Nonlinear waves and polarization in diffusive directed particle flow

TL;DR

This paper studies nonlinear wave phenomena and polarization effects in a one-dimensional particle flow model with reaction, diffusion, and advection, demonstrating the existence of traveling waves and their dependence on parameters.

Contribution

It establishes the existence of traveling wave solutions in a reaction-diffusion-advection system with superimposed diffusion and mutual alignment, using singular perturbation techniques.

Findings

Existence of traveling wave solutions for weak diffusion

Evidence for wave solutions at stronger diffusion levels

Dependence of wave velocities on model parameters

Abstract

We consider a system of two reaction-diffusion-advection equations describing the one dimensional directed motion of particles with superimposed diffusion and mutual alignment. For this system we show the existence of traveling wave solutions for weak diffusion by singular perturbation techniques and provide evidence for their existence also for stronger diffusion. We discuss different types of wave fronts and their composition to more complex patterns and illustrate their emergence from generic initial data by simulations. We also investigate the dependence of the wave velocities on the model parameters.