The thesis concerns linear systems and scattering theory. In particular, it presents lineal' systems for some integrable systems and finds discrete analogues for many well known results for continuous variables. It introduces some new tools from linear systems and applies them to standard integrable systems. We begin by expressing the first Painleve equation as the compatibility condition of a certain Lax pair and introduce the Korteweg-de Vries partial differential equation. We introduce the spectral curve for algebraic families and the Toda lattice. The Fredholm determinant of a trace class Hankel integral operator gives rise to a tau function. Dyson used the tau function to solve an inverse spectral problem for Schrodinger operators. When a plane wave is subject to Schrodinger's equation and scattered by a potential u, the output is described at great distances by a scattering function. The spectral problem is to find the spectrum of Schrodinger's operator in L2 and hence the scattering function. The inverse spectral problem is to find the potential given the scattering function. The scattering and inverse scattering problems are linked by the Gelfand- Levitan equation. In this thesis, for a discrete linear system, we introduce a scattering function and Hankel matrix and a version of the Gelfand-Levitan equation for discrete linear systems. We introduce the discrete operator ∑∞/k=n AkBCAk and use it to solve the Gelfand-Levitan equation and compute Fredholm determinants of Hankel operators. We produce a discrete analogue of a calculation of Poppe giving a solution to the Korteweg-de Vries equation and via the methods of linear systems find an analogous solution in terms of Hankel matrices. We then produce a discrete analogue of the Miura transform. Thus the main new contributions of this thesis are the discrete analogues of the R operator, the Gelfand- Levitan equation, the Lyapunov equation and the Miura transform.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:663238 |
| Date | January 2013 |
| Creators | Newsham, Samantha |
| Publisher | Lancaster University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
Page generated in 0.0018 seconds