Integrable PDEs and hyperelliptic functions

We present a short review comparing Baker-Akhiezer and secant identity approaches of the appearence of multidimensional theta functions in the theory of integrable partial differential equations. As examples we discuss the Kadomtsev-Petviashvili equation, a two-dimension generalization of the celebrated Korteweg-de Vries equation, and the Ernst equation which is equivalent to the stationary axisymmetric Einstein equations in vacuum. Solutions to the latter are given on a family of hyperelliptic curves which makes an efficient numerical treatment of modular functions associated to such curves necessary in order to discuss these solutions. We present an efficient numerical approach to hyperelliptic curves based on Clenshaw-Curtis integration which computes all needed quantities even in almost degenerate situations to machine precision.