Distributions Supported on Fractal Sets and Solutions to the Kadomtsev--Petviashvili Equation

Abstract

In this note we will discuss a potentially interesting extension of some recent results on primitive solutions to completely integrable partial differential equations. We will discuss a family distributions that are holomorphic on the Riemann sphere except on the singular sets homeomorphic to a Cantor set or Sierpinski gasket. These distributions allow us to produce solutions to the Kadomtsev--Petviashvili equation. These distributions are limits of families of rational functions that can also be associated with holomorphic line bundles on surfaces with a finite number of doubly degenerate singular points. We conjecture that a subset of these distributions can be used to formulate a definition of a holomorphic line bundle on some surfaces that are homeomorphic to spheres except where they become doubly degenerate on singular sets homeomorphic to a Cantor set or Sierpinski gasket.