Extensions of the nonlinear Schr?dinger equation using Mathematica

Abstract

The aim of this thesis is to investigate the theory of the extensions of the Nonlinear Schrödinger Equation (NLSE), concentrating on the following main points: Developing further analytical techniques and properties under relativistic conditions. This thesis demonstrates numerical techniques that can be used to form numerical codes that can be applied to the very recent need for source and industrial application of laser-driven ion sources for ion implantation. The analytical and numerical evaluations of the nonlinear mechanisms are measured utilising various techniques that include computer packages such as Mathematica(R) [Wolfram 2003]1, Maple™ 9 [2003] and C++© [Strousop 2003]. This project expands the author's present undergraduate honours research work on the theory of Schrödinger equations. The breaking of light waves: in the course of this research the breaking of light waves was the first new phenomenon to be encountered. The highest authority on this subject [Zakharov and Shabat 1972], Prof. Zakharov, advised me [Zakharov 2004] that this topic was at that time not researched in any detail. It was envisaged that entering more fully into this area of research using Mathematica version 5 [Wolfram 2003], which had been recently released (June 2003) and which is uniquely adapted for such research, would be the most profitable direction to go. The intention was to research the behaviour of radiation from the soliton in respect of the higher order (dispersion) term in the NLSE. This research was expected to reveal its properties and consequences and possibly new ways in which this radiation can be predicted, controlled, eliminated or otherwise profitably manipulated. These results are considered vital to the uses of solitons, particularly in optical fibre telecommunications. Numerical artefacts: At this juncture the direction of the research changed in a way that had not been anticipated. The compilation and execution of Mathematica codes, now advanced to the use of new techniques and iterative methods such as the Split-Step Method, had been anticipated to clearly show the existence of secondary and possibly tertiary radiation attending the soliton. It had also been anticipated that this would confirm the theory that this radiation attended only solitons resulting from the cubic, and odd numbered, higher-order NLSE. The first assumption simply did not materialise and the second was not at all up to expectations. At best, the results coming from this line of investigation could only show that solitons derived from the even numbered, or quadratic higher-order NLSEs were in some ways fundamentally different from the odd numbered or cubic ones. These setbacks all resulted from a phenomenon, hitherto unanticipated as a problem to this program of research, namely 'numerical artefact' in Mathematica [Beech and Osman 2005: 1369; See Appendix I Paper 3]. This reduced Paper 3 [ibid] 'Effects of higher order dispersion terms in the nonlinear Schrödinger Equation' from a serious contribution in this field to a scathing criticism of the use of iterative methods in computerised mathematics.

Date of Award	2009
Original language	English