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

2009/09/18 by Robert L. Jerrard, Jerrard, Robert L.
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.0909.3548

Abstract

We study semilinear wave equations with Ginzburg-Landau type nonlinearities multiplied by a factor ε^-2, where ε>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.

Citations

• Weak solutions and development of singularities of the SU(2) σ‐model

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