Barge, S. (1999) Twistor theory and the K.P. equations. PhD thesis, University of Oxford.
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)|
|Subjects:||O - Z > Partial differential equations|
O - Z > Quantum theory
O - Z > Relativity and gravitational theory
|Research Groups:||Mathematical Physics Group|
|Deposited By:||Eprints Administrator|
|Deposited On:||10 Mar 2004|
|Last Modified:||20 Jul 2009 14:18|
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