  Using the matrix Riemann-Hilbert factorisation approach for non-linear
evolution systems which take the form of Lax-pair isospectral deformations, the
higher order asymptotics as $t \to \pm \infty$ $(x/t \sim {\cal O}(1))$ of the
solution to the Cauchy problem for the modified non-linear Schr\"{o}dinger
equation, $i \partial_{t} u + {1/2} \partial_{x}^{2} u + | u |^{2} u + i s
\partial_{x} (| u |^{2} u) = 0$, $s \in \Bbb R_{> 0}$, which is a model for
non-linear pulse propagation in optical fibres in the subpicosecond time scale,
are obtained: also derived are analogous results for two gauge-equivalent
non-linear evolution equations; in particular, the derivative non-linear
Schr\"{o}dinger equation, $i \partial_{t} q + \partial_{x}^{2} q - i
\partial_{x}(| q |^{2} q) = 0$.
