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

SIGMA 20 (2024), 100, 30 pages      arXiv:2405.12837      https://doi.org/10.3842/SIGMA.2024.100

Lagrangian Multiform for Cyclotomic Gaudin Models

Vincent Caudrelier a, Anup Anand Singh a and Benoît Vicedo b
a) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
b) Department of Mathematics, University of York, York YO10 5DD, UK

Received May 28, 2024, in final form November 07, 2024; Published online November 15, 2024

Abstract
We construct a Lagrangian multiform for the class of cyclotomic (rational) Gaudin models by formulating its hierarchy within the Lie dialgebra framework of Semenov-Tian-Shansky and by using the framework of Lagrangian multiforms on coadjoint orbits. This provides the first example of a Lagrangian multiform for an integrable hierarchy whose classical $r$-matrix is non-skew-symmetric and spectral parameter-dependent. As an important by-product of the construction, we obtain a Lagrangian multiform for the periodic Toda chain by choosing an appropriate realisation of the cyclotomic Gaudin Lax matrix. This fills a gap in the landscape of Toda models as only the open and infinite chains had been previously cast into the Lagrangian multiform framework. A slightly different choice of realisation produces the so-called discrete self-trapping (DST) model. We demonstrate the versatility of the framework by coupling the periodic Toda chain with the DST model and by obtaining a Lagrangian multiform for the corresponding integrable hierarchy.

Key words: Lagrangian multiforms; integrable systems; classical $r$-matrix; Gaudin models.

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