Showalter, Kenneth

Zykov, Vladimir S.

English

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Faculty Scholarship2005Wave Front Interaction Model of StabilizedPropagating Wave SegmentsVladimir S. ZykovKenneth ShowalterFollow this and additional works at: https://researchrepository.wvu.edu/faculty_publicationsThis Article is brought to you for free and open access by The Research Repository @ WVU. It has been accepted for inclusion in Faculty Scholarshipby an authorized administrator of The Research Repository @ WVU. For more information, please contact ian.harmon@mail.wvu.edu.Digital Commons CitationZykov, Vladimir S. and Showalter, Kenneth, "Wave Front Interaction Model of Stabilized Propagating Wave Segments" (2005). FacultyScholarship. 432.https://researchrepository.wvu.edu/faculty_publications/432Wave Front Interaction Model of Stabilized Propagating Wave SegmentsVladimir S. Zykov1 and Kenneth Showalter21Institut fu¨r Theoretische Physik, Technische Universita¨t Berlin, D-10623 Berlin, Germany,2Department of Chemistry, West Virginia University, Morgantown, WV 26506-6045, USA(Received 14 July 2004; published 14 February 2005)A wave front interaction model is developed to describe the relationship between excitability and thesize and shape of stabilized wave segments in a broad class of weakly excitable media. These wavesegments of finite size are unstable but can be stabilized by feedback to the medium excitability; theydefine a separatrix between spiral wave behavior and contracting wave segments. Unbounded wavesegments (critical fingers) lie on the asymptote of this separatrix, defining the boundary between excitableand subexcitable media. The model predictions are compared with results from numerical simulations.DOI: 10.1103/PhysRevLett.94.068302 PACS numbers: 82.40.Bj, 05.45.–a, 05.65.+b, 47.54.+rWave front interaction models have been developed todescribe steadily rotating spiral waves [1] and, in weaklyexcitable media, critical fingers [2]. Critical fingers areobserved at the excitability boundary between excitableand subexcitable media. Both the spiral wavelength and thesize of the spiral core increase as excitability decreases inan excitable medium, until, at the excitable-subexcitableboundary, they become infinite. At this critical excitability,the unbounded spiral wave has opened up to form anunbounded, nearly planar wave with a free end, which isthe critical finger [3]. At lower excitabilities, in the sub-excitable regime, this wave contracts tangentially at its freeend. Hakim and Karma have reviewed applications of frontinteraction models to describe spiral waves and criticalfingers [4].Recent studies of the photosensitive Belousov-Zhabotinsky (BZ) reaction [5,6] have shown that, in addi-tion to spiral waves and critical fingers, there is anotherpossible waveform with a free end, a propagating wavesegment that is stationary in size and shape [7–9]. Theseconstant velocity waves have two free ends, and their sizeand shape are uniquely determined by the excitability ofthe medium. They are inherently unstable, but they can beeasily stabilized by applying an appropriate feedback to theexcitability of the medium. Figure 1 shows examples ofstabilized wave segments in the photosensitive BZ reactionat two different excitabilities [7,8].These waves were stabilized by varying the illuminationintensity imposed on the reaction medium (thereby varyingthe excitability) in proportion to the wave size. The locusof wave size as a function of excitability defines a separa-trix that delineates the boundary between spiral wavebehavior and contracting wave segments. For a particularexcitability, there is an unstable wave segment with aparticular size (and shape), which, in the absence of stabi-lizing feedback, will either form counterrotating spiralwaves or contract tangentially at its free ends and even-tually disappear. The asymptote of the locus of wave sizeas a function of excitability, where the wave size is infinite,is exactly the same excitability boundary between excit-able and subexcitable media as defined by the criticalfinger [7,8].We note that similar