Summary: Using the matrix Riemann-Hilbert factorization approach for nonlinear evolution systems which take the form of Lax-pair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, we obtain the leading-order asymptotics as \(t\to \pm \infty\) of the solution to the Cauchy problem for the modified nonlinear Schrödinger equation \[ i \partial_{t} u + \tfrac{1}{2} \partial_{x}^{2} u + |u |^{2} u + i s \partial_{x} (|u |^{2} u)= 0, \quad s\in \mathbb{R}_{> 0}, \] which is a model for nonlinear pulse propagation in optical fibers in the subpicosecond time scale. Also derived are analogous results for two gauge-equivalent nonlinear evolution equations – in particular, the derivative nonlinear Schrödinger equation \[ i \partial_{t} q+ \partial_{x}^{2} q- i \partial_{x}(|q |^{2} q)= 0. \] As an application of these asymptotic results, explicit expressions for position and phase shifts of solitons in the presence of the continuous spectrum are calculated.