KP Solitons from Tropical Limits

TL;DR

This paper explores the connection between tropical geometry and KP soliton solutions, introducing a new framework that links algebraic curve degenerations to explicit soliton solutions via tropical limits.

Contribution

It introduces the Hirota variety and an algorithm to derive KP solitons from tropical degenerations of algebraic curves, advancing the understanding of soliton solutions in algebraic geometry.

Findings

Tau functions become finite exponential sums supported on Delaunay polytopes.

The Hirota variety parametrizes all tau functions from tropical limits.

An algorithm computes soliton solutions from rational nodal curves.

Abstract

We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. We compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and we present an algorithm that finds a soliton solution from a rational nodal curve.