Interface dynamics in semilinear wave equations

Abstract

We consider the wave equation ${\epsilon }^2(-{\partial }_t^2+\Delta )u+f(u)=0$ for $0<\epsilon \ll 1$, where $f$ is the derivative of a balanced, double-well potential, the model case being $f(u)=u-u^3$. For equations of this form, we construct solutions that exhibit an interface of thickness $O(\epsilon )$ that separates regions where the solution is $O({\epsilon }^k)$ close to $\pm 1$, and that is close to a timelike hypersurface of vanishing {\em Minkowskian} mean curvature. This provides a Minkowskian analog of the numerous results that connect the Euclidean Allen-Cahn equation and minimal surfaces or the parabolic Allen-Cahn equation and motion by mean curvature. Compared to earlier results of the same character, we develop a new constructive approach that applies to a larger class of nonlinearities and yields much more precise information about the solutions under consideration.

34 pages

References in corpus (1)

• Lorentzian varifolds and applications to closed relativistic strings

Cited by in corpus (1)

• A variational scheme for hyperbolic obstacle problems