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We study the Cauchy problem on the real line for the nonlocal Fisher-KPPequation 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, $\phi(y) \equivH\left(\frac{1}{4}-y^2\right)$.  After showing that the problem is globally well-posed, we demonstrate thatpositive, spatially-periodic solutions bifurcate from the spatially-uniformsteady state solution $u=1$ as the diffusivity, $D$, decreases through$\Delta_1 \approx 0.00297$. We explicitly construct these spatially-periodicsolutions as uniformly-valid asymptotic approximations for $D \ll 1$, over onewavelength, via the method of matched asymptotic expansions. These consist, atleading order, of regularly-spaced, compactly-supported regions with width of$O(1)$ where $u=O(1)$, separated by regions where $u$ is exponentially small atleading order as $D \to 0^+$.  From numerical solutions, we find that for $D \geq \Delta_1$, permanent formtravelling waves, with minimum wavespeed, $2 \sqrt{D}$, are generated, whilstfor $0 < D < \Delta_1$, the wavefronts generated separate the regions where$u=0$ from a region where a steady periodic solution is created. The structureof these transitional travelling waves is examined in some detail.

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