Nonlinear Stability of nonsingular solitons of the Principal Chiral Field equation

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.