Tropical KP Theory on Banana Curves

Simonetta Abenda; Türkü Özlüm Çelik; Claudia Fevola; Yelena Mandelshtam

Paper

Tropical KP Theory on Banana Curves

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.