Symmetry in Nonlinear Mathematical Physics - 2009

Matthew England (Heriot-Watt University, Edinburgh, UK)

Higher genus Abelian functions associated with algebraic curves

In recent years there has been a revival of interest in the theory of Abelian (multiply periodic) functions associated with algebraic curves. We can define such functions to generalise the classic Weierstrass p-function and we seek to derive the related partial differential equations and addition formula.

After an introduction to the topic I will describe how we have generated new results for the functions associated to higher genus curves, giving examples for a cyclic tetragonal curve of genus six. I will discuss some of the key methods used to compute the results and demonstrate that such functions can give a solution to the KP-equation.