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, $$u_t=D{u}_{xx}+u(1-\phi \ast u),$$ where $\phi \ast u$ is a spatial convolution with the top hat kernel, $\phi (y)\equiv H(\frac{1}{4}-y^2)$. 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 ${\mathrm{\Delta }}_1\approx 0.00297$. 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 $D\ge {\mathrm{\Delta }}_1$, permanent form travelling waves, with minimum wavespeed, $2\sqrt{D}$, are generated, whilst for $0<D<{\mathrm{\Delta }}_1$, 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.