KP line solitons and Tamari lattices

TL;DR

This paper links KP-II line soliton solutions to Tamari lattices, providing a combinatorial framework to understand their evolution and critical transitions on shallow fluid surfaces.

Contribution

It establishes a novel connection between KP-II soliton dynamics and Tamari lattices, enabling detailed analysis of soliton pattern evolution.

Findings

Maximal chains in Tamari lattices correspond to soliton evolutions.

Critical events in soliton patterns are characterized by transitions in Tamari lattices.

The framework allows computation of soliton evolution details.

Abstract

The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "critical" events where a transition to a different rooted binary tree takes place. We prove that these correspond to maximal chains in Tamari lattices (which are poset structures on associahedra). We further derive results that allow to compute details of the evolution, including the critical events. Moreover, we present some insights into the structure of the more general line soliton solutions. All this yields a characterization of possible evolutions of line soliton patterns on a shallow fluid surface (provided that the KP-II approximation applies).