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 Є R. This equation exhibits a transition from pulled to pushed front behavior at β_c D 2. We prove convergence of the solutions to a traveling wave in a reference frame centered at a position m_β (t) and study the asymptotics of the front location m_β (t). When β < 2, it has the same form as for the standard Fisher-KPP equation established by Bramson: m_β (t) D 2t - (3=2) log t C x₁ C o(1) as t → ∞. This form is typical of pulled fronts. When β > 2, the front is located at the position m_β (t) D c_*(β)t C x₁ C o(1) with c_*(β) D β=2 C 2=β, which is the typical form of pushed fronts. However, at the critical value β_c D 2, the expansion changes to m_β (t) D 2t - (1=2) log t C x₁ C 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 D β_c 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 β_c D 2 and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.