Accelerating fronts in semilinear wave equations

Abstract

We study dynamics of interfaces in solutions of the equation $\epsilon \square u+\frac{1}{\epsilon }{f}_{\epsilon }(u)=0$, 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.

28 pages, 2 figures