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Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

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posted on 2024-01-03, 09:51 authored by Mats VermeerenMats Vermeeren

Many integrable hierarchies of differential equations allow a variational description, called a Lagrangian multiform or a pluri-Lagrangian structure. The fundamental object in this theory is not a Lagrange function but a differential $d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for a field to be a solution to the pluri-Lagrangian problem. In this paper we present a procedure to obtain Hamiltonian structures from the pluri-Lagrangian formulation of an integrable hierarchy of PDEs. As a prelude, we review a similar procedure for integrable ODEs. We show that exterior derivative of the Lagrangian $d$-form is closely related to the Poisson brackets between the corresponding Hamilton functions. In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as the Boussinesq hierarchy.

Funding

DFG Research Fellowship VE 1211/1-1

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Open Communications in Nonlinear Mathematical Physics

Volume

1

Pages

94 - 127

Publisher

International Society of Nonlinear Mathematical Physics

Version

  • VoR (Version of Record)

Rights holder

© The author(s)

Publisher statement

This is an Open Access Article. It is published by the International Society of Nonlinear Mathematical Physics under the Creative Commons Attribution 4.0 International Licence (CC BY). Full details of this licence are available at: https://creativecommons.org/licenses/by/4.0/

Acceptance date

2021-08-31

Publication date

2021-09-10

Copyright date

2021

eISSN

2802-9356

Language

  • en

Depositor

Dr Mats Vermeeren. Deposit date: 19 December 2023

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