Lagrangian multiforms for Kadomtsev-Petviashvili (KP) and the Gelfand-Dickey hierarchy

TL;DR

This paper introduces a unified Lagrangian multiform for the entire KP hierarchy, capturing its integrability, and extends this framework to the Gelfand-Dickey hierarchy, including KdV and Boussinesq equations.

Contribution

It provides the first Lagrangian multiform for the complete KP hierarchy and derives related multiforms for the Gelfand-Dickey hierarchy through reduction.

Findings

Unified variational structure for KP hierarchy

Lagrangian multiforms for Gelfand-Dickey hierarchy

Encapsulation of integrability in a single variational object

Abstract

We present, for the first time, a Lagrangian multiform for the complete Kadomtsev-Petviashvili (KP) hierarchy -- a single variational object that generates the whole hierarchy and encapsulates its integrability. By performing a reduction on this Lagrangian multiform, we also obtain Lagrangian multiforms for the Gelfand-Dickey hierarchy of hierarchies, comprising, amongst others, the Korteweg-de Vries and Boussinesq hierarchies.