Numerical simulation of the nonlinear schrodinger equation using stable implicit finite difference methods

Abstract

Nonlinear Schrodinger equation (NLSE) in the context of drift wave packet is a difficult partial differential equation to solve without any approximations or transformations. Numerical computation must be taken into account to solve this complicated problem and the interplay between the first and second-order chromatic dispersions (CDs) and Kerr nonlinear effect needs to be considered. Although the NLSE in the absence of the first-order CD parameter term has been solved using various numerical and analytical methods, but the influential parameters, second-order CD, and self-phase modulation (SPM) have yet to be examined. Therefore, in this thesis the influence of these factors on newwave forms and on related conserved quantitieswas investigated. The NLSE was studied numerically by using finite difference methods. The Crank-Nicolson, which is second-order in time and space, was used. A high accuracy method that is fourth-order in space and second-order in time and known as the Douglas idea was also used to solve the NLSE. The accuracy and stability of the obtained schemes were analyzed. The conserved quantities mass, momentum, and energy were also computed. NLSE solutions were analyzed to illustrate the complex interfered model medium properties such as dispersion, dissipation, and nonlinearity. The impacts of the first and second-order CDs, nonlinearity on the structure of one soliton, interactions between two and three solitons, dark soliton, soliton-like periodic and dissipative, and shock waves were numerically inspected. It was found that these parameters affect not only the width and amplitude of the wave but also shock strength over the time evolution. On the other hand, new important waves existence can propagate by increasing time and become the effective wave propagation. These waves may be periodic, supersoliton, and oscillatory shock forms. Furthermore, a comparison of the obtained results of different techniques in this study confirmed that they are consistent with each other as well as with previous studies, indicating the accuracy of the numerical programming used. The findings of this study have the potential to improve communication performance through the development of physical parameters in the used model.