particlelike waves have been ob-served in previous theoretical studies, such as activeBrownian particles [10,11], moving spots in reaction-diffusion systems [12,13], and filaments in semiconductors[14]. Particlelike waves have also been experimentallyobserved in a gas discharge system [15,16] and in theoxidation of CO on single-crystal Pt [17].In this Letter, we generalize the wave front interactionmodel [2,4] to describe the stabilized wave segments andshow how their size depends on the excitability of themedium. These waves are particularly amenable to thisanalysis because they are stationary in the moving coor-dinate system. Our analysis offers a theoretical character-ization of the separatrix defined by the locus of wave sizeas a function of excitability, and we show that it is con-sistent with descriptions of critical fingers.We use a generic, two-variable reaction-diffusion modelfor excitable media:@u@tDr2uFu;v; @v@t Gu;v It; (1)where the variables ux; y; t and vx; y; t represent, re-spectively, the activator and inhibitor in a two-dimensionalreaction medium, and typically  1. The function ItFIG. 1. Stabilized wave segments in the photosensitive BZreaction. Left and right panels show waves stabilized at lowerand higher excitabilities, resulting from higher and lower lightintensities, respectively. Superimposed snapshots are shown withan interval of 40.0 s between snapshots; panels ca. 0:95 cm	1:45 cm. Figure adapted from Ref. [8].PRL 94, 068302 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 20050031-9007=05=94(6)=068302(4)$23.00 068302-1  2005 The American Physical Societyspecifies a feedback parameter, such as the intensity oflight imposed on the photosensitive BZ medium, which isapplied uniformly over the domain.For the functions Fu; v and Gu; v, we take the formused previously [2]:Fu; v  3u u3  v; Gu; v  u ; (2)with the parameter values D  1,   0:005, and  1:6. This system has a single uniform rest state,u0; v0  ; 3 3, which is stable with respect tosmall perturbations. However, a suprathreshold perturba-tion applied to the rest state (or specific initial conditions)gives rise to an undamped, propagating excitation wave.For It  Ic ’ 0:05942 and these parameters, the excit-ability of the medium corresponds to the boundary be-tween the excitable and subexcitable medium, where thecritical finger is observed. When It> Ic, the medium issubexcitable and propagating wave segments contract anddisappear. For It< Ic, the medium is excitable and prop-agating wave segments of different sizes, corresponding todifferent excitabilities, can be stabilized by feedback to theexcitability of the medium.We assume the wave segment propagates in x direction,and we will characterize it by its width (rather than by itsarea or arc length as in [7,8]). To this end, we follow thelocation ymaxt of the tip of the wave along the y axisorthogonal to the propagation direction. The wave is sta-bilized by negative feedback to the medium excitabilitywith respect to the wave width, where the control parame-ter It is proportional to the half-width ymax of the wavesegmentIt  kfbymaxt  yd  I0: (3)Figure 2 shows the results of feedback stabilization over arange of wave widths, with kfb  0:00005, I0  0:058,and the target width yd taken from the interval 28< yd <110. We see that the wave width, W  ymax1, is amonotonically increasing function of the asymptotic valueof the control parameter Is  I1. The increase in W withIs is very much like the increase in wave size with increasein light intensity in the photosensitive BZ reaction and insimulations using the Oregonator model [7,8].A wave segment propagating in a two-dimensional me-dium can be characterized by the two boundaries thatdefine it: the wave front and the wave back. The problemof wave propagation can be considered as a free-boundaryproblem, where the normal velocity of the boundary cndepends on the value of slow variable v at the boundaryand on the local interfacial curvature k [18]. At the mid-point of the stabilized wave segment, y  0, the normaldirection coincides with the propagation direction of thefull wave along the x