Lagrangian Theory of Integrable Hierarchies: Connections and Applications
Final Report Abstract
A decade before the start of the project, a variational principle for integrable systems was developed in the context of multi-dimensionally consistent difference equations. This variational (i.e. Lagrangian) perspective has proven to be quite useful in the study of discrete integrable systems. On the continuous side, a perfectly analogous variational principle can be formulated for integrable hierarchies of partial differential equations. The overarching goal of this project was to establish how this perspective fits into the existing theory of integrable hierarchies. The first part of the project aimed to establish connections between the recently developed variational perspective and common notions of integrability in the continuous context. In particular, connections were sought to (bi-)Hamiltonian structures (a natural counterpart to Lagrangian structures) and Lax pairs (which in some special cases have been found to be closely related to variational principles). The second part of the project consisted in exploring applications of the Lagrangian perspective to unsolved questions in integrable systems, in particular towards classification of integrable systems and quantum integrability. The project had to adapt to the effects of the pandemic on working practices and collaboration possibilities. In particular, close collaboration with the local hosts was impossible during lockdown, whereas the universal switch to remote working made it easier to collaborate externally. This led to some aspects of the proposal (Lax pairs, classification) being largely dropped, but also to successful international collaborations beyond the scope of the original proposal, mainly on topics concerning discretisation. Connections to Hamiltonian structures were successfully established, though some open questions on this topic remain. Initial steps towards quantisation were made, which exposed unexpected gaps in the classical theory in the context of non-abelian symmetry groups and semidiscrete systems. We were able to fill these gaps. Our results in this area also lead to interesting possibilities of applying our variational principle beyond integrable systems.
Publications
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Hamiltonian structures for integrable hierarchies of Lagrangian PDEs. Open Communications in Nonlinear Mathematical Physics 1: ocnmp:7491 (2021)
Mats Vermeeren
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Lagrangian multiforms on Lie groups and non-commuting flows
Vincent Caudrelier, Frank Nijhoff, Duncan Sleigh, Mats Vermeeren
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Semi-discrete Lagrangian 2-forms and the Toda hierarchy
Duncan Sleigh, Mats Vermeeren
