Gradient estimates and the smooth convergence of approximate travelling waves for reaction-diffusion equations

The space derivatives of Freidlin's travelling wave like solutions of generalized KPP equations are considered in this paper. We give estimates of the first two space derivatives on the wave front and show that the travelling wave is nearly flat on the trough and on the crest. Differentiation of heat semigroups, logarithmic transformation and semi-classical analysis based on stochastic analysis are the main tools used here