Accelerating fronts in semilinear wave equations

TL;DR

This paper investigates the dynamics of interfaces in semilinear wave equations, demonstrating that solutions develop moving interfaces that follow prescribed geometric evolution, including relativistic motion with acceleration.

Contribution

It provides a rigorous analysis of interface motion in semilinear wave equations on Minkowski space and Lorentzian manifolds, linking solutions to geometric evolution laws.

Findings

Interfaces follow timelike hypersurfaces with mean curvature proportional to 

In 1D, interfaces behave like accelerated relativistic particles

Solutions exhibit accelerating front dynamics consistent with geometric laws

Abstract

We study dynamics of interfaces in solutions of the equation $\epsilon \Box u + \frac 1 \epsilon f_\epsilon(u)=0$, for $f_\epsilon$ of the form $f_\epsilon(u) = (u^2-1)(2u- \epsilon\kappa)$, for $\kappa\in {\mathbb R}$, as well as more general, but qualitatively similar, nonlinearities. We consider equations of this form both in $(1+n)$-dimensional Minkowski space, $n\ge 1$, and on certain more general Lorentzian manifolds, and we prove that for suitable initial data, solutions exhibit interfaces that sweep out timelike hypersurfaces of mean curvature proportional to $\kappa$. In particular, in 1 dimension these interfaces behave like a relativistic point particle subject to constant acceleration.