Hyperspherical trigonometry, related elliptic functions and integrable systems

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.