The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension, $$u_t = D u_{xx} + u(1-\phi*u),$$ where $\phi*u$ is a spatial convolution with the top hat kernel, $\phi(y) \equiv H\left(\frac{1}{4}-y^2\right)$. After showing that the problem is globally well-posed, we demonstrate that positive, spatially-periodic solutions bifurcate from the spatially-uniform steady state solution $u=1$ as the diffusivity, $D$, decreases through $\Delta_1 \approx 0.00297$. We explicitly construct these spatially-periodic solutions as uniformly-valid asymptotic approximations for $D \ll 1$, over one wavelength, via the method of matched asymptotic expansions. These consist, at leading order, of regularly-spaced, compactly-supported regions with width of $O(1)$ where $u=O(1)$, separated by regions where $u$ is exponentially small at leading order as $D \to 0^+$. From numerical solutions, we find that for $D \geq \Delta_1$, permanent form travelling waves, with minimum wavespeed, $2 \sqrt{D}$, are generated, whilst for $0 < D < \Delta_1$, the wavefronts generated separate the regions where $u=0$ from a region where a steady periodic solution is created. The structure of these transitional travelling waves is examined in some detail.