Hyperspherical trigonometry, related elliptic functions and integrable systems

• Main Author: Jennings, Paul Richard
• Other Authors: Nijoff, Frank
• Published: University of Leeds 2013
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.617146

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.