Achilles' heel of a traveling pulse subject to a local external stimulus

The response of a traveling pulse to a local external stimulus is considered numerically for a modified three-component Oregonator, which is a model system for the photosensitive Belousov-Zhabotinsky (BZ) reaction. The traveling pulse is traced and constantly stimulated, with the distance between the pulse and the stimulus being kept constant. We are interested in the minimal strength of the spatially localized stimulus in order to eliminate the pulse. The use of a stimulus of small width allows us to detect the point in the pulse most sensitive to the external stimulus, referred to as the “Achilles' heel” of the traveling pulse, at which minimal strength of stimulus causes a collapse of the pulse. Our findings are demonstrated experimentally as well with the photosensitive BZ reaction.

Figure 1

Schematic figures for our problem setting. (a) A stable traveling pulse is stimulated for $t\ge t_{\mathrm{irrad}}$ with a local light irradiation that moves together with the pulse. (b) The irradiation function $\varphi (x,t)$ is changed so that the distance $dev$ between the pulse front $x_{\mathrm{front}}$ and the irradiation center $x_{\mathrm{irrad}}$ is kept constant. Here, the pulse front and the pulse back are defined as the intersections of the $v$ component $v(x,t)$ of a traveling pulse solution with some fixed threshold $v_{th}$. The pulse width ${\chi }_0\left(t\right)$ is also defined as ${\chi }_0\left(t\right)=x_{\mathrm{front}}\left(t\right)-x_{\mathrm{back}}\left(t\right)$. During the irradiation, the width $d_{\mathrm{irrad}}$ and the intensity ${\epsilon }_{\mathrm{irrad}}$ are fixed.

Figure 2

The profile of a stable rightward-traveling pulse solution observed numerically in (1) with (2) for (a) ${\epsilon }_{\mathrm{irrad}}=0$ and (b) ${\epsilon }_{\mathrm{irrad}}\ne 0$. The figures in the middle and on the right are magnifications of those on the left. The $u$, $v$, and $w$ components are represented by the gray solid, dot-dash, and black solid curves, respectively. (a) Before the irradiation, the maximal value $v_{max}$ of the $v$ component is about $0.1274$, such that the threshold $v_{th}$ for determining the pulse front is fixed as $v_{th}=0.01274$. The dashed vertical lines denote the position of the pulse front $x_{\mathrm{front}}$, defined as $v[x_{\mathrm{front}}\left(t\right),t]=v_{th}$. In the middle figure, the edge of the dent in the $w$ component is indicated by the bold arrow. (b) Under the irradiation, the size of the irradiated traveling pulse (ITP) becomes smaller than that from before the irradiation in (a). The parameters for the irradiation in (2) are fixed as $d_{\mathrm{irrad}}=1.0$, $dev=0.6$, and ${\epsilon }_{\mathrm{irrad}}=0.00453$. The irradiated part is indicated by the shaded regions. In (a) and (b), the numerical simulation is performed for $x\in [0,200)$ with the periodic boundary condition, and the other parameters are chosen as ${\epsilon }_1=0.0508,{\epsilon }_2=0.000168,q=0.0001,f=2.5,D_u=D_v=D_w=1.0,$ and ${\varphi }_0=0.0005$.

