Solving nonlinear Schrödinger equation of optical fiber type using inverse scattering transform

Abstract

The main purpose of this study is to solve the nonlinear Schrödinger (NLS) equation of optical fiber type using inverse scattering transform (IST) method. Prior to that, simpler problem on the initial-valued Korteweg-de Vries (KdV) equation is discussed to show how such nonlinear evolution equation can be linearized and solved using IST. Then, a more general IST method based on two schemes known as AKNS (Ablowitz-Kaup-Newell-Sigur) and ZS (Zakharov-Shabat) are discussed. AKNS method is described in terms of scattering theory whereas ZS method is expressed solely based on operators. The NLS equation of optical fibre type should be solved using ZS scheme to avoid any specific calculations of the scattering data. Finally, the solutions obtained are used to demonstrate the occurence of solitons from the constructed graph using Mathematica software. It is found that the solution from the NLS equation is a propagating wave enveloped in a wave packet, called a bright soliton. On the other hand, the existence of dark soliton is also detected when the nonlinear term in the NLS equation is negative. Both of these solitons are able to retain its shape after moving over some distance on the graph. The occurence of solitons are able to be demonstrated based on the contructed graphs from the solutions of the NLS equation.