  The methodology of the Riemann-Hilbert (RH) factorisation approach for
Lax-pair isospectral deformations is used to derive, in the solitonless sector,
the leading-order asymptotics as $t \to \pm \infty$ $(x/t \sim \mathcal{O}(1))$
of solutions to the Cauchy problem for the defocusing non-linear
Schr\"{o}dinger equation (D${}_{f}$NLSE), $\mi \partial_{t}u
+\partial_{x}^{2}u-2(| u |^{2}-1)u=0$, with (finite-density) initial data
$u(x,0)=_{x \to \pm \infty} \exp (\tfrac{\mi (1 \mp 1) \theta}{2})(1+ o(1))$,
$\theta \in [0,2 \pi)$. A limiting case of these asymptotics related to the RH
problem for the Painlev\'{e} II equation, or one of its special reductions, is
also identified.
