The Singularity Threshold of the Nonlinear Sigma Model Using 3D Adaptive Mesh Refinement

TL;DR

This paper investigates the nonlinear sigma model in 3D using adaptive mesh refinement, demonstrating that the spherically symmetric critical solution acts as an intermediate attractor between singularity formation and dispersal, extending previous symmetric results.

Contribution

It generalizes the threshold behavior of the nonlinear sigma model from spherical symmetry to full 3D, showing the critical solution remains an intermediate attractor.

Findings

Spherically symmetric critical solution persists as an intermediate attractor in 3D.

Adaptive mesh refinement effectively captures singularity formation and dispersal.

Threshold behavior similar to gravitating theories observed without symmetry assumptions.

Abstract

Numerical solutions to the nonlinear sigma model (NLSM), a wave map from 3+1 Minkowski space to S^3, are computed in three spatial dimensions (3D) using adaptive mesh refinement (AMR). For initial data with compact support the model is known to have two regimes, one in which regular initial data forms a singularity and another in which the energy is dispersed to infinity. The transition between these regimes has been shown in spherical symmetry to demonstrate threshold behavior similar to that between black hole formation and dispersal in gravitating theories. Here, I generalize the result by removing the assumption of spherical symmetry. The evolutions suggest that the spherically symmetric critical solution remains an intermediate attractor separating the two end states.