KP solitons, higher Bruhat and Tamari orders

TL;DR

This paper explores the relationship between KP-II soliton solutions, Tamari lattices, and higher Bruhat and Tamari orders, revealing a combinatorial structure underlying these integrable systems.

Contribution

It establishes a connection between tree-shaped KP-II soliton solutions and higher Bruhat and Tamari orders, expanding the combinatorial understanding of these solutions.

Findings

Tree-shaped soliton solutions form chains in Tamari lattices.

Analysis involves higher Bruhat and Tamari orders.

Provides a combinatorial framework for KP-II solutions.

Abstract

In a tropical approximation, any tree-shaped line soliton solution, a member of the simplest class of soliton solutions of the Kadomtsev-Petviashvili (KP-II) equation, determines a chain of planar rooted binary trees, connected by right rotation. More precisely, it determines a maximal chain of a Tamari lattice. We show that an analysis of these solutions naturally involves higher Bruhat and higher Tamari orders.