Higher Order Asymptotics of the Modified Non-Linear Schr\"{o}dinger Equation

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, $i \partial_{t} q + \partial_{x}^{2} q - i \partial_{x}(| q |^{2} q) = 0$.