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

TL;DR

This paper proves that solutions to certain semilinear wave equations with Ginzburg-Landau nonlinearities concentrate energy along evolving timelike minimal surfaces in Minkowski space, confirming cosmological and mathematical predictions.

Contribution

It rigorously establishes the link between wave equation solutions and timelike minimal surfaces, validating heuristic and asymptotic predictions in physics and mathematics.

Findings

Energy concentrates along timelike minimal surfaces

Solutions approximate minimal surface evolution when smooth

Supports cosmological models of cosmic strings and domain walls

Abstract

We study semilinear wave equations with Ginzburg-Landau type nonlinearities multiplied by a factor $\epsilon^{-2}$, where $\epsilon>0$ is a small parameter. We prove that for suitable initial data, 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 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.