Lagrangian Theory of Integrable Hierarchies: Connections and Applications

• Applicant: Dr. Mats Vermeeren
• Subject Area: Mathematics
• Term: from 2019 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 428317136

A variational principle for integrable systems has been developed in the past decade 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. It is not yet clear how this perspective fits into the existing theory of integrable hierarchies. The goal of the first part of the project is to establish connections between the recently developed variational perspective and common notions of integrability in the continuous context. In particular, this project aims to connect the variational principle 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 this project consists in exploring applications of the Lagrangian perspective to unsolved questions in integrable systems. Previous work has shown that a Lagrangian approach is useful to study the relation between discrete and continuous integrable systems. The same is expected in the context of quantum integrability. In addition, there are strong indications that this variational principle can help explain why high-dimensional integrable systems are relatively rare.

• DFG Programme: Research Fellowships
• International Connection: United Kingdom
• Hosts: Dr. Vincent Caudrelier; Professor Dr. Frank Nijhoff