Lagrangian 3-form structure for the Darboux system and the KP hierarchy

TL;DR

This paper develops a Lagrangian 3-form structure for the Darboux system and the KP hierarchy, revealing new geometric insights and extending to matrix KP systems, thereby enriching the understanding of integrable systems.

Contribution

It introduces a novel Lagrangian multiform framework for the Darboux system and KP hierarchy, including a generalization to non-Abelian matrix KP systems.

Findings

Established a Lagrangian 3-form structure for the Darboux system.

Connected the Darboux system to the KP hierarchy via Miwa variables.

Extended the framework to matrix (non-Abelian) KP systems.

Abstract

A Lagrangian multiform structure is established for a generalisation of the Darboux system describing orthogonal curvilinear coordinate systems. It has been shown in the past that this system of coupled PDEs is in fact an encoding of the entire Kadomtsev-Petviashvili (KP) hierarchy in terms so-called Miwa variables. Thus, in providing a Lagrangian description of this multidimensionally consistent system amounts to a new Lagrangian 3-form structure for the continuous KP system. A generalisation to the matrix (also known as non-Abelian) KP system is discussed.