Global Existence and Long Time Behavior in the 1+1 dimensional Principal Chiral Model with Applications to Solitons

TL;DR

This paper establishes local and global existence results for the 1+1 dimensional Principal Chiral Field model, analyzes decay properties of solutions, and explores applications to solitons, advancing understanding of its long-term behavior.

Contribution

It provides the first rigorous existence theory for solutions of the PCF model and introduces virial functionals to study decay and stability of solitons.

Findings

Proved local existence of solutions in an energy space.

Established small global solutions under non-degeneracy conditions.

Developed virial functionals to describe decay inside the light cone.

Abstract

In this paper, we consider the 1+1 dimensional vector valued Principal Chiral Field model (PCF) obtained as a simplification of the Vacuum Einstein Field equations under the Belinski-Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step towards the study of its nonlinear stability.