Simple Diffraction Processes for Nagumo‐ and Fisher‐type Diffusion Fronts

Abstract

The diffraction of a diffusion front by concave and convex wedges is studied for Nagumo and Fisher's equations on the limit of fast reaction and small diffusion, using both the asymptotic theory and full numerical solutions. For the case of a convex corner, the full numerical solution confirms that the front evolves according to the asymptotic theories. On the other hand, for the concave corner, it is shown numerically that the diffraction produces at the corner a region of low values of the solution for both the Nagumo and Fisher's equations. Moreover, in both cases, the front eventually evolves, leaving behind a cavity. In the case of the Nagumo equation, it is shown that the long‐term behavior of the diffraction front is just a traveling front, bent at the sloping wall. The bent region maintains its size as the front travels. This behavior is predicted by an exact traveling wave solution of the asymptotic equation for the front propagation. Good agreement is found between the numerical and the asymptotic solutions. On the other hand, behavior of the diffracted front for Fisher's equation is different. In this case, the front is bent at the sloping wall, but, as time passes, the bend becomes smaller and moves toward the sloping wall. This behavior is, again, predicted by the asymptotic solution. The numerics strongly suggest that the final state for the concave corner is a steady cavity‐like solution with low values at the corner and high values away from it. This solution has an angular dependence that varies with the angle of the sloping wall.