Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Duncan Sleigh, Frank Nijhoff, Vincent Caudrelier

TL;DR

This work constructs a complete, local Lagrangian multiform for the Kadomtsev-Petviashvili (KP) hierarchy by assembling Dickey's pseudodifferential Lagrangians within a dressing framework, yielding a 3-form M whose multiform Euler–Lagrange equations reproduce the full KP system. Through a reduction scheme based on L^n_- = 0, the KP multiform induces explicit Lagrangian multiforms for the Gelfand–Dickey (GD) hierarchy, including the KdV and Boussinesq cases, via carefully defined contracted Lagrangians and their residues. The paper provides two closely related KP multiforms—one Dickey-based and one uniform—both yielding the same KP equations while maintaining locality of the constituent Lagrangians. The results establish the integrability of the KP hierarchy at the variational level and offer a pathway to GD multiforms with explicit, local densities, strengthening links between dressing, residues, and multiform variational principles.

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.

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

TL;DR

This work constructs a complete, local Lagrangian multiform for the Kadomtsev-Petviashvili (KP) hierarchy by assembling Dickey's pseudodifferential Lagrangians within a dressing framework, yielding a 3-form M whose multiform Euler–Lagrange equations reproduce the full KP system. Through a reduction scheme based on L^n_- = 0, the KP multiform induces explicit Lagrangian multiforms for the Gelfand–Dickey (GD) hierarchy, including the KdV and Boussinesq cases, via carefully defined contracted Lagrangians and their residues. The paper provides two closely related KP multiforms—one Dickey-based and one uniform—both yielding the same KP equations while maintaining locality of the constituent Lagrangians. The results establish the integrability of the KP hierarchy at the variational level and offer a pathway to GD multiforms with explicit, local densities, strengthening links between dressing, residues, and multiform variational principles.

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.

Paper Structure

This paper contains 14 sections, 15 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Lagrangian multiforms
  3. Pseudodifferential operators
  4. The KP hierarchy and its reduction to Gelfand-Dickey
  5. The KP hierarchy
  6. The Gelfand-Dickey hierarchy as a reduction of KP
  7. A Lagrangian for the KP hierarchy
  8. Lagrangian multiforms for the KP hierarchy
  9. A Lagrangian multiform for KP based on Dickey's Lagrangian
  10. An alternative KP Lagrangian multiform
  11. Reduction to multiforms for the Gelfand-Dickey hierarchy
  12. Conclusion
  13. Multiform Euler-Lagrange equations in terms of variational derivatives
  14. Explicit form of the KP Lagrangian multiform

Key Result

Lemma 1.1

The residue of a commutator of two pseudodifferential operators $X$ and $Y$, for some $h\in \mathcal{A}$, so is a total $x$ derivative.

Theorems & Definitions (28)

  • Lemma 1.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 18 more