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

Jerrard, Robert

doi:10.2140/apde.2011.4.285

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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

DOI: 10.2140/apde.2011.4.285

Abstract

We study semilinear wave equations with Ginzburg–Landau-type nonlinearities, multiplied by a factor of $ε^{- 2}$ , where $ε > 0$ 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

Milestones

Received: 1 December 2009

Accepted: 15 April 2010

Published: 18 November 2011

Authors

Robert Jerrard
University of Toronto Toronto M4X 1S9 Canada

Vol. 4, No. 2, 2011