Tropical KP Theory on Banana Curves

TL;DR

This paper explores the deep connections between tropical geometry, matroid theory, and integrable systems by analyzing KP solutions associated with banana curves, revealing combinatorial structures underlying multi-soliton solutions.

Contribution

It introduces a novel framework linking banana curve degenerations, tropical theta divisors, and matroid polytopes to describe KP multi-solitons explicitly.

Findings

Tropical theta divisor encodes matroid and Grassmannian structures.

Delaunay polytopes are combinatorially equivalent to uniform matroid polytopes.

Explicit parametrization of tau functions as multi-solitons in tropical limit.

Abstract

The Kadomtsev-Petviashvili (KP) equation is the cornerstone of integrable systems, whose solutions reflect deep connections in algebraic geometry. Banana curves are reducible rational curves obtained as a degeneration of hyperelliptic curves. In this work, we relate the family of KP multi-solitons arising from banana curves together with non-special divisors of fixed degree to the combinatorics of the tropical theta divisor of the curve. We describe the Voronoi and Delaunay polytopes and show that the latter are combinatorially equivalent to uniform matroid polytopes. As a consequence, the combinatorics of the tropical theta divisor canonically encodes the matroid and Grassmannian structures underlying the associated KP multi-soliton solutions. We define the Hirota variety of a banana graph, which parametrizes all tau functions arising from such a graph. Starting from the matroid arising from Delaunay polytopes and the periods in the tropical limit, we construct an explicit parametrization of this variety which realizes the tau function as a multi-soliton. Our framework specializes naturally to real and positive settings.