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

Abstract

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 $t\to \pm \infty$ of the solution to the Cauchy problem for the modified nonlinear Schrödinger equation, $i{\partial }_{t}u+1/2{\partial }_x^{2}u+|u{|}^{2}u+is{\partial }_x(|u{|}^{2}u)=0$, $s\in {\mathbb{R}}_{>0}$, 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, $i{\partial }_{t}q+{\partial }_x^{2}q-i{\partial }_x(|q{|}^{2}q)=0$. 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.

38 pages, 1 figure, LaTeX