  In this work we report preliminary results on the relaxational dynamics of
one dimensional Bose gases, as described by the Lieb-Liniger model, upon
release from a parabolic trap. We explore the effects of integrability and
integrability breaking upon these dynamics by placing the gas post-release in
an integrability breaking one-body cosine potential of variable amplitude. By
studying the post-quench evolution of the conserved charges that would exist in
the purely integrable limit, we begin to quantify the effects of the weak
breaking of integrability on the long time thermalization of the gas.
