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Hyperspherical trigonometry, related elliptic functions and integrable systems

Jennings, Paul Richard (2013) Hyperspherical trigonometry, related elliptic functions and integrable systems. PhD thesis, University of Leeds.

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Abstract

The basic formulae of hyperspherical trigonometry in multi-dimensional Euclidean space are developed using multi-dimensional vector products, and their conversion to identities for elliptic functions is shown. The basic addition formulae for functions on the 3-sphere embedded in four-dimensional space are shown to lead to addition formulae for elliptic functions, associated with algebraic curves, which have two distinct moduli. Application of these formulae to the cases of a multi-dimensional Euler top and Double Elliptic Systems are given, providing a connection between the two. A generalisation of the Lattice Potential Kadomtsev-Petviashvili (LPKP) equation is presented, using the method of Direct Linearisation based on an elliptic Cauchy kernel. This yields a (3 + 1)-dimensional lattice system with one of the lattice shifts singled out. The integrability of the lattice system is considered, presenting a Lax representation and soliton solutions. An associated continuous system is also derived, yielding a (3 + 1)- dimensional generalisation of the potential KP equation associated with an elliptic curve.

Item Type: Thesis (PhD)
ISBN: 978-0-85731-861-9
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.617146
Depositing User: Repository Administrator
Date Deposited: 12 Sep 2014 13:16
Last Modified: 06 Oct 2016 14:42
URI: http://etheses.whiterose.ac.uk/id/eprint/6892

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