axis, as shown in Fig. 3. Let  ()specify the angle between the x axis and the outward(inward) pointing normal on the front (back) boundaryand s (s) be the arc length measured from the top ofthe stabilized segment. Then the normal velocity of thefront and of the back should be written as cn  cv Dk and cn  cv Dk, where the local curvatureof the front (back) is expressed ask  d=ds: (4)Let km be the local curvature at the midpoint of thestabilized wave. The propagation velocity of the wave istherefore cs  c0 Dkm, where c0  cv  cv0.FIG. 2. Width W of stabilized waves as a function of theasymptotic value of the control parameter Is, computed for thesystem (1) and (2) subjected to feedback stabilization accordingto (3) (solid line) and from approximation (22) (dashed line). Acomputational grid of 500	 500 elements was used, comovingwith the propagating wave, with time and space steps t  0:02and x  0:3.FIG. 3. Boundaries of a propagating wave segment computedas the solution of the free-boundary problem (5)–(7) for Km 0:0402 and Bk  0:5108. The wave front (back) is shown by athick (thin) solid line. Solutions computed for slightly deviatedvalues Bk  0:5101 and Bk  0:5110 are shown by dotted anddashed lines, respectively.PRL 94, 068302 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 2005068302-2Note that within the excited region u  uev, where uevis the largest root of equation Fu; v  0. Then the free-boundary problem for the stabilized wave segment can bewritten asc0 Dk  cs cos; (5)cv Dk  cs cos; (6)csdv=dx  Guev; v Is: (7)Equations (5) and (6) describe the geometry of the frontand back of the wave, respectively, while Eq. (7) describesthe evolution of the slow variable v between the front andthe back of the wave.Using Eq. (4), one can transform Eq. (5) into an ordinarydifferential equation for the angle :Ddds cs cos  c0: (8)Since dy  ds cos and dx  ds sin, it isstraightforward to obtain equations for the Cartesian coor-dinates x; y of the front [7,19]:dxd D sincs cos  c0 ; (9)dyd  D coscs cos  c0 : (10)For the conditions x  0,   =2 and y  0,  0, the solutions of Eqs. (9) and (10) have the formxD 1cslnc0c0  cs cos ; (11)yD cs 2c0csc20  c2sq arctanc0  cs tan2c20  c2sq : (12)These expressions specify the shape of the wave front,x  xy, for any km within the interval 0< km <c0=D.In order to obtain the shape of the wave back, thefunction c  cv must be specified. We first assume thismonotonically decreasing function vanishes at a value v v. We can then say, in the vicinity of v, the wave velocityis a linear function of v: cv  v  v. The constant is uniquely determined by Fu; v. For the case we con-sider here, where Fu; v is the cubic function of u definedin Eq. (2),   1= 2p [2]. We follow Karma [2] andintroduce the quantity   v  v0, which characterizesthe excitability of the medium. Note that the propagationvelocity can be written as c0  , provided that  1.For  1, we can use the approximation Guev; v Is  G, where G  Guev; v  Is [2]. With thisapproximation, the value of the slow variable at the waveback v can be obtained from Eq. (7):v  v0 Gcsxy  xy: (13)Substitution of Eq. (13) into Eq. (6) results in an equationfor the back of the waveDdds c0 Gcsx  x  cs cos: (14)After rescaling of space (S  c0s=D, X  c0x=D,and Km  Dkm=c0), Eq. (14) becomesddS 1 BkX  X1 Km  1 Kmcos; (15)whereBk  GD23: (16)In order to integrate this equation we rewrite Eqs. (11)and (12) in dimensionless form and fix 0<Km < 1.Numerical integration of Eq. (15) starting with  =2, X  0, and Y  Wc0=D at S  0, and takinginto account that dY=dS  cos anddX=dS  sin, produces a solution depending onBk. Using a trial and error method one must vary the valueof Bk until the corresponding solution satisfies the secondboundary condition, Y  0,   0 (see Fig. 3).Repetition of this process for different Km yields the de-pendence Bk  BkKm shown in Fig. 4. For Km  1 thisdependence is well approximated by the linear relationshipBk  B0  #Dkmc0 ; (17)where #  0:63. We note that for Km  0, the wave seg-ment is identical to the critical finger studied in Refs. [2,4],and the value of BkKm ! 