Lagrangian multiform structure of discrete and semi-discrete KP systems

TL;DR

This paper develops a Lagrangian multiform framework for discrete and semi-discrete KP systems, unifying their variational structure and revealing new integrability properties and hierarchies.

Contribution

It introduces a novel Lagrangian multiform formalism for the potential AKP system, encompassing discrete, semi-discrete, and differential-difference variants, along with a generating PDE for the KP hierarchy.

Findings

Establishment of a variational structure using Lagrangian multiforms.

Identification of a double-zero structure in the exterior derivative of the Lagrangian 3-form.

Development of a variational formulation for Lax systems via eigenfunction representations.

Abstract

A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semi-discrete variants including several differential-difference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semi-discrete) variants of the KP system, and their associated Lax representations, including a novel `generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3-form for the lattice potential KP equation is shown to exhibit a double-zero structure, which implies the corresponding generalised Euler-Lagrange equations. Alongside the 3-form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme.