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

TL;DR

This paper explores distributions supported on fractal sets that enable solutions to the Kadomtsev--Petviashvili equation, extending the understanding of integrable PDEs with fractal singularities.

Contribution

It introduces a new class of distributions supported on fractal sets that produce solutions to the KP equation, linking complex analysis, fractal geometry, and integrable systems.

Findings

Distributions supported on Cantor and Sierpinski gasket sets can generate KP solutions.

These distributions are limits of rational functions related to holomorphic line bundles.

A conjecture links these distributions to defining holomorphic line bundles on singular surfaces.

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.