Cross-overs of Bramson's shift at the transition between pulled and pushed fronts

TL;DR

This paper investigates the transition between pulled and pushed fronts in monostable traveling wave equations, deriving a crossover function for the front position shift at critical non-linearity, with implications for more general models.

Contribution

It provides an explicit crossover function for the Bramson shift at the pulled-pushed transition in a solvable model, potentially applicable to broader classes of equations.

Findings

Derived the crossover function for the front position shift.

Analyzed the dependence on initial conditions and cut-offs.

Identified the universality of the shift modification at the transition.

Abstract

The Bramson logarithmic shift of the position of pulled fronts is a universal feature common to a large class of monostable traveling wave equations. As one varies the non-linearities it so happens that one can observe, at some critical non linearity, a transition from pulled fronts to pushed fronts. At this transition the Bramson shift is modified. In the limit where time goes to infinity and the non-linearity becomes critical, the position of the front exhibits a cross-over. The goal of the present note is to give the expression of this cross-over function, for a particular model which is exactly soluble, with the hope that this expression would remain valid for more general traveling wave equations at the transition between pulled and pushed fronts. Other cross-over functions are also obtained, for this particular model, to describe the dependence on initial conditions or the effect of a cut-off.