Stochastic generalised KPP equations

We classify multiplicative white noise perturbations k(-) dw, of generalised KPP equations and their effects on deterministic approximate travelling wave solutions by the behaviour of integral(0)(t) k(2)(s) ds. If integral(0)+(infinity) k(2)(s) ds < + infinity, the solutions of the stochastic generalised KPP equations converge to deterministic approximate travelling waves and 1/2 lim/sigma-->+infinity 1/sigma integral(0)(sigma) k(2)(s) ds > sup/(t,x)is an element of D 1/t integral(0)(t) (c) over bar(Phi(s)(Phi(t)(-1)(x))) ds, (c) over bar(.) being an associated potential energy, Phi(s) a solution of the corresponding classical mechanical equations of Newton, D being a certain domain in R(1) x R(r), then the white noise perturbations essentially destroy the wave structure and force the solutions to die down. For the case 1/2 lim sigma-->+infinity 1/sigma integral(0)(sigma) k(2)(s) ds less than or equal to sup/(t,x)is an element of D 1/t integral(0)(t) (c) over bar(Phi(s)(Phi(t)(-1)(x))) ds (suppose the existence of the limit) we show that there is a residual wave form but propagating at a different speed from that of the unperturbed equations. Numerical solutions are included and give good agreement with theoretical results.