Matrix KP: tropical limit and Yang-Baxter maps

TL;DR

This paper investigates soliton solutions of matrix KP equations in a tropical limit, revealing connections to Yang-Baxter maps, R-matrices, and solutions to the tetrahedron equation, thus linking integrable systems with algebraic structures.

Contribution

It introduces a novel link between tropical soliton solutions of matrix KP equations and Yang-Baxter maps, including new solutions to the tetrahedron equation.

Findings

Polarizations are determined by Yang-Baxter maps in the tropical limit.

The vector KP R-matrix satisfies the Yang-Baxter equation.

New solutions to the tetrahedron (Zamolodchikov) equation are derived.

Abstract

We study soliton solutions of matrix Kadomtsev-Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attached to its constituting lines. There is a subclass of "pure line soliton solutions" for which we find that, in this limit, the distribution of polarizations is fully determined by a Yang-Baxter map. For a vector KP equation, this map is given by an R-matrix, whereas it is a non-linear map in case of a more general matrix KP equation. We also consider the corresponding Korteweg-deVries (KdV) reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain a new solution of the tetrahedron (or Zamolodchikov) equation. Moreover, a solution of the functional tetrahedron equation arises from the parameter-dependence of the vector KP R-matrix.