Matrix KP: tropical limit, Yang-Baxter and pentagon maps

TL;DR

This paper investigates the tropical limit of matrix KP-II solitons, revealing how their support graphs relate to Yang-Baxter and pentagon equations, and introduces new solutions connecting integrable systems with polygon equations.

Contribution

It introduces a subclass of soliton solutions with tree-like tropical limit graphs and links their polarization distributions to Yang-Baxter and pentagon equations, providing new solutions and generalizations.

Findings

Polarization distributions are governed by a parameter-dependent binary operation.

A solution of the pentagon equation is derived from the binary operation's parameter dependence.

New solutions to the hexagon equation are obtained from the generalized R-matrix.

Abstract

In the tropical limit of matrix KP-II solitons, their support at fixed time is a planar graph with "polarizations" attached to its linear parts. In this work we explore a subclass of soliton solutions whose tropical limit graph has the form of a rooted and generically binary tree, as well as solutions with a limit graph consisting of two relatively inverted such rooted tree graphs. The distribution of polarizations over the constituting lines of the graph is fully determined by a parameter-dependent binary operation and a (in general non-linear) Yang-Baxter map, which in the vector KP case becomes linear, hence is given by an R-matrix. The parameter-dependence of the binary operation leads to a solution of the pentagon equation, which exhibits a certain relation with the Rogers dilogarithm via a solution of the hexagon equation, the next member in the family of polygon equations. A generalization of the R-matrix, obtained in the vector KP case, is found to also solve a pentagon equation. A corresponding local version of the latter then leads to a new solution of the hexagon equation.