#### Vol. 4, No. 2, 2011

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

### Robert Jerrard

Vol. 4 (2011), No. 2, 285–340
##### 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.

##### Keywords
Minkowski minimal surface, semilinear wave equation, topological defects, defect dynamics
##### Mathematical Subject Classification 2000
Primary: 35B40, 35L70, 53C44
Secondary: 85A40