Classification of KPI Lumps

TL;DR

This paper investigates a family of rational solutions to the KP I equation, revealing their structure via Grassmannians, Schur functions, and connections to Painlevé equations, with solutions classified by integer partitions.

Contribution

It introduces a novel classification of KP I rational solutions using Grassmannian points, Schur functions, and partitions, linking solutions to algebraic and combinatorial structures.

Findings

Solutions form well-defined wave patterns at large times

Each solution corresponds uniquely to an integer partition

Solutions are described by classical Schur functions and symmetric group representations

Abstract

A large family of nonsingular rational solutions of the Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions are constructed via the Gramian method and are identified as points in a complex Grassmannian. Each solution is a traveling wave moving with a uniform background velocity but have multiple peaks which evolve at a slower time scale in the co-moving frame. For large times, these peaks separate and form well-defined wave patterns in the $xy$-plane. The pattern formation are described by the roots of well-known polynomials arising in the study of rational solutions of Painlev\'e II and IV equations. This family of solutions are shown to be described by the classical Schur functions associated with partitions of integers and irreducible representations of the symmetric group of $N$ objects. It is then shown that there exists a one-to-one correspondence between the KPI rational solutions considered in this article and partitions of a positive integer $N$.