Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation

Abstract

We study solitary wave solutions of the higher order nonlinear Schr\"odinger equation for the propagation of short light pulses in an optical fiber. Using a scaling transformation we reduce the equation to a two-parameter canonical form. Solitary wave (1-soliton) solutions always exist provided easily met inequality constraints on the parameters in the equation are satisfied. Conditions for the existence of $N$-soliton solutions ( $N\ensuremath{\ge}2$) are determined; when these conditions are met the equation becomes the modified Korteweg--de Vries equation. A proper subset of these conditions meet the Painlev\'e plausibility conditions for integrability.