Interface dynamics in semilinear wave equations

TL;DR

This paper constructs solutions to a semilinear wave equation with a small parameter, revealing interface dynamics that approximate a hypersurface with vanishing Minkowskian mean curvature, extending Euclidean analogs.

Contribution

Develops a new constructive method for solutions with interfaces in wave equations, applicable to broader nonlinearities and providing detailed solution behavior.

Findings

Interfaces of thickness O(ε) separate regions near ±1

Solutions approximate timelike hypersurfaces with zero Minkowskian mean curvature

Method extends to a larger class of nonlinearities

Abstract

We consider the wave equation $\varepsilon^2(-\partial_t^2 + \Delta)u + f(u) = 0$ for $0<\varepsilon\ll 1$, 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 $O(\varepsilon )$ that separates regions where the solution is $O(\varepsilon^k)$ 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.