Combinatorics of KP solitons from the real Grassmannian

TL;DR

This paper explores the combinatorial structure of KP soliton solutions derived from the real Grassmannian, revealing connections to total positivity, cluster algebras, and providing a complete description of their asymptotic behavior.

Contribution

It offers a comprehensive combinatorial description of KP soliton contour plots using Grassmannian stratifications and characterizes regular solutions via total positivity.

Findings

Complete asymptotic description of contour plots for large |y| or |t|

Connection between regular solitons and the totally non-negative Grassmannian

Insights into the inverse problem for KP solitons

Abstract

Given a point A in the real Grassmannian, it is well-known that one can construct a soliton solution u_A(x,y,t) to the KP equation. The contour plot of such a solution provides a tropical approximation to the solution when the variables x, y, and t are considered on a large scale and the time t is fixed. In this paper we give an overview of our work on the combinatorics of such contour plots. Using the positroid stratification and the Deodhar decomposition of the Grassmannian (and in particular the combinatorics of Go-diagrams), we completely describe the asymptotics of these contour plots when |y| or |t| go to infinity. Other highlights include: a surprising connection with total positivity and cluster algebras; results on the inverse problem; and the characterization of regular soliton solutions -- that is, a soliton solution u_A(x,y,t) is regular for all times t if and only if A comes from the totally non-negative part of the Grassmannian.