Defects in semilinear wave equations and timelike minimal surfaces in Minkowski space

Abstract

We study semilinear wave equations with Ginzburg–Landau-type nonlinearities, multiplied by a factor of ${\epsilon }^{-2}$, where $\epsilon >0$ is a small parameter. We prove that for suitable initial data, the solutions exhibit energy-concentration sets that evolve approximately via the equation for timelike Minkowski minimal surfaces, as long as the minimal surface remains smooth. This gives a proof of the predictions made (on the basis of formal asymptotics and other heuristic arguments) by cosmologists studying cosmic strings and domain walls, as well as by applied mathematicians.