### Genus four hyperelliptic solution of KP (by J. Frauendiener and C. Klein)

The Kadomtsev-Petviashvili (KP) equation appears in the
approximate treatment of many weakly two-dimensional wave phenomena
in physics as waves in shallow
water. The equation has explicit solutions in terms of multi-dimensional
theta functions on an arbitrary Riemann surfaces. The picture/animation
describes a solution given on a hyperelliptic Riemann surface of genus 4,
i.e. a compact surface with 4 holes. In general the corresponding solutions
are almost periodic. When the holes of the surface
shrink to points, the solution reduces to the four-soliton solution and
solitary waves appear (see the previous picture).
The theta function solutions are numerically evaluated
using spectral methods.