0 computed for the free-boundary problem (5)–(7) coincides with the value B0 0:535 found for the critical finger [2].Since our ultimate goal is to explain the relationshipbetween the width of the wave segment W and the controlparameter Is, it is convenient to rewrite Eq. (17) asFIG. 4. Function Bk  BkKm obtained from numericalanalysis of Eq. (15).PRL 94, 068302 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 2005068302-3Dkmc0 1#B0  Bk: (18)The value km determines the width of the wave segment Win accordance with Eq. (12) after the substitution  =2. In the case of small curvature (Dkm=c0  1), thisrelationship is given by the expression (cf. [7,8])W2  2D2c0km: (19)Substitution of Eq. (18) into Eq. (19) results in the impor-tant relationshipD2W2c20 2#2B0  Bk: (20)Thus, if the control parameter Is influences the value of Bk,it determines the value of km and, in turn, the width of thewave segment. If the control parameter Is differs onlyslightly from the value Ic, corresponding to the criticalfinger, i.e., Ic  Is  Ic, Eq. (20) can be linearized to giveD2W2c20 2#2dBkdIsjIsIcIc  Is: (21)In order to obtain quantitative results for the reaction-diffusion model (1) and (2), note that the quantities G and in Eq. (16) depend on the control parameter Is and can beexpressed as G  3p   Is and    Is3 3 Is, since v  0. Thus, for the model under con-sideration Eq. (21) can be written asD2W2c20 2B0#29 Ic2  1c 1GcIc  Is: (22)Here values c and Gc correspond to Is  Ic. The com-puted propagation velocity of the stabilized wave is c0 0:277. Substituting this value into Eq. (22), we find thewidth W of the wave segment as a function of Is, as shownin Fig. 2. The dashed line, showing the theoretical result, isin excellent agreement with the direct integration of thereaction-diffusion model (1)–(3), shown by the solid line.In summary, the existence of stabilized propagatingwave segments can be understood by considering the in-teraction of the wave front and wave back boundaries. Wenote that relationship (17) is universal, as it contains onlythe general characteristics of the two-component model(1). It also describes the limiting case of the critical finger[2] when the size of the wave segment goes to infinity.Equation (21) is also general, since the way in which thecontrol parameter Is influences the parameter Bk is notexplicitly specified. In particular, it provides a qualitativeexplanation for the similar linear relationships obtained inexperiments with the light-sensitive BZ reaction and forthe Oregonator model [7,8].Equation (22) demonstrates how these universal resultscan be applied to a specific model of excitable media. Acomparison of the theoretical predictions with the directnumerical integration (see Fig. 2) shows that there is goodquantitative agreement even for the case where  is notextremely small. Therefore, the theoretical approach de-veloped here provides the opportunity to test quantitativestudies of propagating wave segments in a variety of differ-ent active media.We thank the National Science Foundation (CHE-0415392), the Office of Naval Research, the DeutscheForschungsgemeinschaft (SFB 555), and the Alexandervon Humboldt Foundation for supporting this research.K. S. thanks Alexander Mikhailov for helpful discussionsand his hospitality at the Fritz-Haber-Institut der Max-Planck-Gesellschaft.[1] P. Pelce and J. Sun, Physica D (Amsterdam) 48, 353(1991).[2] A. Karma, Phys. Rev. Lett. 66, 2274 (1991).[3] A. S. Mikhailov and V. S. Zykov, Physica D (Amsterdam)52, 379 (1991).[4] V. Hakim and A. Karma, Phys. Rev. E 60, 5073 (1999).[5] A. N. Zaikin and A. M. 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Ertl, Phys. Rev. E, 67, 036208 (2003).[18] J. J. Tyson and J. P. Keener, Physica D (Amsterdam) 32,327 (1988).[19] O. Steinbock, V. S. Zykov, and S. C. Mu¨ller, Phys. Rev. E,48, 3295 (1993).PRL 94, 068302 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 2005068302-4

Wave Front Interaction Model of Stabilized Propagating Wave Segments

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Wave Front Interaction Model of Stabilized Propagating Wave Segments

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