Soliton Solutions of a Variation of the Nonlinear Schrödinger Equation

Abstract

The nonlinear Schrödinger (NLS) equation is a classical field equation that describes weakly nonlinear wave-packets in one-dimensional physical systems. It is in a class of nonlinear partial differential equations (PDEs) that pertain to several physical and biological systems. In this project we apply a pseudo-spectral solution-estimation method to a modified version of the NLS equation as a means of searching for solutions that are solitons, where a soliton is a self-reinforcing solitary wave that maintains its shape over time. We use the pseudo-spectral method to determine whether cardiac action potential states, which are perturbed solutions to the Fitzhugh-Nagumo nonlinear PDE, create soliton-like solutions. We then use symmetry group properties of the NLS equation to explore these solutions and find new ones.