Asymptotics of Solutions to the Modified Nonlinear Schrödinger Equation: Solitons on a Non-Vanishing Continuous Background

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, the leading-order asymptotics as t → ±∞ of the solution to the Cauchy problem for the modified nonlinear Schrödinger equation, i∂tu+ 1 2 ∂ xu+ |u|2u+ is∂x(|u|u) = 0, s ∈ R>0, which is a model for nonlinear pulse propagation in optical fibers in the subpicosecond time scale, are obtained: also derived are analogous results for two gauge-equivalent nonlinear evolution equations; in particular, the derivative nonlinear Schrödinger equation, i∂tq+∂ 2 x q−i∂x(|q|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.