The amplitude equation for weakly nonlinear reversible phase boundaries

TL;DR

This paper derives and analyzes the amplitude equation for weakly nonlinear surface waves at phase boundaries, showing it has symmetry properties ensuring local well-posedness of the evolution problem.

Contribution

It explicitly computes the amplitude equation for weakly nonlinear phase boundary surface waves and demonstrates its symmetry properties lead to well-posedness.

Findings

Explicit amplitude equation derived for weakly nonlinear surface waves.

Symmetry properties ensure local well-posedness of the Cauchy problem.

Builds on previous models to deepen understanding of phase boundary dynamics.

Abstract

This technical note is a complement to an earlier paper [Benzoni-Gavage \& Rosini, Comput. Math. Appl. 2009], which aims at a deeper understanding of a basic model for propagating phase boundaries that was proved to admit surface waves [Benzoni-Gavage, Nonlinear Anal. 1998]. The amplitude equation governing the evolution of weakly nonlinear surface waves for that model is computed explicitly, and is eventually found to have enough symmetry properties for the associated Cauchy problem to be locally well-posed.