Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation

We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength β . This equation exhibits a transition from pulled to pushed front behavior at βc=2 . We prove convergence of the solutions to a traveling wave in a reference frame centered at a position mβ(t) and study the asymptotics of the front location mβ(t) . When β < 2 , it has the same form as for the standard Fisher-KPP equation established by Bramson: mβ(t) = 2t - (3/2)log t + x∞ + o(1) as t→∞ . This form is typical of pulled fronts. When β > 2 , the front is located at the position mβ(t)=c*(β)t+x∞+o(1) with c*(β)=β/2+2/β , which is the typical form of pushed fronts. However, at the critical value βc = 2 , the expansion changes to mβ(t) = 2t - (1/2)log t + x∞ + o(1) , reflecting the “pushmi-pullyu” nature of the front. The arguments for β<2 rely on a new weighted Hopf–Cole transform that allows one to control the advection term, when combined with additional steepness comparison arguments. The case β>2 relies on standard pushed front techniques. The proof in the case β=βc is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at βc=2 and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.