Location of Repository

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

Topics:
Partial differential equations, Quantum theory, Relativity and gravitational theory

Year: 1999

OAI identifier:
oai:generic.eprints.org:33/core69

Provided by:
Mathematical Institute Eprints Archive

Downloaded from
http://eprints.maths.ox.ac.uk/33/1/barge.pdf

- (1993). A self-dual YangMills hierarchy and its reductions to integrable systems in 1+1 and 2+1 dimensions,
- (1985). Determinants of Cauchy-Riemann operators on Riemann surfaces,
- (1986). Discrete and Periodic Spectral Transforms related to the Kadomtsev-Petviashvili equation,
- (1994). dressing and exact solutions for the (2+1)-dimensional Harry Dym equation,
- (1995). Generalized twistor correspondences, d-bar problems and the KP equations, Collection, ‘Twistor Theory’
- (1986). Hamiltonian Structure of the Kadomtsev-Petviashvili-II Equation in the class of Decreasing Caucy Data,
- (1992). Integrable nonlinear equations, Collection: ‘Nonlinear equations in the applied sciences’,
- (1992). Introduction to Multidimensional Integrable Equations:
- (1987). Inverse Scattering for the heat operator and evolutions
- (1990). Inverse Spectral Transform for the Nonlinear Evolution Equation Generating the Davey-Stewartson and Ishimori Equations,
- (1985). Jr: Twistor geometry and field theory,
- (1985). Loop groups and equations of KdV type,
- (1991). On (2+1)-dimensional nonlinear systems of Loewner-type,
- (1987). On a spectral transform of a KdV-like equation related to the Schro¨dinger operator in the plane,
- (1993). On generalized Loewner systems: Novel integrable equations in 2+1 dimension,
- (1977). On self-dual gauge fields,
- (1989). On the dressing method, Collection, ‘Inverse methods
- (1970). On the stability of solitary waves in weakly dispersive media,
- (1974). On three-dimensional packets of surface waves,
- (1991). Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society lecture notes vol. 149,
- Tau Functions and the Twistor Theory of Integrable Systems,
- (1998). The Solution of Certain Initial BoundaryValue Problems for the Linearized Korteweg de Vries Equation,
- (1994). The Twistor Theory of
- (1981). Transformation groups for soliton equations: I-VII ; I:
- (1967). Twistor Algebra,
- (1995). Twistor theory, self-duality and integrability,

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.