Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System

TL;DR

This paper explores the connections between multicomponent heat, Burgers, and KP hierarchies using a generalized Cole-Hopf transformation, enabling the construction of exact solutions through matrix linear systems.

Contribution

It introduces a generalized Cole-Hopf transformation linking multicomponent heat and KP hierarchies and provides a method to generate exact solutions from matrix linear systems.

Findings

Any solution of the multicomponent Burgers hierarchy also solves the multicomponent KP hierarchy.

A generalized Cole-Hopf transformation relates the heat hierarchy to the KP hierarchy.

Exact solutions of the KP hierarchy can be constructed via matrix linear systems.

Abstract

Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy leads to a multicomponent Burgers hierarchy. We show in particular that any solution of the latter also solves a corresponding multicomponent (potential) KP hierarchy. A generalization of the Cole-Hopf transformation leads to a more general relation between the multicomponent linear heat hierarchy and the multicomponent KP hierarchy. From this results a construction of exact solutions of the latter via a matrix linear system.