Equivalence of variational principles to determine the speed of scalar reaction diffusion fronts

TL;DR

This paper demonstrates the logical equivalence and interconnections between various variational principles used to determine the speed of scalar reaction-diffusion fronts, unifying different approaches in the field.

Contribution

It establishes the equivalence and relationships among multiple variational principles for reaction-diffusion front speeds, clarifying their connections and extending their applicability.

Findings

Proves the equivalence of Hadeler and Rothe's principle with Benguria and Depassier's principle for monostable reactions.

Shows that two variational principles for arbitrary reactions are related by a change of variables.

Demonstrates that a principle for monostable reactions is a reformulation of the previous principles in a different variable.

Abstract

The determination of the speed of travelling fronts of the scalar reaction diffusion equation has been the subject of much study. Using different approaches seemingly disconnected variational principles have been established. The purpose of this work is to show the connection between them. For monostable reaction terms, we prove that a principle established by Hadeler and Rothe in 1975 and a second one by Benguria and Depassier in 1996 are logically equivalent, that is, either can be derived from the other. Two variational principles, formulated for arbitrary reaction terms, are shown to be related by a suitable change of variables. Finally a variational principle proven for monostable reaction terms is shown to be a formulation of the two previous ones in yet another independent variable.