Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion

We study waves with exponentially decaying oscillatory tails in a reaction-diffusion system with linear cross diffusion. To be specific, we consider a piecewise linear approximation of the FitzHugh-Nagumo model, also known as the Bonhoeffer-van der Pol model. We focus on two types of traveling waves, namely solitary pulses that correspond to a homoclinic solution, and sequences of pulses or wave trains, i.e., a periodic solution. The effect of cross diffusion on wave profiles and speed of propagation is analyzed. We find the intriguing result that both pulses and wave trains occur in the bistable cross-diffusive FitzHugh-Nagumo system, whereas only fronts exist in the standard bistable system without cross diffusion.

Figure 1

Solitary pulse profiles for the activator $u\left(\xi \right)$ (a, d, g), for the inhibitor $v\left(\xi \right)$ (b, e, h), and in the $u-v$ phase plane (bold lines) (c, f, i). The values of the excitation threshold, the ratio of the time scales, and the self-diffusion coefficient are fixed at $a=1/4, \varepsilon =1$, and $D=1$, respectively. The middle parts, $u_2$ and $v_2$, of the pulses are indicated by the gray color. The null-clines $f(u,v)=-u-v+H(u-a)=0$ and $g(u,v)=u-v=0$ are shown by thin lines in panels (c, f, i). Panels (a, b, c) correspond to the case where the self-diffusion and cross-diffusion coefficients are nearly equal: $h=1.1$. The calculated propagation speed is $c\approx 0.613$. Panels (d, e, f) correspond to the strong cross-diffusion case with $h=5$, where the calculated speed is $c\approx 4.819$, and panels (g, h, i) to the extra-strong cross-diffusion case with $h=50$, where the calculated speed is $c\approx 17.846$.

Figure 2

Space-time plot showing the evolution of the solitary pulse $u(x,t)$ for parameter values from Fig. 1, i.e., $a=1/4, \varepsilon =1, D=1$, and $h=50$, respectively.

Figure 3

Speed of the solitary pulse as a function of the cross-diffusion coefficient, $c=c\left(h\right)$. The values of the excitation threshold, the ratio of the time scales, and the self-diffusion coefficient are fixed at $a=1/4, \varepsilon =1$, and $D=1$, respectively.

Figure 4

Periodic wave train (fast wave) profiles for the activator $u\left(\xi \right)$ (a, d, g), for the inhibitor $v\left(\xi \right)$ (b, e, h), and in the $u-v$ phase plane (bold lines) (c, f, i). The values of the excitation threshold, the ratio of the time scales, and the self- and cross-diffusion coefficients are fixed at $a=1/4, \varepsilon =1, D=1$, and $h=50$, respectively. The first parts, $u_1$ and $v_1$, of the wave trains are indicated by the gray color. The null-clines $f(u,v)=-u-v+H(u-a)=0$ and $g(u,v)=u-v=0$ are shown by thin lines in panels (c, f, i). Panels (a, b, c) correspond to the period $L=20$, where the calculated speed is $c\approx 19.013$, panels (d, e, f) to $L=50$, where the calculated speed is $c\approx 17.328$, and panels (g, h, i) to $L=100$, where the calculated speed is $c\approx 17.851$.

Figure 5

Periodic wave train (slow wave) profiles for the activator $u\left(\xi \right)$ (a, d, g), the inhibitor $v\left(\xi \right)$ (b, e, h), and in the $u-v$ phase plane (bold lines) (c, f, i). The values of the excitation threshold, the ratio of the time scales, and the self- and cross-diffusion coefficients are fixed at $a=1/4, \varepsilon =1, D=1$, and $h=50$, respectively. The first parts, $u_1$ and $v_1$, of the wave trains are indicated by the gray color. The null-clines $f(u,v)=-u-v+H(u-a)=0$ and $g(u,v)=u-v=0$ are shown by thin lines in panels (c, f, i). Panels (a, b, c) correspond to the period $L=20$, where the calculated speed is $c\approx 18.595$, panels (d, e, f) to $L=50$, where the calculated speed is $c\approx 12.551$, and panels (g, h, i) to $L=100$, where the calculated speed is $c\approx 13.780$.

Figure 6

Speed of the periodic wave train as a function of the period (dispersion relation), $c=c\left(L\right)$, for (a) $h=5$, (b) $h=10$, and (c) $h=50$. Wave trains with asymmetric and symmetric profiles are marked by circles and squares, respectively. The values of the excitation threshold, the ratio of the time scales, and the self-diffusion coefficient are fixed at $a=1/4, \varepsilon =1$, and $D=1$, respectively.