# Critical Phenomena in Nonlinear Sigma Models

@article{Liebling1999CriticalPI, title={Critical Phenomena in Nonlinear Sigma Models}, author={Steven L. Liebling and Eric W. Hirschmann and James Isenberg}, journal={Journal of Mathematical Physics}, year={1999}, volume={41}, pages={5691-5700} }

We consider solutions to the nonlinear sigma model (wave maps) with target space S3 and base space 3+1 Minkowski space, and we find critical behavior separating singular solutions from nonsingular solutions. For families of solutions with localized spatial support a self-similar solution is found at the boundary. For other families, we find that a static solution appears to sit at the boundary. This behavior is compared to the black hole critical phenomena found by Choptuik.

## 23 Citations

### Type II Critical Collapse of a Self-Gravitating Nonlinear Sigma Model

- Physics
- 2000

We report on the existence and phenomenology of type II critical collapse within the one-parameter family of SU(2) σ models coupled to gravity. Numerical investigations in spherical symmetry show…

### ec 1 99 9 Dispersion and collapse of wave maps

- Mathematics
- 2000

We study numerically the Cauchy problem for equivariant wave maps from 3+1 Minkowski spacetime into the 3-sphere. On the basis of numerical evidence combined with stability analysis of self-similar…

### Type II Critical Collapse of a Self-Gravitating Nonlinear σ-Model

- Physics
- 2000

We report on the existence and phenomenology of type II critical collapse within the one-parameter family of SU(2) σ-models coupled to gravity. Numerical investigations in spherical symmetry show…

### Singularity/scattering alternative for a discrete family of static critical solutions with various numbers of unstable eigenmodes

- Physics, Mathematics
- 2011

This paper reports on the study of some solutions for a system of coupled nonlinear wave equations of a hyperbolic type, effectively dependent on one spatial (radial) variable and one time variable.…

### Self-similar spherically symmetric wave maps coupled to gravity

- Mathematics
- 2000

We investigate spherically symmetric continuously self-similar (CSS) solutions in the SU(2) sigma model coupled to gravity. Using mixed numerical and analytical methods, we provide evidence for the…

### On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem

- Mathematics
- 2012

We consider co-rotational wave maps from (3 + 1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions.…

### On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem

- MathematicsAnnales Henri Poincaré
- 2011

We consider co-rotational wave maps from (3 + 1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions.…

### Black hole critical phenomena without black holes

- Physics
- 2000

Studying the threshold of black hole formation via numerical evolution has led to the discovery of fascinating nonlinear phenomena. Power-law mass scaling, aspects of universality, and…

### Maxwell-dilaton dynamics

- PhysicsPhysical Review D
- 2019

The dynamics of Maxwell-dilaton theory in Minkowski spacetime are studied using fully nonlinear, numerical evolutions. This model represents the flat-space sector of Einstein-Maxwell-Dilaton theory…

### Spectral Properties and Stability of Self-Similar Wave Maps

- Mathematics
- 2007

In this thesis the Cauchy problem and in particular the question of singularity formation for co--rotational wave maps from 3+1 Minkowski space to the three--sphere $S^3$ is studied. Numerics…

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