Asymptotics of Solutions to the Modified Nonlinear Schrödinger Equation: Solitons on a Nonvanishing 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, we obtain the leading-order asymptotics as $t \! \to \! \pm \infty$ of the solution to the Cauchy problem for the modified nonlinear Schrodinger equation, $i \partial_{t} u + \frac{1}{2} \partial_{x}^{2} u + \vert \, u \vert^{2} u + i s \partial_{x} (\vert u \vert^{2} u) \! = \! 0$, $s \! \in \! \Bbb R_{> 0}$, 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 Schrodinger equation $i \partial_{t} q \! + \! \partial_{x}^{2} q \! - \! i \partial_{x}(\vert q \vert^{2} q) \! = \! 0$. As an application of these asymptotic results, explicit expressions for position and phase shifts of solitons in the prese...