This article is cited in 5 papers

A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations

Mats Vermeeren

Institut für Mathematik, MA 7-1, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany

Abstract: A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the discrete case) or commutativity of the flows (in the continuous case) with a variational principle. Recently we developed a continuum limit procedure for pluri-Lagrangian systems, which we now apply to most of the ABS list and some members of the lattice Gelfand–Dickey hierarchy. We obtain pluri-Lagrangian structures for many hierarchies of integrable PDEs for which such structures where previously unknown. This includes the Krichever–Novikov hierarchy, the double hierarchy of sine-Gordon and modified KdV equations, and a first example of a continuous multi-component pluri-Lagrangian system.

Keywords: continuum limits, pluri-Lagrangian systems, Lagrangian multiforms, multidimensional consistency.

MSC: 37K10, 39A14

Received: November 20, 2018; in final form May 16, 2019; Published online June 3, 2019

Language: English

DOI: 10.3842/SIGMA.2019.044

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ArXiv: 1811.01855