Higher Order Asymptotics of the Modified Non-Linear Schrödinger Equation

Abstract

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 $t\to \pm \infty$ $(x/t\sim \mathcal{O}(1))$ of the solution to the Cauchy problem for the modified non-linear 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 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 Schrödinger equation, $i{\partial }_{t}q+{\partial }_x^{2}q-i{\partial }_x(|q{|}^{2}q)=0$.

54 pages, 7 figures, LaTeX, long appendix