Coherence transition in degenerate diffusion equations with mean field coupling

TL;DR

This paper studies how nonlinear diffusion affects the transition from incoherent to coherent states in a mean-field coupled diffusion model on the circle, revealing new bifurcation scenarios and localization phenomena.

Contribution

It introduces nonlinear diffusion into a classical model, analyzing how it alters the coherence transition and bifurcation structure compared to linear diffusion.

Findings

Super-quadratic diffusion leads to multistability before bifurcation.

Quadratic diffusion results in an infinitely degenerate bifurcation.

Strong advection causes localization of coherent equilibria.

Abstract

We introduce non-linear diffusion in a classical diffusion advection model with non local aggregative coupling on the circle, that exhibits a transition from an uncoherent state to a coherent one when the coupling strength is increased. We show first that all solutions of the equation converge to the set of equilibria, second that the set of equilibria undergoes a bifurcation representing the transition to coherence when the coupling strength is increased. These two properties are similar to the situation with linear diffusion. Nevertheless nonlinear diffusion alters the transition scenari, which are different when the diffusion is sub-quadratic and when the diffusion is super-quadratic. When the diffusion is super-quadratic, it results in a multistability region that preceeds the pitchfork bifurcation at which the uncoherent equilibrium looses stability. When the diffusion is quadratic the pitchfork bifurcation at the onset of coherence is infinitely degenerate and a disk of equilibria exist for the critical value of the coupling strength. Another impact of nonlinear diffusion is that coherent equilibria become localized when advection is strong enough, a phenomenon that is preculded when the diffusion is linear.