Nonlinear Stability of nonsingular solitons of the Principal Chiral Field equation

TL;DR

This paper proves the nonlinear stability of small nonsingular solitons in the Principal Chiral Field model on SL(2,R), extending techniques from Einstein's equations to a new integrable system.

Contribution

It introduces a novel application of vector field methods to establish stability of solitons in the Principal Chiral Field equation.

Findings

Small nonsingular solitons are nonlinearly stable over time.

The method extends previous techniques from Einstein's equations to integrable models.

Null weighted norms of perturbations remain controlled for all times.

Abstract

We consider the Principal Chiral Field model posed in 1+1 dimensions into the Lie group $\text{SL}(2,\mathbb R)$. In this work we show the nonlinear stability of small enough nonsingular solitons. The method of proof involves the use of vector field methods as in a previous work by the second and third authors dealing with the Einstein's field equations under the Belinski-Zakharov formalism, extending for all times the size of suitable null weighted norms of the perturbations at time zero.