Barge, S. (1999) *Twistor theory and the K.P. equations.* PhD thesis, University of Oxford.

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## Abstract

In this thesis, we discuss a geometric construction analogous to the Ward correspondence for the KP equations. We propose a Dirac operator based on the inverse scattering transform for the KP-II equation and discuss the similarities and differences to the Ward correspondence.

We also consider the KP-I equation, describing a geometric construction for a certain class of solutions. We also discuss the general inverse scattering of the equation, how this is related to the KP-II equation and the problems with describing a single geometric construction that incorporates both equations.

We also consider the Davey-Stewartson equations, which have a similar behaviour. We demonstrate explicitly the problems of localising the theory with generic boundary conditions. We also present a reformulation of the Dirac operator and demonstrate a duality between the Dirac operator and the first Lax operator for the DS-II equations.

We then proceed to generalise the Dirac operator construction to generate other integrable systems. These include the mKP and Ishimori equations, and an extension to the KP and mKP hierarchies.

Item Type: | Thesis (PhD) |
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Subjects: | O - Z > Partial differential equations O - Z > Quantum theory O - Z > Relativity and gravitational theory |

Research Groups: | Mathematical Physics Group |

ID Code: | 33 |

Deposited By: | Eprints Administrator |

Deposited On: | 10 Mar 2004 |

Last Modified: | 20 Jul 2009 14:18 |

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