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

We study semilinear wave equations with Ginzburg–Landau-type nonlinearities, multiplied by a factor of ε2, 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.

Minkowski minimal surface, semilinear wave equation, topological defects, defect dynamics
Mathematical Subject Classification 2000
Primary: 35B40, 35L70, 53C44
Secondary: 85A40
Received: 1 December 2009
Accepted: 15 April 2010
Published: 18 November 2011
Robert Jerrard
University of Toronto
Toronto M4X 1S9