Accelerating fronts in semilinear wave equations

We study dynamics of interfaces in solutions of the equation $\epsilon \Box u

\frac 1 \epsilon f_\epsilon(u)=0 $, f o r$ f_\epsilon $o f t h e f o r m$ f_\epsilon(u) = (u^2-1)(2u- \epsilon\kappa) $, f o r$ \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, $n \geq 1$ , 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 $κ$ . In particular, in 1 dimension these interfaces behave like a relativistic point particle subject to constant acceleration.

We study dynamics of interfaces in solutions of the equation $\epsilon \Box u

\frac 1 \epsilon f_\epsilon(u)=0 $, f o r$ f_\epsilon $o f t h e f o r m$ f_\epsilon(u) = (u^2-1)(2u- \epsilon\kappa) $, f o r$ \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, $n \geq 1$ , 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 $κ$ . In particular, in 1 dimension these interfaces behave like a relativistic point particle subject to constant acceleration.

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