The speed of traveling waves in a FKPP-Burgers system

TL;DR

This paper analyzes how the coupling strength in a FKPP-Burgers system influences traveling wave speeds, revealing a transition from no effect to a significant increase in wave speed as the coupling grows.

Contribution

It provides the first precise analysis of how coupling affects wave speeds in a FKPP-Burgers system, identifying a threshold and a transition in wave behavior.

Findings

Wave speed unaffected by advection below threshold coupling 

Wave speeds grow at least as  ho^{1/3} for large coupling 

Transition from pulled to pushed waves as coupling increases

Abstract

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 $\rho$ is large or small. First, it is proved that there is a threshold $\rho_0$ under which the advection has no effect on the speed of traveling waves (although the advection is nonzero). Second, when $\rho$ is large, wave speeds must be at least $\mathcal{O}(\rho^{1/3})$. These results together give that there is a transition from pulled to pushed waves as $\rho$ 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.