Shaping wave patterns in reaction-diffusion systems

We present a method to control the two-dimensional shape of traveling wave solutions to reaction-diffusion systems, such as, interfaces and excitation pulses. Control signals that realize a pregiven wave shape are determined analytically from nonlinear evolution equation for isoconcentration lines as the perturbed nonlinear phase diffusion equation or the perturbed linear eikonal equation. While the control enforces a desired wave shape perpendicular to the local propagation direction, the wave profile along the propagation direction itself remains almost unaffected. Provided that the one-dimensional wave profile of all state variables and its propagation velocity can be measured experimentally, and the diffusion coefficients of the reacting species are given, the new approach can be applied even if the underlying nonlinear reaction kinetics are unknown.

Figure 1

Top: Shaping a Schlögl front according to a protocol $\varphi \left(y,t\right)$ (black line). Yellow (red) corresponds to a high (low) value of $u$. Bottom: Corresponding control function $f\left(\mathbf{r},t\right)$. Blue (black) corresponds to a high (low) value of $f$. See movie in the Supplemental Material [38].

Figure 2

Time-dependent local coordinate system $\left(\rho ,s\right)$ used for the derivation of the linear eikonal equation. Dashed blue lines are contours with $s=\text{const}$, black solid lines are contours with $\rho =\text{const}$. The red dotted line denotes the curve $\gamma \left(s,t\right)$ which corresponds to the contour with $\rho =0$. $\gamma \left(s,t\right)$ defines the position of the traveling wave in two spatial dimensions and outlines the desired shape of the controlled wave pattern.

Figure 3

Shaping a circular FitzHugh-Nagumo pulse to a flower with five petals. (a) activator $u$, (b) inhibitor $v$, (c) activator control $f_u$, (d) inhibitor control $f_v$. The protocol curve $\gamma \left(s,t\right)$ outlining the maximum activator value is shown as the black line (top left). Small numerical artifacts resulting from inverting the coordinate transform $\chi$ from Cartesian $\left(x,y\right)$ to local coordinates $\left(\rho ,s\right)$ can be seen as a dark shadow in the control functions. Domain size is $L_x=L_y=150$. See Supplemental Material [38] for a movie.

Figure 4

Additionally to the bulk control $\mathbf{f}\left(x,t\right)$ acting inside the domain, a boundary control $b\left(t\right)$ is necessary to move a wave from inside the domain to the outside and back again. Top: Space-time plot of the bulk control $\mathbf{f}\left(x,t\right)$ acting inside the domain. Light (dark) corresponds to a high (low) value of $\mathbf{f}$. The white line denotes the contour of the Schlögl front with $u\left(x,t\right)=1/2$. Bottom: Boundary control over time applied at the upper domain boundary at $x=60$. Left: Under bulk control alone, a front leaving the domain cannot be returned. Right: With additional boundary control the front is successfully forced back into the domain. Parameter of the Schlögl model is $a=0.1$. See Supplemental Material [38] for two movies.