev and Biktashev
9680 J. Phys. Chem., Vol. 98, No. 38, 1994 phase (rad)
9.421
0
1.5
1
0.5
2
2.5
3
distance (cm)
Figure 7. Dependence of dqldt (z axis) vs a2q/ax2and laqlax12 (x and y ixes). This dependence is well approximated by a plane, which is required by the Burgers equation (eq 1.1). Axes are scaled in the to following ranges: x, -26.6 to 33.8; y, 0 to 63; z, -6.67 x 1.55 x 10-3.
5.1 shows that the Burgers equation provides a rather good approximation, with a tolerance of about lo%, for the actual phase evolution in the BZ reaction. An analytical solution of eq 1.1 can be obtained by introducing a “logarithmic phase” ly (Cole-Hopf transformation): q = oot
+ Po,-’ log
?J!f
(5.3)
This reduces the nonlinear equation (eq 1.1) to a linear diffusion equation:
To mimic the computations presented above, let us choose stepwise initial conditions:
(5-5) This corresponds to analogous initial conditions for eq 5.4
with
Figure 8. Phase profiles q(x,t) - oot at moments t = 33.5 s (curve 1) through 619.9 s (curve 8) with the interval of 83.8 s. Curve 0 is the phase profile used as initial conditions (t = 0). The curves 1-8 intersect each other at a single point A, as is required by the Burgers equation.
According to eq 5.8, the established phase of oscillations at large t at any point in the medium is
where q
*
pl + qQ = 3 L
vrvo V(x,t) = +v1+vo
2
2
(5.8)
431
-VO
v
-1
log(ch(v))
(5.11)
L
At small v, the established phase of oscillations, q*,is the averaged phase ( q o q11)/2. At large v, q ~ *tends to q1; i.e., the initially most leading phase is established in the system after passing the dephasing wave. Experimental observations of this effect were reported in ref 27. It is easy to note that the value p,defined by eq 5.1 1, is the point of intersection of the curves q(x,t) - mot at various values of t, if q is the analytical solution given by eqs 5.3 and 5.8. Figure 8 represents the numerically obtained phase profiles. It can be seen that all the curves do intersect at a single point. The phase at the point of intersection is 8.3 rad (Figure 8). This value is close to that calculated from eq 5.11, p = 8.06 rad, which provides additional evidence for the validity of the Burgers approximation for this case. The motion of the dephasing wave can be studied by tracing the position of its crest, i.e. the point of maximal local frequency. This point is defined by the equation
+
(5.7) The solution to the problem (eqs 5.4 and 5.6) can be written explicity :
+
cox = qtx= 0
(5.12)
x = 2y(Pt)’”
(5.13)
The substitution
transforms eq 5.12 to an equation for y , independent of t:
ed(+2)
The solution leads to visually different patterns of phase evolution, depending on the value of the dimensionless ratio
(5.9) where A q = - q o is the initial phase shift. At v = 0, the solution is identical to that of a linear diffusion equation; i.e., an initially sharp phase distribution is blurring but remaining symmetric. As v is nonzero but small, the evolution of phase becomes asymmetric, though it still looks like diffusion. When v is large, the behavior becomes significantly asymmetric, resulting in dephasing wave appearance. It is the latter case that is realized in the computations presented above. The values presented in eqs 5.1 correspond to v = 2.36.
2
v1-vo - 3 y e -0 2 (5.14)
This means that the crest of the dephasing wave moves according to the law in eq 5.13, where the value of the constant y can be found for each particular case from the transcendental equation (eq 5.14). Numerical analysis shows that this equation has two roots. At ml > 0 the smaller (negative) root corresponds to the point of maximal local frequency, while the larger one corresponds to the minimum of local frequency. In Figure 9, the dependence of the dephasing wave crest position vs the square root of time is shown (cf.Figure 5b). At t** 70 s this dependence changes qualitatively, being approximately linear at t > t**,in accordance with eq 5.13. Note that this time moment is the same as the moment t*, mentioned
Oscillation Phase Distribution in the BZ Reaction
J. Phys. Chem., Vol. 98, No. 38, 1994 9681
,
position (cm)
1.5,
i l
1.3
n -. -7.
0
5
10
15 20 25 time"' (s"')
30
35
Figure 9. The position of the dephasing wave crest vs the square root of time. The vertical dotted line shows the moment re* = 70 s corresponding to the qualitative change in the wave dynamics. The Burgers equation requires this dependence to be linear, which is satisfied after h.
in the previous section, and that it corresponds to the crossing of the inflection point of the dispersion curve. Therefore, we can state that the qualitative prediction (eq 5.13) derived from the Burgers equation is valid for the waves with (k,w)lying on the OI branch of the dispersion curve (Figure 2a), i.e. just for the phase waves, as they were defined in Section 2.
6. Discussion In this paper, we have suggested and verified a simple criterion for distinguishing between different regimes of wave propagation in an autooscillating medium. This criterion is based on the form of the dispersion curves. As well as the conventional division of chemical waves to trigger waves, diffusion phase waves, and phase waves, this criterion is rather qualitative and relative, because of the following reasons. First, there is no strict boundary between trigger and phase waves. The point of inflection, as it is shown above, separates the two regions, but ordinarily it lies at the valley of the dispersion function (Figure 2); i.e., the second derivative is small in the broad region of parameters. That is why the determination of zero for the second derivative (the point of inflection) is strongly affected by the noise which occurs in any experimental system. Second, we should remember that the trigger waves in autooscillating media have properties close to but not coinciding with those in excitable media. For instance, a solitary pulse, or a finite train of trigger waves initiated in an unbounded autooscillating medium, will transform to phase waves. This is because of bulk oscillations occumng far ahead and back of such waves, which interfere with their propagation, raising the velocity and smoothing the fronts (this phenomenon was mentioned by Zeldovich while studying flame fronts propagati or^).^^ Therefore, the classification of waves should take into account not only initial conditions but also local (in space and time) properties of waves. We have applied this classification to the phenomenon of the dephasing wave described in ref 27, which is a superstructure over the phase waves. We suggest a phenomenological discription of this phenomenon, using the asymptotical theory of oscillation phase dynamics-the Burgers equation, which is formally applicable for long phase waves. We have shown that the Burgers equation adequately describes the numerical results, both in local and global, and quantitative and qualitative aspects. In particular, we have shown that (i) if we substitute the experimental phase into eq 1.1, the difference between the left-hand side and the right-
hand side is small (see Figure 7); (ii) the coefficients of the Burgers equation, obtained by fitting to the experimental phase evolution, are close to those obtained by a pion' estimations; (iii) the precise solution of the Burgers equation shows the same qualitative behavior as the experimentally measured phase (in particular, the dephasing wave is observed both in a numeric experiment and in the precise solution of the Burgers equation); (iv) the qualitative predictions of the Burgers equation about the dynamics of the dephasing wave remain valid while the local frequency and wavenumber correspond to those of phase waves, in the sense of the suggested classification. The dephasing wave motion, as described by the Burgers equation, is the motion with nonuniform velocity (see eq 5.13). Meanwhile, recent natural experiments with the BZ reaction27 are well described in the assumption of the constant velocity of this wave. This seeming contradiction can be explained as follows. According to eq 5.13, the velocity is proportional to t-lR, so at large t it changes slowly, being approximately constant during a long time period. Comparing Figures 5b and 9, we can see that both assumptions are reasonably valid for the numeric phase evolution. To distinguish between these assumptions, experiments of much longer duration are needed.
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