Non-equilibrium quantum mechanics has received a considerable
attention in the last years and in particular the dynamics observed
after a quantum quench, when a global parameter in a many-body
Hamiltonian is rapidly changed. In the one-dimensional case the
problem is even more exiting since there one can create physical
models that do not thermalize canonically due to the presence of
many local integrals of motion: the so called integrable
models. We focus here on the post-quench time evolution of the
simplest interacting integrable models as the Lieb-Liniger
delta-interacting Bose gas and the spin-1/2 XXZ spin chain. We
address the problem of determining the physical properties of the
steady state after a quench and their time evolution. In particular
we consider a sudden switch of the inter-particle interactions in a
free Bosonic gas and the non-equilibrium time evolution induced by a
sudden Bragg pulse applied to a strongly interacting gas, as an
attempt to model the celebrated quantum Newton cradle. Finally we
show how quantum quenches with inhomogenous initial states can
represent ideal settings to study quantum transport in many-body
systems.