Interface Dynamics in Semilinear Wave Equations

We consider the wave equation ε2(-∂t2+Δ)u+f(u)=0 for 0 < ε≪ 1 , where f is the derivative of a balanced, double-well potential, the model case being f(u) = u- u3. For equations of this form, we construct solutions that exhibit an interface of thickness O(ε) that separates regions where the solution is O(εk) close to ± 1 , for k≥ 1 , and that is close to a timelike hypersurface of vanishing 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.