The Speed of Traveling Waves in a FKPP-Burgers System

Jason J. Bramburger, Christopher Henderson

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


We consider a coupled reaction–advection–diffusion system based on the Fisher-KPP and Burgers equations. These equations serve as a one-dimensional version of a model for a reacting fluid in which the arising density differences induce a buoyancy force advecting the fluid. We study front propagation in this system through the lens of traveling waves solutions. We are able to show two quite different behaviors depending on whether the coupling constant ρ is large or small. First, it is proved that there is a threshold ρ under which the advection has no effect on the speed of traveling waves (although the advection is nonzero). Second, when ρ is large, wave speeds must be at least O(ρ1 / 3). These results together give that there is a transition from pulled to pushed waves as ρ increases. Because of the complex dynamics involved in this and similar models, this is one of the first precise results in the literature on the effect of the coupling on the traveling wave solution. We use a mix of ordinary and partial differential equation methods in our analytical treatment, and we supplement this with a numerical treatment featuring newly created methods to understand the behavior of the wave speeds. Finally, various conjectures and open problems are formulated.

Original languageEnglish (US)
Pages (from-to)643-681
Number of pages39
JournalArchive for Rational Mechanics and Analysis
Issue number2
StatePublished - Aug 2021

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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