Figure 3

(a) The phase diagram for the pulse behavior obtained numerically for (1) with (2). The irradiation width is $d_{\mathrm{irrad}}=1.0$. The horizontal and vertical axes denote the deviation of the irradiation center from the pulse front $dev$, and the irradiation intensity ${\epsilon }_{\mathrm{irrad}}$, respectively. A stable traveling pulse is observed in the TP regime, while the traveling pulse eventually collapses for the parameters in the COL regime. (b) (i) The existence of the irradiated traveling pulse solution (black solid) and the spatially uniform solution (gray solid) obtained by numerical simulation for (1) with (2). The deviation $dev$ is fixed at $dev=1.0$, and the irradiation intensity ${\epsilon }_{\mathrm{irrad}}$ is varied. The vertical axis denotes the $L_2$ norm $\sqrt{{\int }_0^L\{u{(x,t)}^2+v{(x,t)}^2+w{(x,t)}^2\}dx\phantom{{}^{l^l}}}$ of a solution, where $L=200$ is the system size for the numerical simulation. The stable irradiated traveling pulse (ITP) solution branch terminates at ${\epsilon }_{\mathrm{irrad}}={\epsilon }_{\mathrm{irrad}}^{SN}$, while the spatially uniform solution (BG) is observed stably for the range of ${\epsilon }_{\mathrm{irrad}}$ we are concerned with. (ii) The ITP solution obtained by numerical simulation for (1) with (2). The deviation $dev$ is fixed at $dev=1.0$, and the irradiation intensity ${\epsilon }_{\mathrm{irrad}}$ is varied. (c) A saddle-node bifurcation curve (gray solid) obtained by changing the value of $dev$ and plotting the saddle-node point ${\epsilon }_{\mathrm{irrad}}^{SN}$ in (b-i). The black dashed curve, which almost coincides with the gray solid curve, represents the TP-COL phase boundary in (a) magnified around the local minimum. Note that the saddle-node point ${\epsilon }_{\mathrm{irrad}}^{SN}$ in (b-i), and hence the saddle-node bifurcation curve, was obtained by increasing the value of ${\epsilon }_{\mathrm{irrad}}$ and continuating the ITP solution, while the TP-COL boundary in (a) was obtained by the direct numerical simulation for each $dev$ and ${\epsilon }_{\mathrm{irrad}}$, with the initial condition fixed as the traveling pulse before the irradiation shown in Fig. 2. The parameters are the same as those in Fig. 2.

Figure 4

(a) (i) The TP-COL phase boundaries obtained numerically for (1) with (2) for the three different irradiation widths of $d_{\mathrm{irrad}}=1.0,0.5$, and 0.25, which are represented by the dashed, dash-dot, and solid curves, respectively. (ii) A magnification of the profile of the stable traveling pulse solution in Fig. 2. The vertical shaded region denotes the range $dev\in [0.3,0.4]$. (b) (i) The magnified phase boundaries of Fig. 4 around the local minima. (ii) The three phase boundaries in (i) plotted with the vertical axis replaced by $E=d_{\mathrm{irrad}}\times {\epsilon }_{\mathrm{irrad}}$. (c) (i) The TP-COL phase boundaries obtained numerically for (1) with (2) for the three different irradiation widths of $d_{\mathrm{irrad}}=0.2,0.1$, and 0.05, which are represented by the dashed, dash-dot, and solid curves, respectively. (ii) The three phase boundaries in (i) plotted with the vertical axis replaced by $E=d_{\mathrm{irrad}}\times {\epsilon }_{\mathrm{irrad}}$. The other parameters are the same as those in Fig. 2.

Figure 5

Schematic illustration of (a) the reaction medium (top view) and (b) the experimental setup (side view).

Figure 6

(a) Experimental results for the propagation of a chemical wave for different values of $dev$: (i) traveling pulse, ${\epsilon }_{\mathrm{irrad}}=51.5\mathrm{klx}$, and $dev=0.59\mathrm{mm}$; (ii) collapse, ${\epsilon }_{\mathrm{irrad}}=51.5\mathrm{klx}$ and $dev=0.25\mathrm{mm}$. The horizontal and vertical axes denote space and time, respectively. The analyzed region was along the $x$ axis, as defined schematically in Fig. 5. (b) Experimental results for ${\epsilon }_{\mathrm{irrad}}$ depending on $dev$ for light pulse irradiation to a chemical wave. The collapse and the traveling pulse are shown by filled and empty circles, respectively. $d_{\mathrm{irrad}}$ was fixed at $0.45\mathrm{mm}$. The shaded region $-0.69<dev<0.23$ corresponds to the range in which the light pulse completely or partly overlapped a chemical wave. An estimated boundary is also superposed with the dashed curve to facilitate comparison with the numerical counterpart in Fig. 3.

Figure 7

The profiles of irradiated traveling pulse solutions observed numerically for $d_{\mathrm{irrad}}=1.0$ and (i) $dev=0.0$, (ii) $dev=0.6$, and (iii) $dev=5.0$. The parameters correspond to the three points in the TP regime labeled (i)–(iii) in the rightmost phase diagram, which is the same one as in Fig. 3. The $u$, $v$, and $w$ components are represented by gray solid, dot-dash, and black solid curves, respectively. The $w$ component of the profile is largely deformed for (ii) and (iii) around the irradiated part indicated by the shaded regions. The other parameters are given in Fig. 2.