KP solitons and the Schottky uniformization
Takashi Ichikawa, Yuji Kodama
TL;DR
The work addresses how real, regular KP solitons, classified by the TNN Grassmannian, can be realized as degenerate limits of real finite-gap (algebraic-geometric) solutions. It constructs Schottky uniformizations from combinatorial data in the TNN setting and shows that the corresponding smooth finite-gap solutions degenerate to KP solitons, formalized via the M-theta function as a limit of Riemann theta functions. The main contribution is the theorem that KP solitons for TNN Grassmannians arise from singular limits of real finite-gap solutions associated with Schottky-uniformized surfaces derived from the Le-/Gamma-diagram data, with explicit formulas for these limits. This establishes a concrete bridge between algebraic-geometric finite-gap theory and tropical soliton pictures, providing a constructive path from combinatorics to explicit soliton solutions and clarifying the role of canonical homology bases in the degeneration.
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.