Renormalization Group Theory And Variational Calculations For Propagating Fronts

TL;DR

This paper develops a renormalization group approach and a variational principle to analyze and accurately predict the propagation speed of fronts into unstable states, combining analytical and numerical methods.

Contribution

It introduces a perturbative RG method for speed calculation and applies a variational principle to improve predictions of front propagation speeds.

Findings

RG approach successfully predicts speed changes for stable fronts.

Variational principle provides tight upper bounds and exact speeds in some cases.

Identifies transition from linear to nonlinear stability regimes.

Abstract

We study the propagation of uniformly translating fronts into a linearly unstable state, both analytically and numerically. We introduce a perturbative renormalization group (RG) approach to compute the change in the propagation speed when the fronts are perturbed by structural modification of their governing equations. This approach is successful when the fronts are structurally stable, and allows us to select uniquely the (numerical) experimentally observable propagation speed. For convenience and completeness, the structural stability argument is also briefly described. We point out that the solvability condition widely used in studying dynamics of nonequilibrium systems is equivalent to the assumption of physical renormalizability. We also implement a variational principle, due to Hadeler and Rothe, which provides a very good upper bound and, in some cases, even exact results on the propagation speeds, and which identifies the transition from ` linear'- to ` nonlinear-marginal-stability' as parameters in the governing equation are varied.