A fith-order Runge-Kutta in the interaction picture method for simulating the nonlinear Schrödinger equation

• Main Author: Ong, Gerald
• Other Authors: Marco Battiato
• Format: Final Year Project
• Language: English
• Published: Nanyang Technological University 2022
• Subjects: Science::Physics::Optics and light
• Online Access: https://hdl.handle.net/10356/156914
• Institution: Nanyang Technological University

The nonlinear Schrödinger equation (NLSE) is a version of the well known Schrödinger Equation that describes complex wave forms in a nonlinear medium. One of the most important applications of this equation is in fibre optics, where data is transferred using light pulses through optical fibres. During pulse propagation, the optical pulse interacts with dispersive and nonlinear properties of the fibre which makes it a good example of a nonlinear medium. Therefore the waveform of the pulse of light that travels through the fibre can be modelled using the NLSE. However, analytical solutions to explain light-pulse propagation exist for only a few specific cases such as solitons for specific ratios of dispersive and nonlinear properties of the fiber. If the conditions were to change even slightly, one would not be able to solve the NLSE analytically. Since the optical fibre has a number of higher-order linear and nonlinear properties such as higher-order dispersion, pulse self-steepening and Raman effects, an analytic solution does not exist to explain pulse dynamics when one includes these effects with the NLSE. In this case, the equation needs to be modified to include additional terms related to the above effects. This new equation is called the perturbed nonlinear Schrödinger equation. To solve the NLSE, the most well known method of solving it is the split-step Fourier method. Using this method allows a method known as the fourth order Runge-Kutta in the interaction picture can be used to solve the NLSE efficiently. However in this project, A higher order Runge-Kutta method called the Dormand-Prince method has been implemented in the interaction picture, and it proves to be even more efficient than the fourth order Runge-Kutta method.