The nonlinear Schrödinger 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 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 nonlinear Schrödinger 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. The pseudo-spectral method estimates solutions by utilizing the Fourier transform to evaluate the spatial derivative within the nonlinear Schrödinger equation. An ode solver is then applied to the resulting ordinary differential equation. We use this method to determine whether cardiac action potential states, which are perturbed solutions to the Fitzhugh-Nagumo nonlinear partial differential equation, create soliton-like solutions. After finding soliton-like solutions, we then use symmetry group properties of the nonlinear Schrödinger equation to explore these solutions. We also use a Lie algebra related to the symmetries to look for more solutions to our modified nonlinear Schrödinger equation.
| Identifer | oai:union.ndltd.org:ETSU/oai:dc.etsu.edu:honors-1070 |
| Date | 11 May 2013 |
| Creators | Middlemas, Erin |
| Publisher | Digital Commons @ East Tennessee State University |
| Source Sets | East Tennessee State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Undergraduate Honors Theses |
| Rights | Copyright by the authors., http://creativecommons.org/licenses/by-nc-nd/3.0/ |
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