  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.
