KP solitons and the Schottky uniformization

TL;DR

This paper explores the connection between KP solitons, the TNN Grassmannian, and Schottky uniformization, revealing how solitons can be derived from degenerations of finite-gap solutions associated with Riemann surfaces.

Contribution

It constructs Schottky groups for each TNN Grassmannian element and demonstrates how KP solitons arise from degenerating finite-gap solutions, linking soliton theory with algebraic geometry.

Findings

KP solitons are singular limits of finite-gap solutions.

Schottky groups are constructed for each TNN Grassmannian element.

KP solitons can be obtained via degeneration of finite-gap solutions.

Abstract

Real and regular soliton solutions of the KP hierarchy have been classified in terms of the totally nonnegative (TNN) Grassmannians. These solitons are referred to as KP solitons, and they are expressed as singular (tropical) limits of shifted Riemann theta functions. In this talk, for each element of the TNN Grassmannian, we construct a Schottky group, which uniformizes the Riemann surface associated with a real finite-gap solution. Then we show that the KP solitons are obtained by degenerating these finite-gap solutions.