### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 044, 35 pages      arXiv:1811.01855      https://doi.org/10.3842/SIGMA.2019.044

### 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

Received November 20, 2018, in final form May 16, 2019; Published online June 03, 2019

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.

Key words: continuum limits; pluri-Lagrangian systems; Lagrangian multiforms; multidimensional consistency.

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