Vera Mikyoung Hur, University of Illinois at Urbana-Champaign

Title: Stable undular bores: rigorous analysis and validated numerics 

Abstract: I will discuss the ‘global’ nonlinear asymptotic stability of the traveling front solutions to the Korteweg-de Vries–Burgers equation, and other dispersive-dissipative perturbations of the Burgers equation. Earlier works made strong use of the monotonicity of the profile, for relatively weak dispersion effects. We exploit the modulation of the translation parameter, establishing a new stability criterion that does not require monotonicity. Instead, a certain Schrodinger operator in one dimension must have exactly one negative eigenvalue, so that a rank-one perturbation of the operator can be made positive definite. Counting the number of bound states of the Schrodinger equation, we find a sufficient condition in terms of the ’width’ of a front. We analytically verify that our stability criterion is met for an open set in the parameter regime including all monotone fronts. Our numerical experiments, revealing more stable fronts, suggest a computer-assisted proof. Joint with Blake Barker, Jared Bronski, and Zhao Yang.

Andrej Zlatoš, University of California San Diego

Title:Homogenization in front propagation models

Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will present results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.

Kevin Zumbrun, Indiana University Bloomington

Title:  Singular amplitude equations and Eckhaus type stability criteria for convective Turing bifurcation with conservation laws

Abstract: In modern biomorphology models like Murray-Oster, the reaction diffusion scenario of Turing is augmented by mechanical forces, leading to reaction convection diffusion equations with conservation laws. This leads to formal Eckhauss-type amplitude equations that are *singular* with respect to the small bifurcation parameter epsilon, possessing a fast-slow time scale structure. We show that nonetheless one may reduce the question of stability to constant coefficient (singular) and, by a detailed 2-parameter matrix perturbation analysis, obtain Eckhaus type stability criteria analogous to those of the classical Turing case that are necessary and sufficient for spectral, linearized and nonlinear stability of associated bifurcating Turing patterns.