Breakdown of the standard perturbation theory and moving boundary approximation for “pulled” fronts

A moving boundary approximation or similar perturbative schemes for the response of a coherent structure like a front, vortex or pulse to external forces and noise can generally be derived if two conditions are obeyed: (i) there must be a separation of the time scales of the dynamics on the inner and outer scale, and (ii) solvability-type integrals must converge. We point out that both of these conditions are not satisfied for pulled fronts propagating into an unstable state: their relaxation on the inner scale is algebraic rather than exponential, and in conjunction with this, solvability integrals diverge. This behavior can be explained by the fact that the important dynamics of pulled fronts occurs in the leading edge of the front rather than in the nonlinear internal front region itself. As a consequence, the dynamical behavior of pulled fronts is often qualitatively different from the standard case in which fronts between two (meta)stable states are considered, as has recently been established for the relaxation, the stochastic behavior and the response to multiplicative noise. We here show that this is also true for the coupling of pulled fronts to other fields.