Perturbation theory for the one-dimensional reaction-diffusion equation with a quadratic nonlinearity of coalescence/annihilation type

We take a reaction-diffusion equation describing two-particle coalescence (or, alternatively, annihilation) in an infinite domain as a starting point to develop a perturbation scheme based in an expansion in terms of the effective reaction constant valid in the slow reaction limit. We compute the first-order correction to the diffusion equation and show that the underlying approximation breaks down at late times, suggesting that higher-order terms become increasingly important in this regime.