PDE Seminar

Speaker: Prof. Atanas Stefanov 

 

https://www.uab.edu/cas/mathematics/people/faculty-directory/atanas-stefanov

 

Title: Kink Solutions for Dispersive-Diffusive PDEs: Three Short Stories

 

Abstract: Traveling kinks are special solutions to PDEs on the real line characterized by distinct limiting values at the infinities. They arise naturally in combustion dynamics, invasion fronts in population biology, water wave theory, and beyond. This talk presents three recent results centered on the existence and stability of such coherent structures.

In the first story, we establish the existence of kink solutions in the fractional $\phi^4$ model and discuss several consequences (joint work with P. Kevrekidis). 

The second and third stories address a fundamental question in the theory of coherent structures: *are kinks dynamically stable?* That is, do solutions initialized near a kink remain close to it — or to one of its translates — for all time?

We answer this question for two models : 

 

1) the damped Boussinesq equation,

2) dispersive-diffusive Burgers models, such as the KdV–Burgers equation. 

 

For the Boussinesq model, we prove that the unique monotone kink is asymptotically stable via energy estimates and elementary spectral theory. 

For KdV–Burgers, we establish a stronger result: the monotone kink is actually an attracting set — every solution, including those arising from a large perturbation, converges to a translate of the kink, with explicit decay rates. This last result builds on recent developments in the theory of asymptotic attractivity, combined with a careful bootstrap argument for the decay rates.