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

  • Jing An

    Duke University, Durham, USA
  • Christopher Henderson

    University of Arizona, Tucson, USA
  • Lenya Ryzhik

    Stanford University, Stanford, USA
Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation cover

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We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength . This equation exhibits a transition from pulled to pushed front behavior at . We prove convergence of the solutions to a traveling wave in a reference frame centered at a position and study the asymptotics of the front location . When , it has the same form as for the standard Fisher-KPP equation established by Bramson: as . This form is typical of pulled fronts. When , the front is located at the position  with , which is the typical form of pushed fronts. However, at the critical value , the expansion changes to , reflecting the “pushmi-pullyu” nature of the front. The arguments for 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  relies on standard pushed front techniques. The proof in the case 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 and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.

Cite this article

Jing An, Christopher Henderson, Lenya Ryzhik, Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1407