Bistable travelling waves for nonlocal reaction diffusion equations

We are concerned with travelling wave solutionsarising in a reaction diffusion equation with bistable andnonlocal nonlinearity, for which the comparison principle does nothold. Stability of the equilibrium $u\equiv 1$ is not assumed. Weconstruct a travelling wave solution connecting 0 to an unknownsteady state, which is 'above and away', from theintermediate equilibrium. For focusing kernels we prove that, asexpected, the wave connects 0 to 1. Our results also apply readilyto the nonlocal ignition case.