The concept of quasi-integrability: a concrete example

TL;DR

This paper explores the concept of quasi-integrability in deformed sine-Gordon models, defining quasi-conserved quantities, analyzing their behavior in soliton scattering, and supporting findings with numerical simulations of long-lived states.

Contribution

It introduces a definition of quasi-integrability, computes quasi-conserved charges perturbatively, and demonstrates asymptotic conservation and long-lived states in deformed sine-Gordon models.

Findings

Charges are asymptotically conserved in two-soliton scattering.

Numerical simulations show long-lived breather-like and wobble-like states.

The model exhibits quasi-integrability with approximate conservation laws.

Abstract

We use the deformed sine-Gordon models recently presented by Bazeia et al to discuss possible definitions of quasi-integrability. We present one such definition and use it to calculate an infinite number of quasi-conserved quantities through a modification of the usual techniques of integrable field theories. Performing an expansion around the sine-Gordon theory we are able to evaluate the charges and the anomalies of their conservation laws in a perturbative power series in a small parameter which describes the "closeness" to the integrable sine-Gordon model. Our results indicate that in the case of the two-soliton scattering the charges are conserved asymptotically, i.e. their values are the same in the distant past and future, when the solitons are well separated. We back up our results with numerical simulations which also demonstrate the existence of long lived breather-like and wobble-like states in these models.