Lagrangian Multiform for Cyclotomic Gaudin Models

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.

References in corpus (11)

• Lax pair for the Adler (lattice Krichever-Novikov) System
• Quantum Backlund transformation for the integrable DST model
• Lagrangian multiforms and multidimensional consistency
• A Variational Approach to Lax Representations
• On the Lagrangian structure of integrable hierarchies
• Lagrangian 3-form structure for the Darboux system and the KP hierarchy
• Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies
• Cyclotomic Gaudin models with irregular singularities
• Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
• Lagrangian multiforms on Lie groups and non-commuting flows
• Generalized classical Yang-Baxter equation and regular decompositions